Geometric Approximation of Curves and Singularities ...

Geometric Approximation of Curves and Singularities of Secant Maps A Differential Geometric Approach

Sunayana Ghosh

This research in this thesis has been supported by the : IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-006413 (ACS - Algorithms for Complex Shapes) Johann Bernoulli Institute for Mathematics and Computer Science Faculty of Mathematics and Natural Sciences University of Groningen The Netherlands Groningen Graduate School of Science University of Groningen The Netherlands

The images/photos used in creating the cover of this thesis are : The M74 Galaxy image courtesy NASA, ESA and Hubble Heritage. The snail shell photo which appears here courtesy of Giuseppe Zito. The staircase photo from Sagrada Familia (Barcelona) courtesy Danny Fay and reproduced here under the terms of the GNU Free Documentation License.

Cover concept and design : Cherian Mathew

RIJKSUNIVERSITEIT GRONINGEN

Geometric Approximation of Curves and Singularities of Secant Maps A Differential Geometric Approach

Proefschrift

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op vrijdag 3 december 2010 om 16.15 uur

door

Sunayana Ghosh geboren op 9 december 1978 te Allahabad, India

Promotores :

Prof. dr. G. Vegter Prof. dr. J. H. Rieger

Beoordelingscommissie :

Prof. dr. J. B. T. M. Roerdink Prof. dr. D. Siersma Prof. dr. ir. H. S. V. de Snoo

ISBN : 978-90-367-4639-7

Contents

Contents

i

Acknowledgements

v

List of Figures

vii

List of Tables

ix

1 Introduction 1.1 Computer Aided Geometric Design . . . . . 1.1.1 Brief history of CAGD . . . . . . . . 1.2 Curve modeling . . . . . . . . . . . . . . . . 1.2.1 B´ezier curves . . . . . . . . . . . . . 1.2.2 Rational curves . . . . . . . . . . . 1.2.3 Implicit curve modeling . . . . . . . 1.3 Approximation Theory . . . . . . . . . . . . 1.3.1 Approximation order and complexity 1.3.2 Curve interpolation . . . . . . . . . 1.3.3 Geometric Hermite interpolation . . 1.4 Classifications in singularity theory . . . . . 1.4.1 Background . . . . . . . . . . . . . . 1.4.2 Singularities of secant maps . . . . . 1.5 Overview and main results . . . . . . . . . .

1 1 2 4 5 7 9 10 12 13 14 20 21 25 28

i

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CONTENTS 2 Differential geometry of curves 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Planar Curves and Conics . . . . . . . . . . . . 2.2.1 Affine curvature . . . . . . . . . . . . . 2.2.2 Affine Frenet-Serret frame . . . . . . . . 2.2.3 Affine curvature of curves with arbitrary 2.3 Space Curves and Helices . . . . . . . . . . . . 2.3.1 The Darboux vector . . . . . . . . . . . 2.3.2 Generalized helices . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Approximation by conic splines 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Related work . . . . . . . . . . . . . . . . . . . 3.1.2 Results of this chapter . . . . . . . . . . . . . . 3.1.3 Overview . . . . . . . . . . . . . . . . . . . . . 3.2 Geometry of Conics . . . . . . . . . . . . . . . . . . . 3.2.1 Conics have constant affine curvature . . . . . 3.2.2 Osculating conic at non-sextactic points . . . . 3.2.3 The five-point conic . . . . . . . . . . . . . . . 3.3 Optimal conic approximation of affine spiral arcs . . . 3.3.1 Intersections of conics and affine spirals . . . . 3.3.2 Displacement function . . . . . . . . . . . . . . 3.3.3 Equioscillation property . . . . . . . . . . . . . 3.3.4 Monotonicity of optimal Hausdorff distance . . 3.4 Near optimal conic approximation of affine spiral arcs 3.4.1 Uniqueness of equisymmetric conic . . . . . . . 3.4.2 Monotonicity of the equisymmetric distance . . 3.5 Affine curvature of offset curves . . . . . . . . . . . . . 3.6 Complexity of conic splines . . . . . . . . . . . . . . . 3.6.1 Hausdorff metric case . . . . . . . . . . . . . . 3.6.2 Symmetric difference distance case . . . . . . . 3.7 Implementation . . . . . . . . . . . . . . . . . . . . . . 3.7.1 A spiral curve . . . . . . . . . . . . . . . . . . . 3.7.2 Cayley’s sextic . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . ii

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33 33 34 34 36 38 39 39 41 45 45 47 48 51 52 52 53 53 55 55 56 56 58 59 60 61 62 66 66 69 75 76 79 82

CONTENTS 4 Helix spline approximation of space curves 4.1

85

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1

Approximation by bihelical splines . . . . . . . . . . . . 85

4.1.2

Results of this chapter . . . . . . . . . . . . . . . . . . . 86

4.2

Circular Helices . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3

Helix segments with constraints . . . . . . . . . . . . . . . . . . 90

4.4

Bihelices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4.1

4.5

4.6

Junction lines and junction cylinders . . . . . . . . . . . 96

Local approximation of space curves . . . . . . . . . . . . . . . 103 4.5.1

Approximation Theorem . . . . . . . . . . . . . . . . . . 103

4.5.2

Lower bounds for sup-norm of constrained functions . . 105

4.5.3

Curvature of offset curves . . . . . . . . . . . . . . . . . 107

4.5.4

Proof of the Approximation Theorem . . . . . . . . . . 110

Asymptotically optimal bihelix-splines. . . . . . . . . . . . . . . 113 4.6.1

Locally optimal bihelices. . . . . . . . . . . . . . . . . . 113

4.6.2

Complexity of bihelix splines . . . . . . . . . . . . . . . 114

4.6.3

An algorithm for near optimal bihelix splines . . . . . . 115

5 Biarc approximation of plane curves 5.1

121

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.1.1

Related Work . . . . . . . . . . . . . . . . . . . . . . . . 122

5.1.2

Results of this chapter . . . . . . . . . . . . . . . . . . . 122

5.1.3

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2

Local approximation of plane curves with biarcs

. . . . . . . . 123

5.3

Geometry of Biarcs . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.4

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Singularities of secant maps

131

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2

Secant and inner projections of immersions . . . . . . . . . . . 133

6.3

Secant maps of generically immersed surfaces . . . . . . . . . . 140

6.4

Classification of Z2 stable map-germs R4 → Rn . . . . . . . . . 146 iii

CONTENTS A Appendix to Chapter 3 151 A.1 Divided differences and the Division Property . . . . . . . . . . 151 A.2 Approximation of n-flat functions . . . . . . . . . . . . . . . . . 153 B Appendix to Chapter 4 159 B.1 Claim in the proof of Theorem 4.3.2 . . . . . . . . . . . . . . . 159 B.2 Intersecting planes in three-space . . . . . . . . . . . . . . . . . 160 Bibliography

163

Summary

175

Samenvatting

181

iv

Acknowledgements

“A hundred times everyday I remind myself that my inner and outer life are based on the labours of other men, living and dead, and that I must exert myself in order to give in the same measure as I have received and am still receiving.” – Albert Einstein This thesis would not have been possible without the essential and gracious support of many individuals. First and foremost I offer my sincere gratitude to Gert Vegter, my supervisor in Groningen, whose guidance and support enabled me to expand my understanding and knowledge in the field of Computational Geometry. He has been instrumental in me building a critical thinking approach to research. I am also thankful to him for suffering many iterations of this thesis patiently. I would like to thank Joachim Rieger, my supervisor from Halle, Germany, for giving me the opportunity to work with him in the area of Singularity Theory, which has resulted in Chapter 6, of this thesis. I would also like to thank him for his help during my stay in Halle. My heartfelt thanks to Sylvain Petitjean for coauthoring the paper, ‘Approximation by Conic Splines’. For this dissertation I would like to thank my reading committee members: Dirk Siersma, Henk de Snoo and Jos Roerdink for their time, interest and helpful comments. I am grateful to the European Union project: Algorithms of Complex Shapes, Faculty of Mathematics and Natural Sciences Groningen, Groningen Graduate School of Science and Martin Luther University (Halle-Wittenberg), for financing various parts of this project. Scholarship is nominally a solitary enterprise, but a network of communities and people sustain individual scholars in ways both visible and invisible. I wish to extend my warmest thanks to all the PhD students (former and present) and staff of the Johann Bernoulli Institute for Mathematics and v

Computer Science for all their help during the last few years. I wish to thank Alef for translating the summary, that appears in this thesis, into Dutch. I am grateful to my friends and colleagues from Halle for making my stay there a memorable one. On a personal note I would like to take this opportunity to thank many people (teachers and colleagues) who have taught me mathematics and influenced me greatly. I would also like to thank all my friends from India for their support and camraderie. My sincere thanks to Bert for translating a short summary of this thesis into Dutch. Many thanks to my friends in Groningen who have helped in more ways than one during my stay here. Special thanks are due to my parents for believing in me and supporting me in all my pursuits. Paramita, I thank for her unconditional emotional support. My brother, Pothik, who has been my teacher in more ways than one, I will always be grateful to. Last but not the least, I cannot thank Cherian enough for his patience over the years when I spoke endlessly about my work. For always being ready to discuss and argue, I thank him, I could not have asked for a better sparring partner. Sunayana Ghosh October 2010, Groningen.

List of Figures

1.1 1.2 1.3 1.4

Photo: Westlawn Institute of Marine www.westlawn.edu (used with permission) . . . General spline element . . . . . . . . . . . . . . Saddle, Pig Trough and Monkey Saddle . . . . Secant singularities of space curves . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

. . . .

Technology, . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 4 . 11 . 22 . 27

Symmetric difference between two curves. . . . . . . . . . . . . The curve and its osculating conic . . . . . . . . . . . . . . . . One parameter family of bitangent conics . . . . . . . . . . . . Graphs of the family of displacement functions. . . . . . . . . . Notations for symmetric difference distance . . . . . . . . . . . Area between a curve and its chord. . . . . . . . . . . . . . . . Symmetric difference for parabolas and general conics . . . . . Spiral arc approximation by conic and parabolic arcs. . . . . . Spiral arc approximation, the symmetric difference distance case. Parabolic and conic spline approximation of Cayley’s sextic with respect to Hausdorff metric. . . . . . . . . . . . . . . . . . 3.11 Approximation of Cayley’s sextic with respect to symmetric difference distance. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3

46 53 57 57 60 70 71 77 78 80 82

One parameter family of junction cylinders . . . . . . . . . . . 103 Monotonicity of the distance function between curve and bihelices116 Nested bihelices lying on a surface . . . . . . . . . . . . . . . . 117 vii

4.4 4.5

Distance functions for a one parameter family of bihelices . . . 118 Bihelix spline approximation of a conical spiral . . . . . . . . . 119

5.1 5.2 5.3 5.4 5.5

Nested osculating circles of a spiral arc . . Vogt’s Theorem . . . . . . . . . . . . . . . Junction circle of a spiral arc. . . . . . . . Biarcs lying side by side on a spiral arc . Family of distance functions between biarc

. . . . . . . . . . . . . . . . . . . . . . . . and spiral

. . . . . . . . . . . . arc.

. . . . .

. . . . .

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125 126 127 127 129

A.1 Graph of error function . . . . . . . . . . . . . . . . . . . . . . 156 6.1 6.2

Biarc and Family of biarcs . . . . . . . . . . . . . . . . . . . . . 176 Family of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.3 6.4

Twee-boog en een Familie van Twee-bogen . . . . . . . . . . . 182 Familie van kegelbogen . . . . . . . . . . . . . . . . . . . . . . . 183

List of Tables

1.1

Seven catastrophes by Thom . . . . . . . . . . . . . . . . . . . 25

3.1 3.2

Complexity results for spiral arc, the Hausdorff metric case . . Complexity for spiral arc in the symmetric difference distance case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexity results for Cayley’s sextic, the Hausdorff metric case. Complexity results for Cayley’s sextic with respect to the symmetric difference distance. . . . . . . . . . . . . . . . . . . . . .

3.3 3.4

76 79 81 83

4.1 4.2 4.3

Complexity table for a conical spiral curve . . . . . . . . . . . . 119 Complexity table for the moment curve . . . . . . . . . . . . . 119 Complexity table for a spherical helix curve . . . . . . . . . . . 120

6.1 6.2

Generic secant germs of immersions . . . . . . . . . . . . . . . 142 A(H)-stable germs R4 → Rn . . . . . . . . . . . . . . . . . . . 148

ix

Chapter 1

Introduction

1.1

Computer Aided Geometric Design

Computer Aided Geometric Design (CAGD) was coined by Barnhill and Riesenfeld in 1974 and involves the study of geometric algorithms. The problems addressed in this field are related to curve and surface modeling. Its primary objective is to represent real-world objects in forms suitable for computations in CAD/CAM systems. Computer-Aided Design (CAD) involves the use of computer technology for the design of objects, real and virtual. CAD is used to design curves in two dimensional space, and curves, surfaces, and solids in three-dimensional space. CAGD has extensive applications in automotive, shipbuilding and aerospace industries and more recently in computer animation for special effects in movies. This thesis considers problems in curve approximation which fall within the scope of CAGD. Being application oriented in nature, several fields of mathematics come into play in tackling design problems. Tools from approximation theory, geometric modeling and differential geometry are used to solve the curve approximation problems in this thesis. The key goal is the approximation of curves and the asymptotic analysis of the rates of convergence of different approximation schemes with respect to various metrics. To measure the error, one usually uses the notion of approximation order. In this thesis we go a step further and look at the complexity of approximation. The central idea is to approximate a sufficiently smooth parametric curve α : [0, σ] → Rm , m = 2, 3, by a class of tangent continuous spline curves S, such that the Hausdorff distance between α and a spline curves S ∈ S is given by δH (α, S) = max{sup inf d(x, y), sup inf d(x, y)}, x∈α y∈S

y∈S x∈α

2

1. Introduction

where d is the distance between two points in Rm . Thus the Hausdorff distance between α and S is given by δH (α, S) = inf δH (α, S) = cσ n + O(σ n+1 ), S∈S

where σ is the arc length of the curve α and n ∈ N denotes the approximation order of α by the class of spline curves S and c is a positive constant of approximation. In our case c depends on the differential invariants like curvature and affine curvature, cf. Chapter 2. The expression of approximation order is used to compute the complexity of the approximating spline. Complexity describes the minimum number of components of an optimal spline (with respect to Hausdorff distance and symmetric difference distance in our case) approximating a given curve α and is used in the same sense as by Fejes T´oth [42], McClure and Vitale [82] and Ludwig [74]. If the approximation order is n, the complexity of the approximating spline is of the form cε−1/n + O(1), where ε is the error (e.g., Hausdorff distance) of approximation. To find the optimal spline we rely on differential geometric properties of the approximating spline curve.

1.1.1

Brief history of CAGD

Curves were used in the manufacturing environment since Roman times, for the purpose of shipbuilding. A ship’s ribs - wooden planks emanating from the keel - were produced based on templates which could be reused many times [41]. The form of the ribs was defined in terms of tangent continuous circular arcs. Geometric design found application in aeronautics. In his book Analytical Geometry with Application to Aircraft [69] R. Liming combined classical drafting methods with computational techniques for the first time. Liming worked for the NAA (North American Aviation) during World War II, a company which built the famous Mustang fighter planes. Conics, based on the constructions of Pascal and Monge, were used in aircraft and shipbuilding industries. Traditionally these constructions were used by draftsmen as the basic product information. Liming translated the classical drafting constructions into numerical algorithms. This was advantageous because numbers can be stored in unambiguous tables.

1.1. Computer Aided Geometric Design

3

Liming’s conic constructions were an exception but were not widely available outside the aircraft industry. Thus the problem of communicating the information stored on blueprints to computers (driving milling machines) remained. Milling machines were used for production of dies and stamps for sheet metal parts. In France, de Casteljau and B´ezier helped designers to abandon the blueprint process altogether. In the U.S., J. Ferguson at Boeing and S. Coons at M.I.T. provided alternative techniques. General Motors developed its first CAD/CAM system DAC-I (Design Augmented by computer). It used the fundamental curve and surface techniques developed at General Motors by researchers such as C. de Boor and W. Gordon. M. Sabin worked for British Aircraft Corporation and was instrumental in developing their CAD system, Numerical Master Geometry. All these developments took place in the sixties. All these methods existed in isolation for a while, until the seventies started to see a union of different research approaches, culminating in the creation of a new discipline, CAGD. Without the advent of computers a discipline such as CAGD would not have emerged. In order to solve some of the theoretical problems that arose from the blueprint-to-computer challenge, the French car company Citro¨en hired de Casteljau in the year 1959. He developed a system which was primarily aimed at design of curves and surfaces instead of focusing on the reproduction of existing blueprints. From the very beginning he adopted the use of Bernstein polynomials for his curve and surface definitions, together with what is now known as the de Casteljau algorithm. W. B¨ohm was the first to give de Casteljau recognition for his work in the research community. During the early sixties, Pierre B´ezier headed the design department and also realized the need for computer representations of mechanical parts. B´ezier’s work in this area was widely published in [7, 8, 9]. The Renault CAD/CAM system UNISURF was based entirely on B´ezier curves and surfaces [11]. Curves have been employed by draftsmen for centuries. The majority of these curves were circles and some were ‘free-form’. Such curves arose from applications such as ship hull design and architecture. For drawing purposes carefully designed wooden curves consisting of pieces of spirals and conics were used. Another mechanical tool, called a spline was also used. A mechanical spline was a flexible strip of wood that was held in place and

4

1. Introduction

shape by metal weights, known as ducks. A spline bends as little as possible, resulting in shapes that are both aesthetically pleasing and physically optimal. The mathematical counterpart to a mechanical spline is a spline curve, one of the fundamental parametric curve forms. The differential geometry of curves was well understood since the late 1800s after the work of FrenetSerret, Darboux and others. On the other hand the focus of the research in numerical analysis and approximation theory was entirely on functions. Both of these areas were brought together when they became important building blocks of CAGD [41].

Figure 1.1: Spline with ducks (Photo: Westlawn Institute of Marine Technology, www.westlawn.edu)

1.2

Curve modeling

Geometric modeling of curves provides an invaluable tool for representation or visualization, analysis and manufacture of any machine part by providing the basis for the representation of surfaces and solids and thus of real world

1.2. Curve modeling

5

objects. Curve design is fundamentally significant in Computer Aided Design because it allows the user to manipulate the shape of a curve locally (at a small region around a point) without altering it significantly. The way in which active control on curve’s shape can be sought, is by choosing a set of data points and requiring to interpolate a best fitting curve through it. Curve interpolation and curve fitting methods are two of the oldest methods available in curve design. A different problem which is closely related to curve interpolation is the approximation of a parametric or implicit curve by a simpler curve. Various curve approximation methods exist to approximate a given set of data points, of a parametric curve or an implicit curve with respect to some error bound. The motivation behind most curve approximation techniques is to give an efficiently computable and numerically stable algorithm which preserves the shape of the given input. The bisection algorithms presented in this thesis efficiently computes the spline curve with some limitations on shapes. Nonetheless these algorithms can handle shapes under certain constraints on the input curve and are numerically stable (cf. Chapters 3 and 4). In this section we survey some of the curve approximation techniques, without making any claims of being exhaustive.

1.2.1

B´ ezier curves

B´ezier curves are widely used in computer graphics to model smooth curves. Industrial applications of B´ezier techniques are described by B´ezier [10] at Renault, de Casteljau [26] at Citro¨en, Farin [38] at Daimler-Benz and Hochfeld and Ahlers [58] at Volkswagen. The Bernstein basis functions are the building blocks for B´ezier curves. A B´ezier curve of degree n is given by B(t) = Σni=0 bi Bin (t), t ∈ [0, 1]

(1.1)


where Bin (t) = ni (1 − t)n−i ti , and b0 , · · · , bn are control points. For n = 2 we get a quadratic B´ezier curve which is a parabolic segment. The polygon formed by connecting the B´ezier points with line segments, starting with b0 and finishing with bn , is called a control polygon. The convex hull of the

6

1. Introduction

control polygon contains the B´ezier curve. De Casteljau devised a recursive method to evaluate polynomials in Bernstein form or B´ezier curves. The de Casteljau’s algorithm can be used to split a single B´ezier curve into two B´ezier curves at an arbitrary parameter value. The key idea of de Casteljau’s algorithm is that the control polygons generated by recursive subdivison converge to the original B´ezier curve. From equation (1.1) the following properties of B´ezier curves are inferred: - A B´ezier curve passes through the control polygon at the endpoints b0 and bn and hence the curve interpolates the endpoints. - The B´ezier curve generated by an affine map Φ applied to the control polygon is the same as the curve under the same map depending on the original control polygon. More precisely, Σni=0 Φ(bi ) Bin (t) = Φ(Σni=0 bi Bin (t)), thus we conclude that the curve is invariant under affine maps. We also infer that B´ezier curves are independent of the choice of the origin of the coordinate system. Other than these two properties relevant in the context of this thesis the B´ezier curves also have the symmetry property, convex hull property and the variation diminishing property. For more details of these properties we refer to Farin [40]. B´ezier curves provide a powerful tool in curve design, but they have some limitations. If the curve to be modeled has a complex shape, then its B´ezier representation will have a very high degree (for practical purposes, degrees exceeding 10 are considered high [40]), in such cases the curves are modeled using composite B´ezier curves. Such piecewise polynomial curves are an example of spline curves. In describing a shape using free-form curves, it is common to use several curve segments which are joined together with some degree of continuity. There are two types of continuity that can be imposed on two adjacent B´ezier curves: parametric continuity and geometric continuity. In general, two curves which are parametrically continuous to a certain degree are also geometrically continuous to that same degree, but the converse is not true. Parametric spline curves are typically constructed so that the first n parametric derivatives are the same where the curve segments join. This type of continuity condition is called C n or n-th order parametric continuity. E.g., C 0 curves are joined, C 1 curves are C 0 and tangent vectors at the join

1.2. Curve modeling

7

are equal. The concept of geometric continuity was an early attempt at describing, through geometry, the concept of continuity as expressed through a parametric function. In general Gn continuity exists if a curve can be reparametrized to have C n continuity [3]. E.g., G0 continuity implies two curves meet at a point, G1 implies G0 and that the tangent vectors of the two curves at their junction point are proportional. In this thesis the concept of G1 continuity, also known as tangent continuity, is used in curve approximation (cf. Chapters 3 and 4). Modern imaging systems like Postscript, Asymptote and Metafont use B´ezier splines composed of cubic B´ezier curves for drawing curved shapes and allow for curvature (G2 ) continuity. B-spline curves. A B-spline curve is a generalization of a B´ezier curve, where any B-spline curve is parametrized by spline functions which are linear combinations of B-splines. B-splines provide a different approach to representing piecewise polynomial curves. In 1946, I. J. Schoenberg [106] coined the term B-spline which is short for basis spline. He used B-splines for statistical data smoothing, and his paper started the modern theory of spline approximation. B-splines were used in industry after de Boor published his work [24] in the early 1970s. At around the same time Cox published similar results in [21]. Cox and de Boor found separately a recurrence relation to compute B-splines and showed that this is a perfect tool for practical and stable evaluation of B-splines.

1.2.2

Rational curves

Rational techniques and representations were at the root of geometric modeling. Curves like conics and circular arcs can be expressed exactly as rational B´ezier curves, cf. Lee [67]. Conic sections and quadric surfaces were initial building blocks of early CAD systems. In [69, 70] Liming gives many geometric constructions for aircrafts using conics. Later Coons at Ford introduced conics into a CAD environment. Independently, conics were used by engineers at Boeing. A rational B´ezier curve of degree m is a parametric curve which is described by control points, bi ∈ Rn , n = 2, 3, weights wi and the parameter t. Without loss of generality we let t vary from 0 to 1. The curve has the

8

1. Introduction

form c(t) =

m Σm i=0 wi bi Bi (t) , m Σm i=0 wi Bi (t)

(1.2)

where Bim are the Bernstein B´ezier basis functions. If we set all the weights wi to 1 we obtain a polynomial B´ezier curve. The rational B´ezier curves inherit affine invariance, the convex hull property, endpoint interpolation and the variation diminishing property from their polynomial counterpart. One important property that rational B´ezier curves have but do not share with polynomial B´ezier curves is projective invariance, which means that a rational B´ezier curve is invariant under general projective transformations. A conic section can be represented as a rational quadratic B´ezier curve, and when w0 , w2 6= 0, we can assume the two weights w0 and w2 to be 1. For a proof cf. [40, Page 221]. Hence, a conic section is given by c(t) =

b0 B02 (t) + w1 b1 B12 (t) + b2 B22 (t) , B02 (t) + w1 B12 (t) + B22 (t)

with t ∈ [0, 1]. Just like in the B´ezier curves case we can generalize polynomial B-spline curves to rational B-spline curves. This generalization defines non-uniform rational B-splines (NURBS) which are the current industry standard. The NURBS curves also display affine invariance, the convex hull property, endpoint interpolation and the variation diminishing property. One of the problems of interpolation with rational curves is that they are very sensitive to their parametrization as compared to the polynomial spline curves. Approximation with conics is an example of rational B´ezier curve approximation. The most straightforward approach is to fit conics locally through five data points each and then piece together these conic segments. In general the resulting conic spline curve in this case will only be C 0 continuous. To get a smooth conic spline curve one prescribes, three data points p0 , p1 , p2 and two end tangents q0 and q1 . Chapter 3 analyzes the complexity of tangent continuous conic spline approximation of a planar curve. Osculatory interpolation is another practical interpolation method in which one pieces together locally fitted arcs with at least G2 continuity. For an example of osculatory interpolation suppose we are given data points pi , unit tangents qi and

1.2. Curve modeling

9

Euclidean curvature values κi . We can now fit a rational cubic B´ezier curve segments ci such that ci (0) = pi , ci (1) = pi+1 , the unit tangents c0i (0) = qi , c0i (1) = qi+1 and curvatures κi . H¨ollig proved in [60] that such a curve exists under the restrictions on the geometry induced by the data, where the data corresponds to a sufficiently smooth curve with non-vanishing curvature and torsion in space. Furthermore, in [60] he gives a method to compute such a curvature continuous cubic rational spline. In [104] Shaback shows that, given an odd number of points in general position in R2 , if they can be interpolated at all by a smooth curve of non-vanishing curvature then this would be a unique G2 interpolant consisting of pieces of conics under some restrictions on the data. This method of osculatory interpolation leads us to a more general method of approximation known as Geometric Hermite Interpolation, which we describe in Section 1.3.3.

1.2.3

Implicit curve modeling

An overview of various curve modeling methods would be incomplete without considering modeling of implicit curves. Implicit objects have gained increasing importance in geometric modeling, visualization, animation and computer graphics due to their nice geometric properties which give some advantages over traditional modeling methods. The bisection algorithms presented in this thesis give good approximation of a given parametric curve where, the theoretical and algorithmic results on the number of optimal patches, match exactly in the parabolic and conic case. But what these algorithms do not consider is a topologically correct approximation of a given curve. The primary challenge in implicit curve modeling is to develop rigorous, robust and tractable methods for guaranteeing that computational approximations of smooth curves and surfaces preserve critical topological characteristics. Implicit planar curves are defined as the zero set of a function F : R2 → R. The approximation of implicit planar curves by line segments is a very classical problem. The approximation schemes for implicit curves can be roughly divided in continuation methods and adaptive enumeration techniques. Plantinga and Vegter [96] present an algorithm which uses interval arithmetic to determine the global properties of the implicit function. They use these properties to generate a piecewise linear approximation of an implicit curve or surface

10

1. Introduction

which is isotopic to the object itself. For curves and surfaces this paper gives one of the first practical algorithms that guarantees correct topology. Vacavant, Coeurjolly and Tougne [111] use tools from discrete geometry to build an original geometrical and topological representation of the implicit curve. We refer to the thesis of Plantinga [95] for algorithms on isotopic approximation of implicit curves and surfaces and for detailed survey of various methods in implicit curve and surface approximation we refer to [52].

1.3

Approximation Theory

The theory of spline functions and their applications is an area of active research in Approximation Theory since the 1900s. The rapid development of spline functions is due primarily to their great usefulness in applications. Since they are easy to store, evaluate and manipulate on a computer, a plethora of applications have been found. These include, for example, data fitting, function approximation, numerical quadrature and the numerical solution of operator equations, integral equations, optimal control problems, to name a few. As a result of these innumerable applications spline theory has become a firmly entrenched part of Approximation Theory and Numerical Analysis. For a detailed review of approximation theory including the theory of splines we refer to [23, 99, 108, 107]. Approximation problem. One of the basic problems in Approximation Theory can be stated in generic form as follows: Given some element g in a metric space X (with metric d), find a best approximation m∗ to g from some given subset M of X, i.e., find m∗ ∈ M s.t. d(g, m∗ ) = inf d(g, m) =: dist(g, M ) m∈M

(1.3)

and we consider the set of all such elements m∗ . The basic problems that arise are existence, uniqueness and characterization of a best approximation m∗ . The best approximation is defined implicitly by (1.3). A characterization is an explicit description, that can be turned into an algorithm for the computation of this best approximations. See Chapters 3, 4 and 5 in this thesis, where we

1.3. Approximation Theory

11

solve this approximation problem for curves in the plane with sets of biarcs, parabolic arcs and conic arcs and for curves in space with bihelical arcs.

Figure 1.2: General spline element(blue) of a given curve (red)

Figure 1.2 illustrates the approximation of a curve by a spline element. The important general steps (taken in the conic case) for approximating a curve with a spline element are given below. In the conic case, X is the space of all arcs of planar curves with monotone, non-vanishing Euclidean curvature, M is the space of all bitangent conic arcs. Furthermore, d in this case is the Hausdorff distance, between two curves. Since the curves are compact subsets of R2 , the Hausdorff distance in this case is a metric. We also consider the case when the distance function is the symmetric difference distance, for the definition of symmetric difference distance, we refer to Section 3.1 in Chapter 3. • An offset curve β to a planar parametric curve α, defined over interval I, with arc length parameter s, cf. Chapter 2, is given by β(s) = α(s) + f (s)N (s). Here N (s) denotes the unit normal to α at α(s) (cf. Chapter 2). • The distance function f is used to compute the Hausdorff distance δH (α, β) = ||f ||∞ = maxs∈I |f (s)|. • If α and β have n + 1 intersections, such that α(si ) = β(si ), where s0 ≤ s1 ≤ · · · ≤ sn , then the distance function is expressed in terms of a divided-difference formula, cf. Chapter 3 f (s) = (s − s0 ) · · · (s − sn )[s0 , · · · , sn , s]f,

12

1. Introduction where [s0 , · · · , sn ]f is the n−th divided difference and is defined as the coefficient of sn in the polynomial of degree n that interpolates f at s0 , · · · , sn . For more information we refer to Appendix A and the references in it. • Furthermore, denoting sn − s0 by σ and using the Hermite-Genocchi identity as explained in Appendix A, the function f has the following form f (s) =

(n+1) 1 (s0 )(s (n+1)! f

− s0 ) · · · (s − sn ) + O(σ n+2 ).

One of the goals of this thesis is to express f (n+1) (s0 ) in terms of differential invariants of the curve, like (affine) arc length, (affine) curvature and its derivatives.

1.3.1

Approximation order and complexity

The approximation order is characterized by the properties of the approximation scheme, the metric of approximation as well as by a definite property of the approximated object. In numerical analysis, the approximation order of a numerical method having error O(hn ), where h is the step-size,( i.e., the size h in which an interval is subdivided) is the exponent n. Thus, e.g., given a real valued function f over an interval [a, b], a function g is an n-th order approximation of f if and only if the interval [a, b] can be subdivided into intervals of size h, such that as h ↓ 0, sup |f (x) − g(x)| ≤ C hn , a≤x≤b

where C is a positive constant. We then say that g approximates f with an error O(hn ) and n is the approximation order. Note that n also depends on the approximation scheme. In this thesis we make explicit the constant C of approximation. An associated notion is that of complexity. In this thesis complexity refers to space complexity, i.e., the number of elements in a spline of a certain type the approximates the curve to within distance ε. We express this space complexity in terms of ε and differential invariants of the curve. Another notion

1.3. Approximation Theory

13

of complexity is that of time complexity. The paper by Drysdale, Rote and Sturm [34] is an example where curve approximation algorithms with respect to time complexity are addressed. They consider approximation of polygonal curve with a minimum number of circular arcs and biarcs. Their algorithm takes O(n2 log n) time to compute a series of circular arcs to approximate a polygonal line with n vertices. We discuss the relevance of their and similar results in greater detail in applications of our work in Section 1.3.3. In general, an algorithm with better time complexity implies a faster algorithm, but apart from this its space complexity is also important, this is essentially the number of memory cells which an algorithm needs. A good algorithm keeps this number as small as possible. There is often a time-space-tradeoff involved in a problem, that is, it cannot be solved with less computing time and low memory consumption. One then has to make a compromise and to exchange computing time for memory consumption or vice-versa, depending on which algorithm one chooses. Although, in this thesis we focus on algorithms which look at complexity in terms of space, a possible extension of our work is to use the theoretical results on complexity to determine the time complexity of algorithms. We discuss this probable application of our work in greater detail in the context of geometric Hermite interpolation and refer reader to Section 1.3.3.

1.3.2

Curve interpolation

Curve interpolation, as already discussed earlier is the constructive (possibly approximate) recovery of a curve of a certain class by its known values, or by known values of its derivatives, at given points. In this section we briefly describe some of the existing curve interpolation methods. In Section 1.3.3 a more general interpolation scheme, Geometric Hermite interpolation is discussed in some detail as the curve approximation problems of this thesis lie within the scope of geometric Hermite interpolation. Interpolation of curves are described by points which either come from some mathematical construction, or are obtained as data using some mechanical device such as a digitizer. Curve interpolation using polynomials. The curve interpolation problem, using polynomials of degree n can be solved uniquely, if we are given n+1

14

1. Introduction

pairwise distinct points Pi ∈ R3 , i = 0, 1, · · · , n associated with appropriately selected parameter values ti . Therefore depending on the basis of Pn , where Pn is the space of polynomials of degree at most n, one gets, • interpolation using monomials. In which case the polynomial basis is given by {1, t, · · · , tn }. • Interpolation with Lagrange polynomials. In this case the polynomial basis is given by {L0 , L1 , · · · , Ln }, where Li (tk ) = δik :=

( 1, if i = k, 0, if i 6= k

where δik is the Kronecker delta. The condition assumes that Li has value 1 at t = ti and vanishes at other parameter values. • Interpolation with Newton polynomials. The Newton polynomial basis is given by {N0 , N1 , · · · , Nn }, where Ni (t) = (t − t0 ) (t − t1 ) · · · (t − ti−1 ), and N0 (t) = 1.

1.3.3

Geometric Hermite interpolation

Geometric Hermite Interpolation (GHI) is a generalization of different approximation schemes that have been developed in the past for parametric curves to be approximated by polynomial or rational curves. The central idea is to consider the curve independent of its actual parametrization. In general a curve in this situation is not approximated by a single polynomial or rational curve, but by a geometric spline. There are two analogous problems that one considers in this case. • The first problem involves the computation of complexity of the best approximating spline curve, given an error bound ε w.r.t. a metric and a parametric curve. Furthermore, a part of this problem is to give an algorithm to compute this spline and compare the theoretical and algorithmic results. This is the problem of curve approximation that is considered in this thesis.

1.3. Approximation Theory

15

• The second analogous problem is to distribute knots over a given parametric curve, given a knot count n such that the approximating spline is the optimal one, i.e., one would like to distribute n knots such that the error ε is minimized. A knot is a point on the parametric curve where two adjoining spline elements meet with prescribed conditions of continuity. Although the second part of the problem is not considered in this thesis, we can nonetheless reduce it to the first problem by using our theoretical results. Given knot count n we can compute an asymptotically optimal error bound ε using our theoretical results and hence generate the spline with n knots. In fact work on approximation of convex discs with polytopes and its asymptotic error analysis was done in late 1940s. Fejes T´oth [42] gave asymptotic formulae for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices ( as n → ∞), where the distance is in the sense of the Hausdorff metric. In our case, given a regular curve α : [a, b] → R2 , let d denote the distance function between α and a conic C which is tangent to α at α(a) and α(b). The distance function d is given by C(s) = α(s) + d(s) N (s), where s ∈ [a, b] and N is the unit normal vector to the curve α. Thus C(s) is a point on the conic C which is the intersection of the normal line of α at α(s), and d(s) denotes the distance of the point C(s) from α(s) along the normal N (s). The Hausdorff distance between α and C is thus given by the maximum of the function d attained in the interval [a, b] and is denoted by δH (α, C). In fact it can be easily verified that the Hausdorff distance is attained at a point s ∈ [a, b] where the tangent to α at α(s) and tangent to conic C at C(s) are parallel. Let Pni denote the set of all inscribed polygons of the plane convex curve C and let Pnc denote the set of all circumscribed polygons of C, then the error with respect to the Hausdorff distance is given by Z l 1 i c 1 δH (C, Pn ) = δH (C, Pn ) = 8 ( κ1/2 (s) ds)2 2 + O( n14 ), n 0 here κ is the Euclidean curvature of the curve C and s its arc length parameter and l is the arc length. Furthermore in [42], Fejes T´oth computes the error

16

1. Introduction

of approximation with respect to the symmetric difference distance :
3 R l 1/3 1 δS (C, Pni ) = 12 (t) dt n12 + O( n14 ) 0κ
3 R l 1/3 1 δS (C, Pnc ) = 24 (t) dt n12 + O( n14 ) 0κ

(1.4)

Ludwig in [74] extends the result by Fejes T´oth in [42] for approximation of plane curves with polygons by including second terms of the asymptotic expansions in (1.4). Z 1 λ4 λ 1 λ i k(s) ds n14 + O( n14 ), − δS (C, Pn ) = 12 n2 2 5! 0 where λ is affine arc length and k is affine curvature, cf. Chapter 2. McClure and Vitale give sharp estimates of the order of convergence of best approximations of a convex curve by circumscribed and inscribed polygons with n edges. They compute these results with respect to Hausdorff distance, area and perimeter metrics. They first obtain the asymptotic characterizations of best approximations and use this to give a construction of a distribution function which then is utilized in distributing n knots optimally on a given convex curve. The case of asymptotic error analysis of approximation of convex bodies in d-space by inscribed and circumscribed polytopes with respect to the symmetric difference metric is considered by Gruber in [53, 54, 55]. Given a convex body C, Pni = Pni (C), (n = d + 1, d + 2, · · · ) denotes the family of convex polytopes P having at most n vertices inscribed into C, then the asymptotic error expansion with respect to symmetric difference distance metric is given by δS (C, Pni ) =

c n2/(d−1)

+ O(1) as n → ∞.

The error with respect to symmetric difference metric for n vertices is thus given by Z i 1 δS (C, Pn (C)) = 4√3 ( K 1/4 ds)2 n1 + O(1), ∂C

as n → ∞, where ∂C denotes the boundary of C and K is the Gaussian R curvature. In 3-d, ( ∂C K 1/4 ds)2 ≤ 12πV (C), where V (.) denotes the volume.

1.3. Approximation Theory

17

Schneider in [105] shows that for a convex C 3 −hypersurface in Rd the minimal number of vertices of an inscribed, ε-approximating polytope with respect to Hausdorff distance is Z √ (1−d)/2 Nmin (ε) = Cd ε KdF + O(1), F

where Cd is a constant which depends only on dimension, and K is the Gaussian curvature of F . For a definition of Gaussian curvature cf. [30]. Note that for the case, d = 2, the above result reduces to the case of polygonal approximation of plane curves and coincides with Fejes Toth’s result in [42] Geometric design saw the advent of the theory of GHI after some elementary attempts with quadratic or cubic curves - with the famous paper of de Boor, H¨ olling and Sabin [25] in 1987. In [25] the scheme of parametric cubic spline interpolation for planar curves is considered where in addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is C 2 with respect to arc length and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and the order of approximation is 6. Meek and Walton in [84, 83] present an arbitrarily close arc spline approximation of a smooth curves in the geometric Hermite sense. They show that, for a particular choice of a biarc, the asymptotic error between biarc and spiral 1 is given by 324 κ0 h3 + O(h4 ), where κ0 is derivative with respect to arc length of the Euclidean curvature of the given spiral curve and h is the maximum length of the parameter intervals. The error of approximation in this case is of order 3. A biarc is a pair of circular arcs joined with a continuous unit tangent. A biarc can join two points and match the unit tangents at those two points. However, Meek and Walton do not prove the optimal error of approximation of planar curves with biarcs. In Chapter 4 we consider the approximation of spiral arcs with biarcs and give a proof for the optimal error of approximation of such an arc with biarcs. Our expression for the error matches the expression given by Meek and Walton in [83]. A spiral curve in the plane is a curve which has monotonically increasing/decreasing Euclidean curvature. In [84] Meek and Walton give new method for construction of planar osculating arc splines, they are G1 arcs called triarcs. A triarc is a set of three circular arcs joined with a continuous unit tangent. A triarc can join

18

1. Introduction

a pair of points, matching unit tangents and curvatures at the two points. The triarc spline is a more controlled curve, than arc splines produced by biarcs √
1 2 3−3 3/2 0 3 and the asymptotic error of approximation is 24 κ h + O(h4 ). 3 The paper by Degen [27] deals with approximation of parametric curves. As discussed earlier on in this section, in [25] deBoor, H¨ollig and Sabin consider approximation of planar curves with curvature continuous cubic polynomial spline curve. Degen in [27] generalizes their work to consider rational cubic spline approximation of plane curves and imposes third order contact at the end points, he shows that this raises the approximation order to 8. In 1995 H¨ ollig and Koch [59] introduced a general conjecture for GHI of smooth curves in Rn . They conjecture that, under suitable assumptions, splines of degree ≤ d can interpolate points on a smooth curve in Rn with approximation order d−1 m = d + 1 + b n−1 c.

A lot of work has been done on this conjecture, but only special cases of it have been solved till date. A. Rababah was one of the first authors to compare GHI approximants with Taylor polynomials in [100]. He showed, for an arbitrary degree n, there exists a polynomial one-point approximant having approximation order b4n/3c. Other special case proofs of the conjecture by H¨ollig and Koch can be found in the works of M. Floater. Floater [43, 44] obtained the optimal approximation order O(h2n ) approximating conics with polynomial curves of degree n. The approximation of circles with polynomial curves of degree n was studied earlier by Dokken, et al. in [32]. In a recent paper [45] in 2006, Floater gave a construction of a polynomial curve of degree at most n + k − 2 for rational curves of degree k and he showed that the order of approximation was 2n. A. Lin and M. Walker treat Hermite interpolation of space curves in [71]. They give a construction of a degree 5 B´ezier curve which interpolates given positional, tangent, curvature and torsion data at the endpoints and show that the order of approximation is O(h8 ). Note that A. Lin and M. Walker’s result satisfies the conjecture of H¨ollig and Koch in [59] for R3 . In fact the H¨ ollig-Koch conjecture is also satisfied by parabolic arcs and proved in Chapter 3. Parabolic arcs are polynomial curves of degree 2 in R2 and approximate a planar curve with approximation order 4. Furthermore, the cubic spline approximation of plane curves, which approximates a plane

1.3. Approximation Theory

19

curve with approximation order 6 as shown in [25] is also a special case of H¨ollig-Koch conjecture. Open Problems. A recent paper by Degen [28], published in 2005 provides an interesting survey of results in geometric Hermite interpolation using tools from differential geometry. Moreover, this paper is a source of current open problems in this area. Besides the special case of the H¨ollig-Koch conjecture, which states that in planar case there exists a polynomial approximant of degree n having approximation order 2n, there are few more accessible open problems in this field. 1. Find explicit error bounds for instance for the method of deBoor, H¨ollig and Sabin [25]. In particular, if the method admits several solutions: derive a criterion for the one which yields the lowest error. 2. Are GHI approximants ‘shape preserving’, i.e., having the same signs of curvature, not more inflection points and not more vertices than the curve to be approximated? - In case they are not, derive additional conditions to guarantee that property. 3. Compare results of time and space complexity algorithms for curve approximations. Furthermore, investigate the question whether it is better to use low degree together with a finer subdivision of the whole curve or to take higher degrees of the approximant and less segments, taking into account existence problems, stability of algorithms etc. and comparing cost of computing under the same tolerance requirements. Applications for curve approximation. Among the various applications of the complexity results of geometric Hermite interpolation of different schemes some are briefly outlined here. In [34] Drysdale, Rote and Sturm present an algorithm for approximating a polygonal curve with G0 -spline consisting of a minimum number of circular arcs within a tolerance region ε. They also present a similar algorithm to approximate a polygonal curve with a series of tangent continuous biarcs given points and tangent directions, with a runtime of O(n log n). Similarly, in [57] Held and Eibl present an algorithm for approximating a simple planar polygon by tangent continuous

20

1. Introduction

biarc spline. Their experimental results demonstrate that they compute close to a minimum number of biarcs of an n vertex polygon is roughly O(n log n) time. This kind of algorithms find applications in fields like robot motion planning and computer aided manufacturing environments. To assess the quality of this kind of algorithms theoretical bounds are necessary and hence the complexity results described previously in this section and the ones obtained in this thesis are useful. Furthermore, the expression for the Hausdorff distance can be used to develop algorithms which are useful in distributing n points optimally on a curve. This kind of algorithms gives time complexity results which can be helpful in comparing the time complexity results of algorithms obtained in [34, 57]. The complexity results of this thesis may be of use in applications like determining the intersection of high-degree curves and the building of arrangements of algebraic curves. For example, in [5] Berberich et al. give an exact geometry kernel for conic arcs and present a sweep line algorithm for computing arrangements of curved arcs and in [36, 37] Emiris et al. investigate the Voronoi diagrams of ellipses. A possible application of our work in such algorithms is to compute conic arc approximations, of a given arrangements of curves. Although certified approximation of curves in the plane by conic arcs is still an open problem, by using the methods suggested in this thesis it is possible to find sharper bounds between a curve and its approximating conic. This in turn might be useful in robust computation of geometric structures of curved objects, like its Voronoi diagram, or its medial axis. In this approach the number of elements of a conic spline approximation would be orders of magnitude smaller than the number of line segments needed to approximate a curved object with the same accuracy. Whether this feature outweighs the added complexity of the geometric primitives in the computation of Voronoi diagrams would have to be the goal of extensive experiments.

1.4

Classifications in singularity theory

Classifying objects in mathematics is a fundamental activity. Each branch has its own notion of equivalence and it is equally natural to list the objects in respective branches up to the equivalence under study. The final part of

1.4. Classifications in singularity theory

21

this thesis lies in the area of classifications of singularities of secant maps. In this section the background and motivation for studying classifications in singularity theory are provided, following which we give an overview of the classification problem considered in this thesis. Singularity theory provides some useful tools for the study of the local geometry of curves and surfaces, i.e., study of local properties of a single point or of a suitably small neighborhood of a point. It is a rigorous body of mathematics which arose from the interactions between topology, algebraic geometry and differential geometry. Both these fields owe their origins to theory of smooth functions of one variable. For example by using methods from differential calculus we can recognize the shape of graphs of smooth functions. The following algorithm helps us to recognize the shape of the graph. - Given a function f : I → R, compute f 0 (x) and find a point x0 such that f 0 (x0 ) = 0. - Calculate the second derivative f 00 (x), then the point x0 is a minimum if f 00 (x0 ) > 0 and is a maximum if f 00 (x0 ) < 0. Then x0 is called a critical point or a singular point of f if f 0 (x0 ) = 0. The shape of the graph y = f (x) changes drastically around the singular point. Therefore the singular points of f (x) gives crucial information on the shape of the graph y = f (x) (locally, i.e., close to x = x0 ). Moreover x0 is a degenerate singular point if f 0 (x0 ) = f 00 (x0 ) = 0. Thus the nondegenerate critical points are the maximal or minimal points. We say that x0 is an Ak -type singular point if f 0 (x0 ) = f 00 (x0 ) = · · · = f (k) (x0 ) = 0 and f (k+1) (x0 ) 6= 0. A major mathematical source of singularity theory is the classification of the types of critical points or singularities.

1.4.1

Background

Let f : Rn → R be a smooth function. A point u ∈ Rn is a critical point of f if ∂f ∂x1 |u

= ··· =

∂f ∂xn |u

= 0.

The value f (u) at a critical point u is called a critical value of f . Critical points occur when the graph of f has a horizontal tangent. We have already mentioned briefly the cases for n = 1, for n = 2 there are more possibilities.

22

1. Introduction

The most common are (local) maxima, minima and saddles. However, there are a wide variety of more complicated types e.g., the monkey-saddle and fold. An important distinction between the different critical points is the following.

Saddle, nondegenerate

Pig Trough, degenerate Monkey Saddle, degenercorank 1 ate corank 2

Figure 1.3: Some singularities in R2 .

We say that f has a nondegenerate critical point at u if Df |u=0 = 0 and if D2 f |u is a nondegenerate quadratic form i.e., the Hessian matrix Hf (u) =


∂2f (u) ∂xi ∂xj

is non-singular. Note that saddle (given by x2 − y 2 ) has a nondegenerate critical point, the pig trough (given by x2 + y 2 ) is a degenerate critical point of corank 1, whereas the monkey saddle (given by x3 − 3xy 2 ) has a degenerate critical point of corank 2. Thus we conclude that for a real valued function f a singular point x0 ∈ Rn is nondegenerate if det Hf (x0 ) 6= 0. The Morse lemma (cf. Milnor [87]) is useful in classifying the nondegenerate critical points of a real valued function. Before stating the lemma formally, we introduce the notion of germs for convenience, it is a term signifying a localization of various mathematical objects e.g., germs of functions, germs of mappings, germs of analytic sets, etc. Defintion 1.4.1. Given a point x of a topological space X, and two maps f, g : X → Y , then f and g define the same germ at x if there is a neighborhood U of x such that restricted to U , f and g are equal i.e., f |U = g|U . This defines an equivalence relation and a germ is an equivalence class.

1.4. Classifications in singularity theory

23

Then we have the following result Lemma 1.4.2 (Morse lemma). Let 0 ∈ Rn be a nondegenerate singular point of f (x1 , · · · xn ). Then there exists a local diffeomorphism (germ) φ : (Rn , 0) → (Rn , 0) such that f ◦ φ(x1 , · · · , xn ) = ±x21 ± · · · ± x2n + f (0). By the Morse lemma we can recognize the shape of the graph y = f (x1 , · · · xn ) around a nondegenerate singular point very well. On the other hand, if the Hessian Hf of f is singular then we get a degenerate singular point, the natural question to ask is how does one study degenerate singular points. We can measure how critical a degenerate singular point is by computing its corank corank (f)(x0 ) = n − rank Hf(x0 ). Therefore, a singular point x0 of f is nondegenerate if and only if corank (f)(x0 ) = 0. The Morse lemma is a classification theorem of smooth functions near a corank zero singular point. Another version of the Morse lemma allows us to tidy up somewhat, a degenerate critical point, by ‘splitting’ the function into a Morse piece over one set of variables and a degenerate piece on a different set, whose number is equal to the corank. Lemma 1.4.3 (Thom’s splitting lemma (cf. [78, 97])). Let 0 ∈ Rn be a singular point of f : (Rn , 0) → R with corank (f)(0) = r. Then there exists a local diffeomorphism φ : (Rn , 0) → (Rn , 0) and a smooth local function g : (Rr , 0) → R at 0 such that f ◦ φ(x1 , · · · , xn ) = g(x1 , · · · , xr ) ± x2r+1 ± · · · ± x2n + f (0) and rank H(g)(0) = 0. We call g a residual singularity of f . By the splitting lemma the behavior of a function near a degenerate singular point can be found by studying its residual singularity.

24

1. Introduction

Stability. The notion of stability of different singular points is considered, where the effect of a small deformation on a critical point is studied. A deformation p is considered to be small if the function and all its partial derivatives are small for all x in a neighborhood of 0. Assuming that the derivative of p, Dp|0 = 0, suppose f is Morse, i.e., the critical point at 0 is nondegenerate, then for a small enough (here C 2 small is sufficient) p, f + p is also Morse, If det Hf |0 6= 0 then det H(f + p)|0 6= 0. Then f is stable and for all sufficiently small smooth functions p, the critical points of f and f + p have the same type. For degenerate critical points the situation is completely different. In fact every degenerate critical point is unstable [97]. Families of functions. Two functions f, g : Rn → R are right equivalent around 0 if there is a diffeomorphism germ y : Rn → Rn and a constant term γ, such that g(x) = f (y(x)) + γ. Similarly, there is a notion of equivalence for families of functions f, g : Rn × Rr → R which is given by the following definition. Defintion 1.4.4. If there exists a diffeomorphism e : Rr → Rr , a smooth map y : Rn × Rr → Rn , such that for each s ∈ Rr the map ys : Rn → Rn , ys (x) = y(x, s) is a diffeomorphism and a smooth map γ : Rr → R, such that g(x, s) = f (ys (x), e(s)) + γ(s) for all (x, s) ∈ Rn × Rr , then f and g are equivalent. As there is the notion of stability of functions, similarly there is the notion of stability for families of functions in a natural way. If f : Rn × Rr → R is equivalent in the above sense to any family f + p : Rn × Rr → R, where p is a sufficiently small perturbation family p : Rn × Rr → R, then f is stable. This brings us to a much celebrated result due to Thom, which states that a generic k-parameter family, k ≤ 4, of functions is stable and in the neighborhood of a critical point it behaves, up to sign and change of variable, like one of the seven cases as( cf. Table 1.4.1) The term codimension (codim) serves as a measure of complexity of a critical point. Any small

1.4. Classifications in singularity theory Notation A2 A3 A4 D4− D4+ A5 D5

Codim 1 2 3 3 3 4 4

Germ x3 + y 2 x4 + y 2 x5 + y 2 x3 + xy 2 x3 − xy 2 x6 + y 2 x4 + xy 2

25 Name Fold Cusp Swallow-tail Hyperbolic umbilic Elliptic umbilic Butterfly Parabolic umbilic

Table 1.1: Seven classes of singularities due to Thom

perturbation of a function f of codim r leads to a function with at most r critical points. Apart from an algebraic definition, a germ has codim k if it is the smallest k such that the germ occurs in a generic k-parameter families of local functions. Thom considered the classification of degenerate singularities. However, in many cases, one is interested not in an individual object, but in a collection of them, depending on some parameters. Towards the end of the 1960’s Mather closed the gaps in Thom’s work by generalizing the classification of stable map-germs, he devised a method of producing classes of stable map-germs for any source and target dimension, making crucial use of the concept of contact equivalence. For a collection of Mather’s works we refer to cf. [77, 79, 78, 81, 80] There is a rather huge collection of literature of work on classification of singularities in various kinds of cases. For our purpose we will focus on Z2 -equivariant maps in the following section.

1.4.2

Singularities of secant maps

In Chapter 6 our goal is to study the local structure of the secant map of an immersion. We do this by classifying the singularities of such maps. The principal contribution is the classification of the germs of secant maps of generically immersed surfaces in Rn , for n ≥ 3. In this section, we elaborate the concept of secant maps and illustrate the classification problem with an example from Bruce [16]. The classification problem considered in Chapter 6 is an extension of the classification problem of space curves considered by Bruce.

26

1. Introduction

Secant maps of space curves. Suppose γ : R → R3 ,is a regular, smooth embedding, the projectivized secant map S˜ : R × R → P2 is defined by ˜ 1 , t2 ) = p(γ(t1 ) − γ(t2 )), if t1 6= t2 S(t ˜ t) S(t, = p(γ 0 (t)), where P2 is the real projective two space and p : R3 \ {0} → P2 , is the natural projection onto a real projective space P2 , assigning to each point the unoriented line through that point. Thus, the points on the diagonal ∆ = {(t, t) | t ∈ R}, get mapped to the corresponding tangent line by the secant ˜ We consider the local behavior of the secant map for points on the map S. ˜ 1 , t2 ) = diagonal. Observe that the secant map has the following property, S(t ˜ 2 , t1 ), so we conclude that S˜ has Z2 symmetry along the diagonal. Thus S(t ˜ we work in the space of Z2 to obtain local models of the secant map S, equivariant maps, where a function h : R2 , 0 → R2 is said to be Z2 equivariant if h(−x) = h(x), for all x ∈ R2 . The parametrized form of a generic regular space curve γ, upto a local change of coordinates in R and R3 , is given by, γ : (R, 0) → R3 , γ(t) = t,

N X

a2j tj + O(tN +1 ),

j=2

N X


a3j tj + O(tN +1 ) .

j=2

Thus the secant germ is given by Sˆ : (R2 , 0) → R3 , such that ˆ 1 , t2 ) = S(t

t1 − t2 ,

j j=2 a2j (t1

PN

j j=2 a3j (t1

PN

+1 +1 ), − tN − tj2 ) + O(tN 2 1


+1 +1 − tN ) . − tj2 ) + O(tN 1 2

˜ 1 , t2 ) spans the same line as 1, P a2j σj + Since t1 6= t2 , S(t
P ψ2 (t1 , t2 ), a3j σj + ψ3 (t1 , t2 ) , where σj (t1 , t2 ) = (tj1 − tj2 )(t1 − t2 ) and +1 +1 ψj (t1 , t2 ) = O(tN − tN ). Hence the projectivized secant germ S˜ has 1 2 ˆ Now since S˜ is also a Z2 a representation as the last two coordinates of S. equivariant germ, composing it with the symmetry preserving map such that x := 21 (t1 + t2 ) and y := 12 (t1 − t2 ), we obtain a smooth map S : R2 , 0 → R2 , 0, with S(x, y) = S(x, −y). Proposition 1.4.5. Let γ be a space curve with nowhere vanishing curvature.

1.4. Classifications in singularity theory

27

1. If the torsion of the curve γ at t = 0 , τ (0) 6= 0, then Sˆ : R × R → R3 has a local representation at (0, 0), of the form S(x, y) = (x, y 2 ), which is a fold 1.4. 2. If τ (0) = 0, but τ˙ (0), 3κ(0)¨ τ (0) − κ(0) ˙ τ˙ (0) 6= 0, then Sˆ has a local representation at (0, 0), of the form S(x, y) = (x, xy 2 +y 4 ), see figure 1.4.

Fold

Pitch Fork

Figure 1.4: Singularities of secant maps of space curves

Proof. We give a proof sketch for the first case for Proposition 1.4.5. Thus note that σ2 = 2x, σ3 = 3x2 +y 2 , σ4 = 4(x3 +xy 2 ) and σ5 = 5x4 +10x2 y 2 +y 4 . The 2 jet of S at (0, 0) is (2a22 x + a23 (3x2 + y 2 ), 2a32 x + a33 (3x2 + y 2 )). This jet is Z2 equivalent to (x, 3x2 +y 2 ) and hence (x, y 2 ) (under Z2 equivariant coordinate changes in the source and arbitrary coordinate changes in the target) if and only if a22 a33 − a23 a32 6= 0. Furthermore, the torsion of the 2-jet of the curve γ at t = 0 is given by τ (0) =

1 a222 +a232


a22 a33 − a23 a32 ,

thus using Malgrange preparation theorem we conclude that the secant germ (x, y 2 ) is equivalent to the condition that that τ (0) 6= 0. For a detailed discussion on Z2 equivariant mappings from (R2 , 0) → (R2 , 0) we refer to Bruce [16, Appendix].

28

1.5

1. Introduction

Overview and main results

In this section we give a brief overview of chapters in this thesis and highlight the main contributions. Chapter 2 provides a review of the basic concepts of differential geometry of plane and space curves. In this chapter the affine differential geometry of plane curves is treated in some detail as it is used extensively in approximation with conics in Chapter 3. In particular, we introduce affine arc length and affine curvature, which are invariant under equiaffine transformations. Conic arcs are the only curves in the plane having constant affine curvature, which explains the relevance of these notions from affine differential geometry for our work. Moreover, in this chapter we review the differential geometry of generalized circular helices in three space. A derivation of another normal frame for a space curve is given other than the Frenet-Serret frame, this normal frame contains the normalized Darboux vector. The conditions for a space curve to be a generalized helix is also reviewed. In Chapter 3, we study the approximation of curves in the plane by arcs of conics. We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ε is c1 ε−1/4 + O(1), if the spline consists of parabolic arcs, and c2 ε−1/5 + O(1), if it is composed of general coinc arcs of varying type. The constants c1 and c2 are expressed in the Euclidean and affine curvature of the curve, c1 = c2 =


RL 1 1/4 κ(s)5/12 ds √ 4 0 |k(s)| 128
RL 0 1 √ |k (s)|1/5 κ(s)2/5 ds , √ 5 0 2000 5

where L is the length of the given arc α, k denotes its affine curvature, κ the Euclidean curvature and s is the arc length parameter of α. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline. Furthermore, we also derive the complexity result for approximation of plane curves by parabolic and conic arcs with respect to the symmetric difference distance. We show

1.5. Overview and main results

29

that the complexity of an optimal parabolic or conic spline approximating a smooth curve with non-vanishing curvature to within symmetric difference distance ε is c3 ε−1/4 + O(1) for parabolic arcs and c4 ε−1/5 + O(1), if the spline is composed of general conic arcs, where (240)−1/4

R%

c4 = (7680)−1/5

R%

c3 =

0 0

|k(r)|1/5 dr


5/4

|k 0 (r)|1/6 dr


6/5

,

r denotes the affine arc length parameter. We define an equisymmetric conic arc tangent to a curve at its endpoints, to be the (unique) conic such that the areas of the two moons formed by this conic and the given curve are equal, and show that its complexity is asymptotically equal to the complexity of an optimal conic spline. We also show that the symmetric difference distance between a curve and an equisymmetric conic arc tangent at its endpoints is increasing with affine arc length, provided the affine curvature along the arc is monotone. This property yields a simple and an efficient bisection algorithm for the computation of an optimal parabolic or equisymmetric conic spline. Chapter 4 deals with approximation of curves in space with bihelical splines. In this chapter we show that the complexity of bihelical spline approximating a sufficiently smooth space curve with non-vanishing, monotonically increasing/decreasing curvature to within Hausdorff distance ε is cε−1/3 + O(1), such that c =

1 √ 3 324

Z

L


|κ0 (s)|1/2 ds ,

0

and κ is the Euclidean curvature, L length of the arc of the space curve and s is the arc length parameter. A bihelical spline is composed of bihelical arcs. A bihelical arc is composed of two helical arcs which are tangent to each other at the junction point and tangent to the curve at the endpoints. We also conclude that the expression for complexity of a bihelical spline is the same as that of biarcs in the planar case. Furthermore, given points p0 , p1 and tangents T0 , T1 , respectively we show that there exists a unique helical arc through these points with the given tangents if and only if hp0 − p1 , T0 − T1 i = 0.

30

1. Introduction

Given two points pL , pR and two tangents TL , TR , if we fix a direction T0 ∈ S2 \ {TL , TR } and set it to be the junction tangent, then the locus of points p0 , such that there is a bihelical arc through pL , pR and p0 with tangents TL , TR and T0 is a line denoted by l(T0 ). Now as T0 traces a circle in S2 , we get a family of lines l(T0 ), we show that the union of all such lines l(T0 ) is a cylinder and we call it the junction cylinder. Thus as T0 varies in S2 we get a one parameter family of junction cylinder. From our local computations on bihelices we get an expression of the optimal junction tangent, we use this expression to define a near optimal junction tangent Tnopt , we use this expression for the near optimal junction tangent to obtain the corresponding line of junction points l(Tnopt ). From this line of junction points we choose the one that is closest to the given curve and define the bihelical arc associated with it to be near optimal. Our experimental results suggest that the monotonicity property is true and thus we implement a bisection algorithm for computing a near optimal bihelical spline approximation of a given space curve. The complexity results of our algorithm are promising, the experimental and theoretical results of complexity are almost the same for the curves that were tested. Moreover, in the planar case of approximation with biarcs, we show that in case of approximation of a planar spiral with biarcs, there is a unique biarc minimizing the Hausdorff distance. Fixing two points pL and pR and the tangents at these points TL and TR , there exists a one parameter family of biarcs. The locus of the junction points of this one parameter family is a circle and we call it the junction circle. Finally in Chapter 6 we study the singularities of secant maps of immersions of surfaces in Rn for n ≥ 3. We define the projectivized secant map of an immersion X : Rn → Rm+1 , and relate the secant map to n parameter family of inner projections of X(Rn ) from centers in X(Rn ). We show that the secant map at a point outside the diagonal is a versal n parameter deformation of a germ Rn → Rm for a residual set of embeddings. Moreover, under some restrictions on the pair of dimensions we show that the secant map germ along the diagonal is AZ2 equivalent to some Z2 -stable germ for a residual set of immersions X : Rn → Rm , m > n. We also give the classification of germs of secant maps at the diagonal of surfaces generically immersed in Rn , for n ≥ 3. Also we give the classification of Z2 stable germs from R4 → Rn , for n ≥ 2 ,

1.5. Overview and main results

31

this classification is relevant for the secant maps germs of generic immersions X : R2 → Rn+1 . Publications. Parts of Chapter 2 and Chapter 3 are published as Approximations by conic splines in the journal Mathematics in Computer Science [47]. Parts of Chapter 3 appeared as Minimizing the symmetric difference distance in conic spline approximation in the proceedings of MACIS [50] and as extended abstracts in [94, 51]. Chapter 4 is a preprint Approximation of Space Curves by Helix Splines [49]. Chapter 6 is published as Singularities of secant maps of immersed surfaces in journal Geometriae Dedicata [48].

Chapter 2

Differential geometry of curves

2.1

Introduction

Geometric modeling which includes computer graphics and computer-aided geometric design draws on ideas from differential geometry. This chapter presents the mathematical preliminaries of differential geometry of curves in plane and space which are used in our results that we present in Chapters 3 and 4. For a detailed discussion and introduction to this area we refer to [30, 56, 109]. Differential geometry of curves is an area of geometry that involves the study of smooth curves in the plane and in Euclidean space using the methods of differential and integral calculus. One of the most important tools used to analyze a curve is the Frenet-Serret frame, a moving frame that provides a coordinate system at each point of the curve that describes the curve at a point completely. The theory of regular curves in a Euclidean space has no intrinsic geometry. A curve is said to be regular if its derivative is well defined and non zero on the interval on which the curve is defined. Moreover, different space curves are distinguishable from each other by the way they bend and twist. Formally, these are the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves [30] asserts that the knowledge of these invariants completely determines the curve. Another area of differential geometry that we are concerned with is called affine differential geometry. The differential invariants in this case are invariants under area preserving affine transformations. The name affine differential geometry follows from Klein’s Erlangen program. The basic difference between affine and Euclidean differential geometry is that in the affine case we are concerned with area preserving transformations in the plane whereas in the Euclidean case we are concerned with rigid motions. The affine differential geometry of curves is the study of curves in an affine

34

2. Differential geometry of curves

space, and specifically the properties of such curves which are invariant under the area preserving affine transformations in the plane. As stated above, in Euclidean geometry of curves, the fundamental tool is the Frenet-Serret frame, in affine geometry, the Frenet-Serret frame is no longer well defined, but it is possible to define another canonical moving frame, which we call the affine Frenet-Serret frame, which plays a similar decisive role. The theory was developed in the early 20th century, largely due to the efforts of Blaschke [13]. Overview. In Section 2.2 we present the preliminaries of differential geometry of curves in the plane. In particular, we introduce affine arc length and affine curvature, which are invariant under equiaffine transformations. Moreover, we introduce the notion of affine Frenet-Serret frame followed by derivation of affine curvature of curves with arbitrary parametrization. As stated above conic arcs are the only curves in the plane having constant affine curvature, which explains the relevance of these notions from affine differential geometry. Section 2.3 reviews the differential geometry of generalized and circular helices in three-space. There is a derivation of the Darboux vector. Another normal frame for a space curve is derived other than the FrenetSerret frame, this is a more natural frame for the parametric expression of a helix. Furthermore, the condition for a space curve to be a generalized helix is reviewed.

2.2

Planar Curves and Conics

Circular arcs and straight line segments are the only regular smooth curves in the plane with constant Euclidean curvature. Conic arcs are the only smooth curves in the plane with constant affine curvature. The latter property is crucial for our approach, so we briefly review some concepts and properties from affine differential geometry of planar curves. See also Blaschke [13].

2.2.1

Affine curvature

Recall that a regular curve α : J → R2 defined on a closed real interval J, i.e., a curve with non-vanishing tangent vector T (u) := α0 (u), is parameterized according to Euclidean arc length if its tangent vector T has unit length. In

2.2. Planar Curves and Conics

35

this case, the derivative of the tangent vector is in the direction of the unit normal vector N (u), and the Euclidean curvature κ(u) measures the rate of change of T , i.e., T 0 (u) = κ(u) N (u). Euclidean curvature is a differential invariant of regular curves under the group of rigid motions of the plane, i.e., a regular curve is uniquely determined by its Euclidean curvature, up to a rigid motion. The larger group of equi-affine transformations of the plane, i.e., affine transformations with determinant one (in other words, area preserving linear transformations), also gives rise to a differential invariant, called the affine curvature of the curve. To introduce this invariant, let I ⊂ R be an interval, and let γ : I → R2 be a smooth, regular plane curve. We shall denote differen2 tiation with respect to the parameter u by a dot: α˙ = dα ¨ = dduα2 , and so on. du , α Then regularity means that α(u) ˙ 6= 0, for u ∈ I. Let the reparameterization u(r) be such that γ(r) = α(u(r)) satisfies [γ 0 (r), γ 00 (r)] = 1.

(2.1)

Here [v, w] denotes the determinant of the pair of vectors {v, w}, and derivatives with respect to r are denoted by dashes. The parameter r is called the affine arc length parameter. If [α, ˙ α ¨ ] 6= 0, in other words, if the curve α has non-zero curvature, then α can be parameterized by affine arc length, and (2.1) implies that [α(u(r)), ˙ α ¨ (u(r))] u0 (r)3 = 1.

(2.2)

ϕ(u) = [α(u), ˙ α ¨ (u)]1/3 ,

(2.3)

Putting

we rephrase (2.2) as u0 (r) =

1 . ϕ(u(r))

(2.4)

From (2.1) it also follows that [γ 0 (r), γ 000 (r)] = 0, so there is a smooth function k such that γ 000 (r) + k(r) γ 0 (r) = 0.

(2.5)

36

2. Differential geometry of curves

The quantity k(r) is called the affine curvature of the curve γ at γ(r). It is only defined at points of non-zero Euclidean curvature. A regular curve is uniquely determined by its affine curvature, up to an equi-affine transformation of the plane. From (2.1) and (2.5) we conclude k = [γ 00 , γ 000 ]. The affine curvature of α at u ∈ I is equal to the affine curvature of γ at r, where u = u(r).

2.2.2

Affine Frenet-Serret frame

The well known Frenet-Serret identity for the Euclidean frame, namely α˙ = T,

T˙ = κ N,

N˙ = −κ T,

(2.6)

where the dot indicates differentiation with respect to Euclidean arc length, have a counterpart in the affine context. More precisely, let α be a strictly convex curve parameterized by affine arc length. The affine Frenet-Serret frame {t(r), n(r)} of α is a moving frame at α(r), defined by t(r) = α0 (r), and n(r) = t0 (r), respectively. Here the dash indicates differentiation with respect to affine arc length. The vector t is called the affine tangent, and the vector n is called the affine normal of the curve. The affine frame satisfies α0 = t,

t0 = n,

n0 = −k t.

(2.7)

Furthermore, we have the following identity relating the affine moving frame {t, n} and the Frenet-Serret moving frame {T, N }. Lemma 2.2.1. 1. The affine arc length parameter r is a function of the Euclidean arc length parameter s satisfying dr = κ(s)1/3 . ds

(2.8)

2. The affine frame {t, n} and the Frenet-Serret frame {T, N } are related by t = κ−1/3 T,

n = − 13 κ−5/3 κ˙ T + κ1/3 N.

(2.9)

Here κ˙ is the derivative of the Euclidean curvature with respect to Euclidean arc length.

2.2. Planar Curves and Conics

37

Proof. 1. Let γ(r) be the parametrization by affine arc length, and let α(s) = γ(r(s)) be the parametrization by Euclidean arc length. Then α˙ = T and α ¨ = κ N . Again we denote derivatives with respect to Euclidean arc length by a dot. Since γ 0 = t and t0 = γ 00 = n, we have and N = κ−1 α ¨ = κ−1 (¨ rt + (r) ˙ 2 n)

T = α˙ = rt, ˙

(2.10)

Since [T, N ] = 1, and [t, n] = 1, we obtain 1 = κ−1 r˙ 3 . This proves the first claim. 2. The first part of the lemma implies r¨ = 13 κ−2/3 κ. ˙ Plugging this into the identity (2.10) yields the expression for the affine Frenet-Serret frame in terms of the Euclidean Frenet-Serret frame. The affine Frenet-Serret identities (2.7) yield the following values for the derivatives—with respect to affine arc length—of α up to order five, which will be useful in the sequel: α(4)

α0 = t, α00 = n, α000 = −k t, = −k 0 t − k n, α(5) = (k 2 − k 00 ) t − 2k 0 n.

(2.11)

Combining these identities with the Taylor expansion of α at a given point yields the following affine local canonical form of the curve. Lemma 2.2.2. Let α : I → R2 be a regular curve with non-vanishing curvature, and with affine Frenet-Serret frame {t, n}. Then α(r0 + r) = α(r0 ) + r −

1 3!

k0 r3 − +


k00 r4 + O(r6 ) t0

1 4!

1 2

r2 −

1 4!

k0 r4 −

2 5!


k00 r5 + O(r6 ) n0 ,

where t0 , n0 , k0 , and k00 are the values of t, n, k and k 0 at r0 . Furthermore, in its affine Frenet-Serret frame the curve α can be written locally as x t0 + y(x) n0 , with y(x) =

1 2

x2 + 81 k0 x4 +

1 40

k00 x5 + O(x6 ).

The first identity follows directly from (2.11). As for the second, it follows from the first by a series expansion. Indeed, write x=r−

1 3!

k0 r3 −

1 4!

k00 r4 + O(r6 ).

38

2. Differential geometry of curves

Computing the expansion of the inverse function gives r =x+ Plugging in y =

2.2.3

1 2

r2 −

1 4!

1 3!

k 0 x3 +

k0 r4 −

2 5!

1 4!

k00 x4 + O(x6 ).

k00 r5 + O(r6 ) gives the result.

Affine curvature of curves with arbitrary parametrization

The following proposition gives an expression for the affine curvature of a regular curve in terms of an arbitrary parameterization. See also [13, Chapter 1.6]. Proposition 2.2.3. Let α : I → R2 be a regular C 4 -curve with non-zero Euclidean curvature. Then the affine curvature k of α is given by k=

1 ϕ¨ ϕ − 3ϕ˙ 2 ... α] [¨ α , + , ϕ5 ϕ4

(2.12)

where ϕ = [α, ˙ α ¨ ]1/3 . 1 Proof. Identity (2.4) implies γ 0 (r) = Γ(u(r)), where Γ(u) = ϕ(u) α(u). ˙ We denote differentiation with respect to u by a dot, like in α, ˙ and differentiation 0 00 ˙ with respect to r by a dash, like in γ . Then γ (r) = u0 (r) Γ(u(r)), and 000 00 0 2 ˙ ¨ γ (r) = u (r) Γ(u(r)) + u (r) Γ(u(r)). From the definition of Γ we obtain  2  ϕ˙ ϕ¨ ϕ˙ 1 ... ˙Γ = − ϕ˙ α˙ + 1 α ¨ ¨ , and Γ = 2 3 − 2 α˙ − 2 2 α ¨ + α. 2 ϕ ϕ ϕ ϕ ϕ ϕ

Furthermore, since u0 (r) = fore, γ 00 (r) = −

ϕ˙ 1 α˙ + 2 α ¨, ϕ3 ϕ

1 ϕ(u(r)) ,

ϕ(u(r)) ˙ it follows that u00 (r) = − ϕ(u(r)) 3 . There-

and γ 000 (r) =

 3

ϕ˙ 2 ϕ¨ − ϕ5 ϕ4

 α˙ − 3

ϕ˙ 1 ... α ¨ + 3 α, ϕ4 ϕ

where we adopt the convention that ϕ, α, and their derivatives are evaluated at u = u(r). Hence, the affine curvature of α at u ∈ I is given by k(u) = [γ 00 , γ 000 ]  2  ϕ˙ ϕ¨ ϕ˙ 2 ϕ˙ 1 ... ... α] = [¨ α , − 3 − [ α, ˙ α ¨ ] + 3 [α, ˙ α ¨ ] − 6 [α, ˙ α] ϕ5 ϕ7 ϕ6 ϕ7 ϕ 1 ϕ¨ ϕ˙ ... ... = [¨ α, α] + 6 [α, ˙ α ¨ ] − 6 [α, ˙ α]. 5 ϕ ϕ ϕ

2.3. Space Curves and Helices

39

... From (2.3) it follows that [α, ˙ α ¨ ] = ϕ3 and [α, ˙ α] = 3 ϕ2 ϕ. ˙ Using the latter identity we obtain expression (2.12) for the affine curvature of α. Remark 2.2.4. Proposition 2.2.3 gives the following expression for the affine curvature k in terms of the Euclidean curvature κ: k=

9 κ4 − 5 (κ) ˙ 2 + 3κκ ¨ , 8/3 9κ

where κ˙ and κ ¨ are the derivatives of the Euclidean curvature with respect to arc length. This identity is obtained by observing that, for a curve parameterized by Euclidean arc length, the function ϕ is given by ϕ = κ1/3 . This follows from the Frenet-Serret identities (2.6) and the definition (2.3) of ϕ. Substituting this expression into (2.12) yields the identity for k in terms of κ.

2.3

Space Curves and Helices

In Chapter 4 we consider the approximation of curves in space with circular helices. In this section we present the principal notions about differential geometry of space curves and review the fact that circular helices are regular curves in space with constant curvature, and non-zero constant torsion. Generalized and circular helices in three-space have nice properties, that are easily derived by considering a special moving frame based on the Darboux vector. The equations of motion of this moving frame are similar to the wellknown Frenet-Serret frame associated with regular space curves.

2.3.1

The Darboux vector

Let α : I → R3 be a regular curve, parametrized by arc length, and let {T, N, B} be its standard Frenet-Serret frame. Recall that this moving frame satisfies the equations T0 = N 0 = −κ T B0 =

κN −τ B τN

(2.13)

40

2. Differential geometry of curves

Here κ and τ are the curvature and torsion of the curve α, and they are the differential invariants of α. Let A be the skew-symmetric Frenet-Serret matrix 

 0 κ 0   A = −κ 0 −τ  . 0 τ 0 The vector (τ, 0, −κ)T , corresponding to τ T − κB, spans the kernel of A. Normalization gives the unit vector V = τ T − κ B,

(2.14)

√ with β = κ2 + τ 2 , κ = κ/β and τ = τ /β. Note that κ and τ only depend on the ratio τ /κ. Since V is orthogonal to the normal N , we introduce the orthonormal frame {V, N, W }, with W = V × N = κ T + τ B.

(2.15)

A straightforward calculation shows that T = τ V + κ W and B = −κ V + τ W.

(2.16)

The vector Ω = βV is the Darboux vector, so Ω = τ T −κ B, cf [65]. It satisfies the identities T 0 = T × Ω, N 0 = N × Ω and B 0 = B × Ω. These identities are easily derived from the Frenet-Serret equations. Lemma 2.3.1. The system {N, W, V } is a positively oriented orthonormal moving frame, i.e., N × W = V . The motion of this frame is determined by N0

−β W

=

W0 = β N V0

=

where γ=

−γ V γW

(τ /κ)0 τ0 = . 1 + (τ /κ)2 κ

(2.17)

2.3. Space Curves and Helices

41

Proof. Since V and N are orthogonal unit vectors, and W = V × N , it follows that the system {N, W, V } is a positively oriented orthonormal frame. The third identity of (2.17) is a straightforward consequence of the second Frenet-Serret identity in (2.13). Using τ T 0 − κ B 0 = 0 we see that V 0 = τ 0 T − κ0 B = τ 0 (τ V + κ W ) − κ0 (−κ V + τ W ) = (τ 0 τ + κ0 κ) V + (τ 0 κ − κ0 τ ) W κτ 0 − κ0 τ W κ2 + τ 2 = γW.

=

The fourth equality in the latter derivation follows from κ2 + τ 2 = 1, so κ κ0 + τ τ 0 = 0. The expression for W 0 follows by differentiating both sides of the identity W = V × N : W0 = V 0 × N + V × N0 = γ W × N − βV × W = −γ V + β N. τ0 follows from κ d τ κτ 0 − κ0 τ √ τ0 = =κ 3 = κ γ. ds κ2 + τ 2 (κ2 + τ 2 ) 2

Finally, the equality γ =

2.3.2

Generalized helices

As a first illustration of the versatility of the frame {N, W, V } in our further study of helices we consider generalized helices, which are space curves with non-zero curvature for which the ratio of torsion and curvature is constant. We first derive an expression for the derivative of this ratio. Lemma 2.3.2. Let α : I → R3 is a space curve parametrized by arc length, then the derivative of its torsion to curvature ratio is given by  τ 0 Det(α00 , α000 , α(4) ) =− . κ κ5

42

2. Differential geometry of curves

In particular, α is a generalized helix iff Det(α00 , α000 , α(4) ) = 0. Proof. Since α0 = T , α00 = κ N and α000 = κ0 N + κ N 0 = κ0 N − βκ W , we see that α00 × α000 = −βκ2 V . Therefore, Det(α00 , α000 , α(4) ) = hα00 × α000 , α(4) i = −βκ2 hV, α(4) i. Furthermore, hV, α000 i = 0, so hV, α(4) i = − hV 0 , α000 i = − hγ W, κ0 N − βκ W i = βκγ. (τ /κ)0 , the claim fol1 + (τ /κ)2

Therefore, Det(α00 , α000 , α(4) ) = −β 2 κ3 γ. Since γ = lows.

Remark 2.3.3. Since γ = 0, the Frenet-Serret equations (2.17) imply that V is constant for generalized helices. This can also be concluded directly from 1 1 the observation that κ = p and τ = p are constant for 2 1 + (τ /κ) 1 + (κ/τ )2 a generalized helix. We show that a generalized helix c lies on a generalized cylinder. To this end we introduce the function ϕ, which is the primitive function of β with Rsp Rs ϕ(0) = 0, i.e., ϕ(s) = 0 β(u) du = 0 κ(u)2 + τ (u)2 du, and the functions C and S, given by Z C(s) =

s

Z cos ϕ(u) du and S(s) =

0

s

sin ϕ(u) du.

(2.18)

0

Lemma 2.3.4. Let α : I → R3 be a generalized helix, parametrized by arc length. Then the normalized Darboux-vector V0 is constant, and α lies on a generalized cylinder with ruling lines parallel to the normalized Darboux vector V0 . More precisely, α(s) = s τ V0 + γ(s), where γ is the curve in the plane perpendicular to V0 through α(0), given by
γ(s) = α(0) + κ(s) C(s) W0 + S(s) N0 .

2.3. Space Curves and Helices

43

Proof. Let {N, W, V } be the moving frame of α. Since the ratio of torsion and curvature is constant, the function γ, introduced in Lemma 2.3.1, is zero. Therefore, V is a constant vector, say V = V0 , and the vectors W and N satisfy the set of differential equations N 0 = −β W, W 0 = β N.

(2.19)

Then the system (2.19) has solution W (s) = cos ϕ(s) W0 + sin ϕ(s) N0 N (s) = − sin ϕ(s) W0 + cos ϕ(s) N0 .

(2.20)

Since α0 (s) = T (s) = τ V0 + κ W (s), it follows that α(s) = sτ V0 + γ(s), where γ is the curve in the plane through p0 = α(0), perpendicular to V0 , given by Z s γ(s) = α(0) + κ W (u) du = α(0) + κ (C(s) W0 + S(s) N0 ) . 0

Therefore, α is a curve on the generalized cylinder with base curve γ and axis direction V0 . Lemma 2.3.4 yields an expression for the generalized helix in terms of the frame {T0 , V0 , T0 × V0 }, which will be useful in later applications. Note, however, that this frame is not necessarily orthogonal. Corollary 2.3.5. Let α : I → R3 be a generalized helix, parametrized by arc length. Then α(s) = γ(s) + s τ V0 , where γ is a curve given by γ(s) = α(0) + C(s) (T0 − τ V0 ) + S(s) T0 × V0 . The curve γ lies in the plane through α(0) perpendicular to V0 . Furthermore, the unit tangent vector of the generalized helix is given by T (s) = T0 + (cos ϕ(s) − 1) (T0 − τ V0 ) + sin ϕ(s) T0 × V0 .

44

2. Differential geometry of curves

Proof. Since κ W0 = T0 − τ V0 and κ N0 = T0 × V0 , the expression for α(s) follows from the identity in Lemma 2.3.4. Since hT0 , V0 i = τ , we see that hγ(s) − α(0), V0 i = 0, so the curve γ lies in the plane through α(0), perpendicular to V0 . The expression for the tangent vector follows from C 0 (s) = cos ϕ(s) and S 0 (s) = sin ϕ(s). Remark 2.3.6. Note that frame {V0 , T0 − τ V0 , T0 × V0 } is orthogonal, though not necessarily orthonormal. In fact, T0 − τ V0 = κ W0 and T0 × V0 = κ N0 . Circular helices. Consider a generalized helix with constant curvature κ and constant torsion τ . This special case of generalized helix is called a circular helix, since it lies on the circular cylinder. In Chapter 4 we consider approximation of space curves with circular helices hence we present them in greater detail in the same chapter.

Chapter 3

Approximation by conic splines

3.1

Introduction

In the field of computer aided geometric design, one of the central topics is the approximation of complex objects with simpler ones. An important part of this field concerns the approximation of plane curves and the asymptotic analysis of the rate of convergence of approximation schemes with respect to different metrics, the metrics used in this chapter being the Hausdorff metric and symmetric difference distance. Various error bounds and convergence rates have been obtained for several types of (low-degree) approximation primitives. For the approximation of plane convex curves by polygons with n edges, the order of convergence is O(n−2 ) for several metrics, including the Hausdorff metric [42, 74, 75, 82]. When approximating a tangent continuous conic spline, the order of convergence, for a strictly convex curve, is O(n−5 ), where n is the number of elements of the conic spline, with respect to the Hausdorff distance metric [104]. Ludwig [73] considers optimal parabolic spline approximation of strictly convex curves having monotone affine curvature with respect to the symmetric difference metric. For the approximation of a convex curve by a piecewise cubic curve, both curves being tangent and having the same Euclidean curvature at interpolation points (knots), the order of approximation is O(h6 ), where h is the maximum distance between adjacent knots [25]. As expected, the approximation order increases along with the degree of the approximating (piecewise-) polynomial curve. As approximants, conic splines represent a good compromise between flexibility and modeling power. They have a great potential as intermediate representation for robust computation with curved objects. Some applications that come to mind are the implicitization of parametric curves (see

46

3. Approximation by conic splines

works on approximate implicitization [31, 33]), the intersection of high-degree curves, the building of arrangements of algebraic curves (efficient solutions are known for sweeping arrangements of conic arcs [5]) and the computation of the Voronoi diagram of curved objects (the case of ellipses has been investigated in [36, 37]). While these applications necessitate a tight hold on the error of approximation, no previous work provides a sharp asymptotic error bound (i.e., the constant of the leading term in the asymptotic expansion) for the Hausdorff metric when the interpolant is curved. In this chapter, we study the optimal approximation of a sufficiently smooth curve with non-vanishing curvature by a tangent continuous interpolating conic spline, which is an optimal approximant with respect to Hausdorff distance. We present the first sharp asymptotic bound on the approximation error (and, consequently, a sharp bound on the complexity of the approximation) for both parabolic and conic interpolating splines. Our experiments corroborate this sharp bound: the complexity of the approximating splines we algorithmically construct exactly matches the complexity predicted by our complexity bound. Furthermore, we also consider approximation with conic splines with respect to the symmetric difference distance. Recall that the symmetric difference distance of two closed curves is the total area of the settheoretic symmetric difference of the regions enclosed by these curves. The symmetric difference distance of two curves that are not closed, but have common endpoints, is the total area of the regions enclosed by the two curves. See Figure 3.1.

Figure 3.1: The symmetric difference of the two curves is the total area of the shaded regions.

3.1. Introduction

47

We present the first sharp asymptotic bound on the approximation error in terms of the symmetric difference distance (and, consequently a sharp bound on the complexity of the approximation). We implemented the approximation algorithm, and our experiments corroborate this sharp bound for optimal parabolic spline approximation and near optimal conic spline approximation. This near-optimal approximation scheme will be explained later in this chapter.

3.1.1

Related work

Fejes T´ oth [42] considers the problem of approximating a convex C 2 -curve C in the plane by an inscribed n-gon. Fejes T´oth proves that, with regard to the Hausdorff distance, the optimal n-gon Pn satisfies 1 δH (C, Pn ) = 8

Z

l 1/2

κ 0

2 1 1 (s) ds + O . n2 n4

(3.1)

Here δH (A, B) is the Hausdorff distance between two sets A and B, l is the length of the curve, s its arc length parameter, and κ(s) its curvature. An asymptotic expression for the complexity of the piecewise linear spline can easily be deduced: the number of elements is c ε−1/2 (1 + O(ε)), where Rl 1 c = 2√ κ(s)1/2 ds. Ludwig [75] extends this result by deriving the 2 s=0 second term in the asymptotic expansion (3.1). If one considers the symmetric difference metric δS instead, one can prove that δS (C, Pn ) = R 3 l 1/3 1 (s)ds n12 + O( n14 ) [82]. Again, this asymptotic expression can 12 0κ be refined, cf. [74]. Schaback [104] introduces a scheme that yields an interpolating conic spline with tangent continuity for a curve with non-vanishing curvature, and achieves an approximation order of O(h5 ), where h is the maximal distance of adjacent data points on the curve. A conic spline consists of pieces of conics, in principle of varying type. This result implies that approximating such a curve by a curvature continuous conic spline to within Hausdorff distance ε requires O(ε−1/5 ) elements. However, the value of the constant implicit in this asymptotic expression of the complexity is not known. Ludwig [73] considers the problem of optimally approximating a convex C 4 -curve with respect to the symmetric difference metric by a tangent continuous parabolic spline Qn

48

3. Approximation by conic splines

R 5 λ 1/5 1 1 with n knots. She proves that δS (C, Qn ) = 240 κ (s)ds + o( n14 ), 0 n4 Rl where λ = 0 κ1/3 (s)ds is the affine length of the convex curve C. These problems fall in the context of geometric Hermite interpolation, in which approximation problems for curves are treated independent of their specific parameterization. The seminal paper by De Boor, H¨ollig and Sabin [25] fits in this context. Floater [44] gives a method that, for any conic arc and any odd integer n, yields a geometric Hermite interpolant with 2n contacts, counted with multiplicity. This scheme gives a Gn−1 -spline, and has approximation order O(h2n ), where h is the length of the conic arc. Ahn [1] gives a necessary and sufficient condition for the conic section to be the optimal approximation of the given planar curve with respect to the maximum norm used by Floater. This characterization does not however yield the best conic approximation obtained by the direct minimization of the Hausdorff distance. Degen [28] presents an overview of geometric Hermite interpolation, also emphasizing differential geometry aspects. The problem of approximating a planar curve by a conic spline has also been studied from a more practical standpoint. Farin [39] presents a global method and discusses at length how curvature continuity can be achieved between conic segments. Pottmann [98] presents a local scheme, still achieving curvature continuity. Yang [115] constructs a curvature continuous conic spline by first fitting a tangent continuous conic spline to a point set and fairing the resulting curve. Li et al. [68] show how to divide the initial curve into simple segments which can be efficiently approximated with rational quadratic B´ezier curves. These methods have many limitations, among which the dependence on the specific parameterization of the curve, the large number of conic segments produced or the lack of accuracy and absence of control of the error.

3.1.2

Results of this chapter

Complexity of conic approximants. Hausdorff metric case. We show that the complexity – the number of elements – of an optimal parabolic spline approximating the curve to within Hausdorff distance ε is of the form c1 ε−1/4 + O(1), where we express the

3.1. Introduction

49

value of the constant c1 in terms of the Euclidean and affine curvatures (see Theorem 3.6.1, Section 3.6). An optimal conic spline approximates the curve to fifth order, so its complexity is of the form c2 ε−1/5 + O(1). Also in this case the constant c2 is expressed in the Euclidean and affine curvature. These bounds are obtained by first deriving an expression for the Hausdorff distance of a conic arc that is tangent to a (sufficiently short) curve at its endpoints, and minimizes the Hausdorff distance among all such bitangent conics. Applying well-known methods like those of [25] it follows that this Hausdorff distance is of fifth order in the length of the curve, and of fourth order if the conic is a parabola. However, we derive explicit constants in these asymptotic expansions in terms of the Euclidean and affine curvatures of the curve. Symmetric difference distance case. We consider the problem of optimally approximating a convex curve with respect to the symmetric difference distance by parabolic and conic splines. Our derivation follows in the same lines as in the Hausdorff metric case. We show that the complexity – the number of elements – of an optimal parabolic spline approximating the curve to within symmetric difference distance ε, is given by c3 ε−1/4 +O(1), where c3 depends on the affine curvature and affine arc length of the given curve. An optimal conic spline approximates the curve to within fifth order, with respect to the symmetric difference distance and the constant of approximation in this case is given by c4 ε−1/5 + O(1), where c4 depends on the derivative of affine curvature and affine arc length. Our method for computing the asymptotic error bound of an optimal parabolic spline are different from those of [73], and allow us to determine the optimal asymptotic error bound in case of general conic splines as well. Obviously, our result for parabolic splines match those of Ludwig [73]. Algorithmic issues. Hausdorff metric case. For curves with monotone affine curvature, called affine spirals, we consider conic arcs tangent to the curve at its endpoints, and show that among such bitangent conic arcs there is a unique one minimizing the Hausdorff distance. This optimal bitangent conic arc Copt intersects the curve at its endpoints and at one interior point, but nowhere else. If α :

50

3. Approximation by conic splines

I → R2 is an affine spiral, its displacement function d : I → R measures the signed distance between the affine spiral and the optimal bitangent conic along the normal lines of the spiral. The displacement function d has an equioscillation property: there are two parameter values u+ , u− ∈ I such that d(u+ ) = −d(u− ) = δH (α, Copt ) and the points α(u− ) and α(u+ ) are separated by the interior point of intersection of α and Copt . Furthermore, the Hausdorff distance between a section of an affine spiral and its optimal approximating bitangent conic arc is a monotone function of the arc length of the spiral section. This useful property gives rise to a bisection based algorithm for the computation of an optimal interpolating tangent continuous conic spline. The scheme reproduces conics. We implemented such an algorithm, and compare its theoretical complexity with the actual number of elements in an optimal approximating parabolic or conic spline.

Symmetric difference distance. We conjecture that there is a unique bitangent conic which minimizes the symmetric difference distance to a smooth affine spiral. This property would be the equivalent of the unicity of the bitangent conic minimizing the Hausdorff distance to the affine spiral, and would be of paramount importance for the design of an algorithm computing the optimal approximant. However, there is another conic spline achieving the same asymptotic bound on the symmetric difference metric, that exhibits these features. More precisely, we introduce the equisymmetric bitangent conic of an affine spiral, which is uniquely determined by the fact that the two moons it forms with the affine spiral have equal area. An equisymmetric conic spline is a tangent continuous conic spline all of whose elements are equisymmetric bitangent conics of the affine spiral. The equisymmetric conic spline, has the property that all moons formed by this spline and the affine spiral have equal area, and we denote by Ces the spline that minimizes the symmetric difference distance to the spiral among all equisymmetric conic splines. Furthermore, the complexity of this equisymmetric conic spline as a function of the symmetric difference distance to the affine spiral is asymptotically equal to the complexity of the optimal conic spline with respect to this error metric. Therefore, we call the computation of the optimal equisymmetric conic spline a near-optimal approximation scheme. We implement the near-optimal

3.1. Introduction

51

approximation scheme for affine spirals. The symmetric difference distance between a section of an affine spiral and its equisymmetric bitangent conic arc is a monotone function of the arc length of the spiral section. This useful property gives rise to an efficient, bisection based algorithm computing the equisymmetric conic spline. For several curves we compare the theoretical complexity of an optimal conic spline with the computed number of elements in an equisymmetric tangent continuous conic spline, and find that these numbers match almost exactly.

3.1.3

Overview

Section 3.2 reviews the result about conics being only curves with constant affine curvature, when they are represented by an implicit equation. Furthermore in this section we review the result that the affine curvature of a curve changes under linear transformation A and is a function of determinant of A. If the determinant of A is 1, then the affine curvature of a curve remains unchanged. We also review the result that given a point on a plane curve, with non-zero curvature, there is a unique conic with five-point contact with the curve at that point and we call the corresponding conic as the osculating conic. Moreover, we derive the conditions for a general conic being determined by five points. Section 3.3 introduces affine spirals, a class of curves which have unique optimal bitangent conic. We show that the displacement function, which measures the distance of the curve to its offset curve along its normals, has an equioscillation property in the sense that it has extremes at exactly two points on the curve. Furthermore, the Hausdorff distance between an arc of an affine spiral and its optimal bitangent conic arc is increasing in the length of this arc. This useful property gives rise to a bisection algorithm for the computation of a conic spline approximating a smooth curve with a minimal number of elements. Section 3.7 presents the output of the algorithm for a collection of examples with respect to the Hausdorff metric. Section 3.4 presents the global properties of equisymmetric bitangent conic arcs of an affine spiral, like the monotonicity of the symmetric difference distance to the spiral as a function of arc length. The main result of Section 3.5 is a relation between the affine curvatures of a curve and a bitangent offset curve. We use this result in Section 3.6 to derive an expression for the complexity of optimal

52

3. Approximation by conic splines

parabolic and conic splines approximating a regular curve with respect to the Hausdorff metric. We do so by deriving a bound on the Hausdorff distance between an affine spiral arc and its optimal bitangent conic. Furthermore, in Section 3.6 we derive the complexity of an optimal conic spline with respect to the symmetric difference distance, and show that the optimal equisymmetric conic spline has the same asymptotic complexity. In Section 3.7 we present another bisection algorithm, together with experimental results corroborating our theoretical complexity bounds for the symmetric difference distance. We conclude with topics for future work in Section 3.8.

3.2 3.2.1

Geometry of Conics Conics have constant affine curvature

Solving the differential equation (2.5) shows that a curve of constant affine curvature is a conic arc. More precisely, a curve with constant affine curvature is a hyperbolic, parabolic, or elliptic arc iff its affine curvature is negative, zero, or positive, respectively. We now give expressions for the (constant) affine curvature of conics defined by an implicit quadratic equation. Proposition 3.2.1 ([92], Theorem 6.4). The affine curvature of the conic defined by the quadratic equation ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 is given by k = S T −2/3 , where a b S= , b c

a b d T = b c e . d e f

The next result relates the affine curvatures of a regular curve in the plane and its image under linear transformations. Lemma 3.2.2. Let α be the image of a regular planar curve β under a linear transformation x 7→ Ax. The affine curvatures kα and kβ of the curves α and β are related by kα = (det A)−2/3 kβ .

3.2. Geometry of Conics

53

Figure 3.2: The curve and its osculating conic (dashed). The affine curvature is increasing in the left picture, and decreasing in the right picture.

Proof. Assume that β is parameterized by affine arc length. Since α(u) = Aβ(u), it follows that the function ϕ, defined by (2.3), satisfies ϕ = ˙ Aβ] ¨ 1/3 = (det A)1/3 [β, ˙ β] ¨ 1/3 = (det A)1/3 . According to Proposi[Aβ, ... ¨ Aβ ] = tion 2.2.3 the affine curvature of α is given by kα = (det A)−5/3 [Aβ, (det A)−2/3 kβ .

3.2.2

Osculating conic at non-sextactic points

At a point of non-vanishing Euclidean curvature there is a unique conic, called the osculating conic, having fourth order contact with the curve at that point (or, in other words, having five coinciding points of intersection with the curve). The affine curvature of this conic is equal to the affine curvature of the curve at the point of contact. Moreover, the contact is of order five if the affine curvature has vanishing derivative at the point of contact. In that case the point of contact is a sextactic point. Again, see [13] for further details. At non-sextactic points the curve and its osculating conic cross (see also Figure 3.2): Corollary 3.2.3. At a non-sextactic point a curve crosses its osculating conic from right to left if its affine curvature is locally increasing at that point, and from left to right if the affine curvature is locally decreasing.

3.2.3

The five-point conic

To derive error bounds for an optimal approximating conic we use the property that the approximating conic depends smoothly on the points of intersection with the curve. More precisely, let α : I → R2 be a regular curve without sextactic points, and let si , 1 ≤ i ≤ 5, be points on I, not necessarily distinct. The unique conic passing through the points α(si ) is denoted by Cs , with

54

3. Approximation by conic splines

s = (s1 , s2 , s3 , s4 , s5 ). If one or more of the points coincide, the conic has contact with the curve of order corresponding to the multiplicity of the point. For instance, if s1 = s2 6= si , i ≥ 3, then Cs has first order contact with (is tangent to) the curve at α(s1 ). If si 6= sj , for i 6= j, then the implicit quadratic equation of this conic can be obtained as follows. Let the Veronese mapping Ψ : R2 → R6 be defined by Ψ(x) = (x21 , x1 x2 , x22 , x1 , x2 , 1), x = (x1 , x2 ), then the equation of the conic Cs is f (x, s) = 0, with
f (x, s) = det Ψ(x), Ψ(α(s1 )), Ψ(α(s2 )), Ψ(α(s3 )), Ψ(α(s4 )), Ψ(α(s5 )) . (3.2) However, if si = sj for i 6= j, then f (x, s) = 0. We obtain a quadratic equation of the conic Cs by (formally) dividing f (x, s) by si − sj . More precisely: Lemma 3.2.4. If α is a C m -curve, m ≥ 4, then the conic Cs has a quadratic equation with coefficients that are C m−4 -functions of s = (s1 , s2 , s3 , s4 , s5 ) ∈ R5 . Proof. Put ψ(s) = Ψ(α(s)). The Newton development of ψ in terms of the divided differences of ψ up to order four associated with the points s1 , . . . , s5 P Q is given by ψ(sk ) = ψ(s1 ) + ki=2 i−1 j=1 (sk − sj ) [s1 , . . . , sk ] ψ, for 2 ≤ k ≤ 5. See Appendix A.1. Plugging these identities into (3.2), we see that f (x, s) = Q (sk − sj ) F (x, s), with 1≤j 0 let α% be the sub-arc between α(0) and α(%), and let β% be the (unique) conic arc tangent to α% at its endpoints, and minimizing the Hausdorff distance between α% and the conic arcs tangent to α% at its endpoints. Then the Hausdorff distance between α% and β% is a monotonically increasing function of %, for % ≥ 0. Proof. First we introduce some notation. The unique interior point of intersection of α% and β% occurs at u = u(%) ∈ I. The sub-arcs of α% and β% between α(0) and α(u(%)) are denoted by α%− and β%− , respectively. The complementary sub-arcs of α% and β% are denoted by α%+ and β%+ , respectively. According to the Equioscillation Property (Corollary 3.3.3) the Hausdorff distance between α% and β% is equal to the Hausdorff distances between α%± and β%± , and is attained as the distance between points a± (%) on α%± and b± (%) on β%± , i.e., δH (α% , β% ) = dist(a− (%), b− (%)) = dist(a+ (%), b+ (%)). The complete conic containing β% will be denoted by K% . We will repeatedly use the following consequence of Bezout’s theorem: Intersection Property: For 0 < %1 < %2 , the conics K%1 and K%2 have at most two points of intersection (possibly counted with multiplicity) different from α(0). Let %1 , %2 ∈ I, with 0 < %1 < %2 . The regions bounded by α%±2 and β%±2 are denoted by R± . Since K%1 is either compact or unbounded, and not disjoint from the boundary of R+ , it intersects this boundary in an even number of

3.4. Near optimal conic approximation of affine spiral arcs

59

points (counted with multiplicity). Our strategy is to prove that β%−1 lies inside R− , or that β%+1 lies inside R+ . In the former case, we see that δH (α%1 , β%1 ) = dist(a− (%1 ), b− (%1 )) < dist(a− (%2 ), b− (%2 )) = δH (α%2 , β%2 ), whereas in the latter case δH (α%1 , β%1 ) = dist(a+ (%1 ), b+ (%1 )) < dist(a+ (%2 ), b+ (%2 )) = δH (α%2 , β%2 ). We distinguish two cases, depending on the order of u(%1 ) and u(%2 ). Case 1: u(%1 ) > u(%2 ). Note that the conic K%1 is tangent to α at α(%1 ), a point contained in α%2 . Therefore, in this case K%1 intersects α%+2 in an odd number of points, namely, once at the point α(u(%1 )) and twice at the point of tangency α(%1 ). β%+2 , the other part of boundary of R+ , in an odd number of points. By the Intersection Property, this odd number is equal to one. Since both endpoints of β%1 lie on the same side of β%2 , this point of intersection does not lie on β%1 . In other words, the interior of β%+1 lies inside the region R+ . Case 2: u(%1 ) < u(%2 ). In this case K%1 does not cross α%+2 , since it intersects α%+2 in two coinciding points at the tangency α(%1 ), but at no other point. Therefore, K%1 intersects β%+2 , the other part of the boundary of R+ , in at least two points (at least one entrance and at least one exit point). By the Intersection Property, apart from α(0), these are the only points in which K%1 and K%2 intersect. Therefore, β%−1 intersects neither β%−2 nor α%−2 in an interior point. In other words, the interior of β%−1 lies inside the region R− . Remark 3.3.5. A similar monotonicity property holds for the Hausdorff distance between an affine spiral and a bitangent parabolic arc. The proof is omitted, since it is straightforward, and along the same lines as the proof of Proposition 3.3.4.

3.4

Near optimal conic approximation of affine spiral arcs

The main result of this section concerns the equisymmetry property and the monotonicity property of the symmetric difference distance. Both properties are global, since the affine spiral is not necessarily short.

60

3. Approximation by conic splines

3.4.1

Uniqueness of equisymmetric conic

In this section we will concern ourselves with the global result, that given an affine spiral arc γ : [u0 , u1 ] → R2 , there is a unique bitangent conic Cσ , in the one parameter family of bitangent conics, such that the areas of the two moons formed by γ and Cσ are equal. Here Cσ is conic arc tangent to γ at γ(u0 ) and γ(u1 ), and intersecting it at an interior point γ(σ). Moreover we show that with respect to the equisymmetry property, the symmetric difference distance is an increasing function of the arc length of the given affine spiral curve γ. Even though we do not show the existence of a unique conic, which minimizes the symmetric difference distance between the curve γ and itself, in the next section we prove that asymptotic error expressions for the symmetric difference distance of a conic minimizing area and an equisymmetric conic are the same upto terms of order 6, in the length of a very short arc. Thus we that the approximation with respect to the equisymmetric conic is very close to the optimal conic approximation. Before we state the main result for this section, let us make some notations clear. Let Cσ be the bitangent conic to γ, intersecting it at an interior point γ(σ). Let γσ− be the arc of γ, defined over [u0 , σ] and γσ+ be the arc defined over [σ, u1 ]. Similarly Cσ− is part of the conic arc in the interval [u0 , σ] and Cσ+ is part of the conic arc in the interval [σ, u1 ]. Let A− (σ) be the area between γσ− and Cσ− and let A+ (σ) be the area between γσ+ and Cσ+ . Therefore the symmetric difference between γ and Cσ is given by δS (γ, Cσ ) = A− (σ) + A+ (σ). Figure 3.5, makes these notations clear.

Figure 3.5: Notations for symmetric difference distance for bitangent conics

3.4. Near optimal conic approximation of affine spiral arcs

61

Lemma 3.4.1 (Unicity of the equisymmetric conic). Given an affine spiral arc γ : [u0 , u1 ] → R2 there is a one parameter family of bitangent conics which intersects γ at γ(σ) as σ varies in the interval [u0 , u1 ]. In this family of bitangent conics there is a unique equisymmetric conic Cσ∗ (i.e., A( σ ∗ ) = A+ (σ ∗ )). Proof. We are given that γ : [u0 , u1 ] → R2 is an affine spiral curve, thus from Proposition 3.3.2, we have that the family of bitangent conics are lying side by side, as shown in the figure 3.3. Therefore it follows that the function A− is strictly increasing as σ varies in the interval [u0 , u1 ], moreover A− (u0 ) = 0. A+ is strictly decreasing as σ varies in the interval [u0 , u1 ], moreover A+ (u1 ) = 0. Therefore, we conclude that there exists a unique σ ∗ such that A− (σ ∗ ) = A+ (σ ∗ ).

3.4.2

Monotonicity of the equisymmetric distance

If one endpoint of the affine spiral moves along the curve γ, the symmetric difference between the affine spiral and its equisymmetric conic arc is monotone in the affine arc length of the affine spiral. This result shows that an adaptive method can be used for the computation of a near optimal approximating conic arc. We use this property for the implementation of the algorithm presented in Section 5. Proposition 3.4.2 (Monotonicity of symmetric difference along affine spiral arcs). Let γ : I → R2 be an affine spiral arc, where I is an open interval containing 0. For % > 0 let γ% be the sub-arc between γ(0) and γ(%), and let β% be the (unique) equisymmetric conic arc tangent to γ% at its endpoints. Then the symmetric difference between γ% and β% is a monotonically increasing function of %, for % ≥ 0. The proof of monotonicity of equisymmetric distance proceeds similar to the monotonicity proof of the Hausdorff distance. Therefore, we omit the proof in this section and refer to Section 3.3.4 for the details of the proof.

62

3.5

3. Approximation by conic splines

Affine curvature of offset curves

The main result of this section is a relation between the affine curvatures of a curve and a bitangent offset curve. Let α : I → R2 be a regular curve parameterized by affine arc length, with affine arc length parameter u ∈ I. Here I is an open interval, containing 0. We consider offset curves tangent to α at α(0) and α(%). The affine curvature of such a curve is related to the affine curvature k of α, as indicated in the first part of the following Lemma. In the second part, an analogous result relates these curvatures when there is an additional point of intersection at α(σ). Lemma 3.5.1 (Affine curvature of offset curves). Let α be a C m -regular curve. 1. Let β : I × I → R2 be a C n -function, such that, β(·, %) is a curve tangent to α at α(0) and α(%), for % ∈ I. If m, n ≥ 5, there are C l -functions P, Q : I × I → R, with l = min(m − 5, n − 4), such that
β(u, %) = α(u) + d(u, %) P (u, %)t(u) + Q(u, %)n(u) , (3.3) where d(u, %) = u2 (u − %)2 . Here t(u) and n(u) are the affine tangent and the affine normal of α, respectively. Furthermore, the affine curvature kβ (u, %) of β(·, %) at 0 ≤ u ≤ % is given by kβ (u, %) = k(0) + 8 Q(0, 0) + O(%).

(3.4)

2. Let β : I × I × I → R2 be a C n -function, such that, β(·, σ, %) is a curve tangent to α at α(0) and α(%) and intersecting α at α(σ), for σ, % ∈ I and 0 ≤ σ ≤ %. If m, n ≥ 6, and, moreover, β also intersects α at α(σ), with 0 ≤ σ ≤ %, then there are C l -functions P, Q : I ×I → R, with l = min(m−6, n−5), such that
β(u, σ, %) = α(u) + d(u, σ, %) P (u, σ, %)t(u) + Q(u, σ, %)n(u) , (3.5) where d(u, σ, %) = u2 (u − %)2 (u − σ). Furthermore, the affine curvature kβ (u, σ, %) of β(·, σ, %) at 0 ≤ u ≤ % is given by kβ (u, σ, %) = k(0) + k 0 (0) u + 8 (5u − σ − 2%) Q(0, 0, 0) + O(%2 ).

3.5. Affine curvature of offset curves

63

Proof. 1. If α is C m , then the functions (u, %) 7→ [β(u, %) − α(u), n(u)] and (u, %) 7→ [β(u, %) − α(u), t(u)] are of class C min(m−1,n) . For fixed %, these functions have double zeros at u = 0 and u = %. The Division Property, cf. Appendix A.1, Lemma A.1.2, guarantees the existence of C min(m−5,n−4) functions P and Q satisfying [β(u, %) − α(u), n(u)] = d(u, %) P (u, %) and [β(u, %) − α(u), t(u)] = d(u, %) Q(u, %). In other words, P and Q satisfy identity (3.3). According to Proposition 2.2.3 the affine curvature of the curve β(·, %) is a C n−4 -function given by kβ =

1 1 [βuu , βuuu ] + 4 (ϕuu ϕ − 3ϕ2u ), 5 ϕ ϕ

(3.6)

ϕ = [βu , βuu ]1/3 . In (3.6), the functions kβ , ϕ, β, and their partial derivatives are evaluated at (u, %). Since n ≥ 5, and 0 ≤ u ≤ %, it follows that kβ (u, %) = kβ (u, 0) + O(%). So, to prove (3.4), it is sufficient to determine β(u, 0) and its derivatives up to order four. Writing β0 (u) = β(u, 0), we see that β0 (u) = α(u) + f (u) (P0 t(u) + Q0 n(u)) + O(u5 ), where f (u) = u4 , P0 = P (0, 0) and Q0 = Q(0, 0). In view of the affine Frenet-Serret identities (2.7) we get β00 = (1 + f 0 P0 ) t + f 0 Q0 n + O(u4 ), β000 = f 00 P0 t + (1 + f 00 Q0 ) n + O(u3 ), β0000

000

000

(3.7) 2

= (−k + f P0 ) t + f Q0 n + O(u ).

Here the functions β0 , f , t, n and k, as well as their derivatives, are evaluated 1

at u. Since ϕ(u, 0) = [β00 (u), β000 (u)] 3 , we use the first two identities of (3.7) to derive ϕ(u, 0) = 1 + 13 f 00 (u) Q0 + O(u3 ) = 1 + 4 u2 Q0 + O(u3 ). Similarly, using the second and third identity of (3.7) we get [β000 (u), β0000 (u)] = k(u) + O(u) = k(0) + 8 Q0 + O(u). Identity (3.4) is obtained by plugging these expressions into (3.6).

64

3. Approximation by conic splines

2. Now we turn to the case where the offset curve not only is tangent to α at its endpoints, but also has an additional point of intersection at α(σ). The existence of functions P and Q satisfying (3.5) is proven as in Part 1, using the Division Property. Again the affine curvature of β is given by (3.6), where this time the functions kβ , ϕ, β, and their partial derivatives are evaluated at (u, σ, %). In (3.5) we have d(u, σ, %) = u5 − (2% + σ) u4 + O(%2 + σ 2 ), P = P0 + O(u), and Q = Q0 + O(u). Focusing on the essential terms only, we rewrite (3.5) as: β = α+(u5 −(2%+σ) u4 ) (P0 t+Q0 n)+O(u6 )+O((%+σ)u5 )+O(%2 +σ 2 ). (3.8) Here α, t and n are evaluated at u, and β at (u, σ, %). For a smoother presentation, we introduce the following terminology. The class Oi (u, σ, %), 0 ≤ i ≤ 4, consists of all C m−i -functions of the form O(u6−i )+O((%+σ)u5−i )+O(%2 +σ 2 ). Using this notation we rewrite (3.8) as β = α + f (P0 t + Q0 n) + O0 (u, σ, %). where f (u, σ, %) = u5 − (2% + σ)u4 . If g ∈ Oi (u, σ, %), then gu ∈ Oi+1 (u, σ, %), for 1 ≤ i ≤ 4. Therefore, we get, as in (3.7): βu = (1 + fu P0 ) t + fu Q0 n + O1 (u, σ, %), βuu = fuu P0 t + (1 + fuu Q0 ) n + O2 (u, σ, %),

(3.9)

βuuu = (−k + fuuu P0 ) t + fuuu Q0 n + O3 (u, σ, %). 1

Since ϕ = [βu , βuu ] 3 , we use the first two identities of (3.9) to derive ϕ = 1 + 13 fuu Q0 + O2 (u, σ, %), so ϕ = 1 + O3 (u, σ, %), ϕ2u = O4 (u, σ, %), and ϕuu = 31 Q0 fuuuu + O4 (u, σ, %). Similarly, using the second and third identity of (3.7) we get [βuu , βuuu ] = k(u) + O4 (u, σ, %). It follows that kβ (u, σ, %) = k(u) + 31 fuuuu Q0 + O4 (u, σ, %) = k(0) + k 0 (0) u + 8 (5u − σ − 2%) Q0 + O(%2 ).

3.5. Affine curvature of offset curves

65

Note that in the last identity we used that O4 (u, σ, %) = O(u2 + σ 2 + %2 ) = O(%2 ), since 0 ≤ u, σ ≤ %. This concludes the proof of the second part. If the offset curves are bitangent conics, the affine curvature of these conics can be expressed in the Euclidean and affine curvature of the curve α at the points of intersection. Furthermore, we can determine the displacement function up to terms of order five if the conic is a parabola, and up to terms of order six in the general case. These results will enable us to determine an asymptotic expression for the Hausdorff distance between a small arc and its optimal bitangent conic. Corollary 3.5.2 (Bitangent conics). Let α be a strictly convex regular C m curve. 1. If m ≥ 8, a parabolic arc tangent to α at α(0) and α(%) has the form β(u, %) = α(u) + u2 (% − u)2 D(u, %) N (u),

(3.10)

where D is a C m−8 -function with D(0, 0) = − 81 k(0) κ(0)1/3 . Here N (u) is the Euclidean normal of α, and κ is its Euclidean curvature. 2. If m ≥ 9, a conic arc tangent to α at α(0) and α(%) and intersecting at α(σ), with 0 ≤ σ ≤ %, has the form β(u, σ, %) = α(u) + u2 (% − u)2 (u − σ) D(u, σ, %) N (u),

(3.11)

1 0 where D is a C m−9 -function with D(0, 0, 0) = − 40 k (0) κ(0)1/3 . Moreover, its affine curvature is of the form

kβ (σ, %) =

1 5

(2k(0) + k(σ) + 2k(%)) + O(%2 ).

Proof. 1. Obviously, the family of parabolic arcs can be written in the form β(u, %) = α(u) + d(u, %) N (u), provided % is sufficiently small. According to Lemma 3.2.4, β is a C m−4 -function, so d = [T, β − α] is a C m−4 -function with double zeros at u = 0 and u = %. According to Lemma 3.5.1, the parabola has a parameterization of the form (3.3), where P and Q are C m−8 -functions. Therefore, d(u, %) = u2 (u − %)2 Q(u, %) [T (u), n(u)], so β is of the form (3.10) with D = Q [T, n]. In particular, D is a C m−8 function. Comparing this expression with identity (3.3) in Lemma 3.5.1,

66

3. Approximation by conic splines

we see that D(u, %) = Q(u, %) [T (u), n(u)]. From (2.9) we conclude that D(0, 0) = κ(0)1/3 Q(0, 0). Since the affine curvature of a parabolic arc is identically zero, Part 1 of Lemma 3.5.1 yields Q(0, 0) = − 81 k(0), yielding the value for D(0, 0) stated in Part 1. 2. As in Part 1 we prove that β has a parameterization of the form (3.11), where D is a C m−9 -function. The affine curvature of a conic arc is constant, 1 0 so Part 2 of Lemma 3.5.1 yields Q(0, 0, 0) = − 40 k (0). Since also in this case 1/3 we have D(0, 0, 0) = κ(0) Q(0, 0, 0), we conclude that D(0, 0, 0) has the value stated in Part 2. Furthermore, (3.4) yields kβ = k(0) + 51 (σ + 2%) k 0 (0) + O(%2 ) =

1 5

(2k(0) + k(σ) + 2k(%)) + O(%2 ).

This concludes the proof of the second part. Remarks 3.5.3. 1. The second part of Corollary 3.5.2 can be generalized in the sense that the affine curvature of a conic intersecting a strictly convex arc at five points is equal to the average of the affine curvatures of the curve at these five points, up to quadratic terms in the affine length of the arc. The proof is similar to the one given above. 2. We conjecture that the ‘loss of differentiability’ is less than stated in Corollary 3.5.2. More precisely, we expect that D is of class C m−4 for a bitangent parabolic arc, and of class C m−5 for a bitangent conic arc.

3.6 3.6.1

Complexity of conic splines Hausdorff metric case

In this section our goal is to determine the Hausdorff distance of a conic arc of best approximation to an arc of α of Euclidean length σ > 0, that is tangent to α at its endpoints. If the conic is a parabola, these conditions uniquely determine the parabolic arc. If we approximate with a general conic, there is one degree of freedom left, which we use to minimize the Hausdorff distance between the the arc of α and the approximating conic arc β. As we have seen in Section 3.3, the optimal conic arc intersects the arc of α in an interior point.

3.6. Complexity of conic splines

67

The main result of this section gives an asymptotic bound on this Hausdorff distance. Theorem 3.6.1 (Error in parabolic and conic spline approximation). Let β be a conic arc tangent at its endpoints to an arc of a regular curve α of length σ, with non-vanishing Euclidean curvature. 1. If α is a C 8 -curve, and β is a parabolic arc, then the Hausdorff distance between these arcs has asymptotic expansion 1 128

δH (α, β) =

5/3

|k0 | κ0 σ 4 + O(σ 5 ),

(3.12)

where κ0 and k0 are the Euclidean and affine curvatures of α at one of its endpoints, respectively. 2. If α is a C 9 -curve, and β is a conic arc, then the Hausdorff distance between these arcs is minimized if the affine curvature of β is equal to the average of the affine curvatures of α at its endpoints, up to quadratic terms in the length of α. In this case this Hausdorff distance has asymptotic expansion δH (α, β) =

1√ 2000 5

|k00 | κ20 σ 5 + O(σ 6 ),

(3.13)

where κ0 is the Euclidean curvature of α at one of its endpoints, and k00 is the derivative of the affine curvature of α at one of its endpoints. Proof. 1. According to Corollary 3.5.2, the parabolic arc has a parameterization of the form (3.10). It follows from Appendix A.2, Lemma A.2.1, applied to the displacement function d(u) = u2 (% − u)2 D(u, %), cf. (3.10), that δH (α, β) =

1 16

|D(0, 0)| %4 + O(%5 ).

(3.14)

From Lemma 2.2.1, part 1, we derive 1/3

% = κ0 σ + O(σ 2 ).

(3.15)

Since D(0, 0) = − 81 k(0) κ(0)1/3 , we conclude from (3.14) and (3.15) that the Hausdorff distance satisfies (3.12). 2. Again, according to Corollary 3.5.2, cf. (3.11), a best approximating conic arc has a parameterization of the form (3.11), with D(0, 0, 0) =

68

3. Approximation by conic splines

1 0 − 40 k (0) κ(0)1/3 . Applying Appendix A.2, Lemma A.2.1 to the displacement function d(u) = u2 (u − σ) (% − u)2 D(u, σ, %), cf. (3.11), we see that

δH (α, β) =

1√ |D(0, 0, 0)| %5 50 5

+ O(%6 ),

(3.16)

where the optimal conic intersects the curve α for σ = σ(%) = 21 % + O(%2 ). Identities (3.15) and (3.16) imply that the Hausdorff distance is given by (3.13). Finally, the affine curvature of this conic is 1 5 (2k(0)

+ k( 12 % + O(%2 )) + 2k(%)) + O(%2 ) = 12 (k(0) + k(%)) + O(%2 ).

This concludes the proof of the main theorem of this section. Remark 3.6.2. It would be interesting to give a direct geometric proof of the fact that the best approximating conic has affine curvature equal to the average of the affine curvatures of α at its endpoints. The preceding result gives an asymptotic expression for the minimal number of elements of an optimal parabolic or conic spline in terms of the maximal Hausdorff distance. Corollary 3.6.3 (Complexity of parabolic and conic splines). Let α : [0, L] → R2 be a regular curve with non-vanishing Euclidean curvature of length L, parameterized by Euclidean arc length, and let κ(s) and k(s) be its Euclidean and affine curvature at α(s), respectively. 1. If α is a C 8 -curve, then the minimal number of arcs in a tangent continuous parabolic spline approximating α to within Hausdorff distance ε is Z L  N (ε) = c1 |k(s)|1/4 κ(s)5/12 ds ε−1/4 (1 + O(ε1/4 )), (3.17) 0

where c1 = 128−1/4 ≈ 0.297. 2. If α is a C 9 -curve, then the minimal number of arcs in a tangent continuous conic spline approximating α to within Hausdorff distance ε is Z L  |k 0 (s)|1/5 κ(s)2/5 ds ε−1/5 (1 + O(ε1/5 )), (3.18) N (ε) = c2 0

√ where c2 = (2000 5)−1/5 ≈ 0.186.

3.6. Complexity of conic splines

69

We only sketch the proof, and refer to the papers by McClure and Vitale [82] and Ludwig [75] for details about this proof technique in similar situations. Consider a small arc of α, centered at α(s). Let σ(s) be its Euclidean arc length. Then the Hausdorff distance between this curve and a 5/3 1 bitangent parabolic arc is 128 |k0 | κ0 σ(s)4 + O(σ(s)5 ), cf. Theorem 3.6.1. Therefore, √ 4 σ(s) = 128 |k(s)|−1/4 κ(s)−5/12 ε1/4 (1 + O(ε1/4 )). Z L 1 The first part follows from the observation that N (ε) = ds. The 0 σ(s) proof of the second part is similar.

3.6.2

Symmetric difference distance case

In this section our goal is to determine the symmetric difference distance of an optimally approximating conic arc of an arc of γ, with affine arc length %. This optimally approximating conic arc is tangent to γ at its endpoints. If the conic is a parabola, these conditions uniquely determine the parabolic arc. If we approximate with a general conic, there is one degree of freedom left, which we use to minimize the symmetric difference distance between the arc of γ and the approximating conic arc β. Moreover, we also give an asymptotic expansion of the symmetric difference distance between the arc of γ and its unique equisymmetric conic arc. We also show that the asymptotic expansion of the symmetric difference distance for an optimal conic spline and an equisymmetric conic spline are equal upto terms of order six in the arc length of the curve. The asymptotic error bound for the parabolic case has already been computed by Ludwig in [73]. We on the other hand use the general formula of symmetric difference distance given by (3.21) and the property that the affine curvature of the parabolic arc is zero. In fact our method allows us to generalize the result for any general conic by using the fact that conics are the only curves in the plane, with constant affine curvature. Theorem 3.6.4 (Error in symmetric difference distance approximation). Let γ : [0, %] → R2 be a sufficiently smooth, regular curve with non-vanishing Euclidean curvature.

70

3. Approximation by conic splines 1. Let β be the parabolic arc tangent to γ at the endpoints, the symmetric difference between the two arcs has the following asymptotic expansion δS (γ, β) =

5 1 240 |k0 |%

+ O(%6 ),

where k0 is the affine curvature of γ at γ(0). 2. Let β be a bitangent conic arc, minimizing the symmetric difference, then the symmetric difference between the two arcs has the following asymptotic expansion δS (γ, β) =

0 6 1 7680 |k0 |%

+ O(%7 ),

(3.19)

where k00 is the derivative of the affine curvature of γ at γ(0). 3. Let β be the equisymmetric bitangent conic arc of γ, then the asymptotic expansion of the symmetric difference between the two curves is given by (3.19).

Figure 3.6: The area of the shaded region is the symmetric difference distance between α and chord α(σ) and α(τ ).

Proof. First we introduce some notation. The symmetric difference distance between a convex curve α and a chord α(σ)α(τ ) is equal to the area of the shaded region in Figure 3.6, and will be denoted by Aα (σ, τ ). Then Z τ Aα (σ, τ ) = 21 [α(u) − α(σ), α0 (u)] du, (3.20) σ

and [v, w] denotes the determinant of two vectors v and w in R2 . 1. Consider the case when the approximating curve β is a parabolic arc. The

3.6. Complexity of conic splines

71

symmetric difference distance between γ and β in the interval [0, %] is given by δS (γ, β) = |Aβ (0, %) − Aγ (0, %)|. (3.21) Also see Figure 3.7 (left). Inserting the Taylor series expansion as given in

Figure 3.7: Shaded region in the first figure shows symmetric difference distance given by (3.21) and shaded region in the second figure shows symmetric difference distance given by (3.23).

Lemma 2.2.2 of β, into (3.20), we obtain Aβ (0, %) =

1 12

%3 +

1 240


−k0 − 8 Q(0, 0) %5 + O(%6 ),

and Aγ (0, %) =

1 12

%3 −

1 240

k0 %5 + O(%6 ).

Therefore, in view of (3.21) δS (γ, β) = |Aβ (0, %) − Aγ (0, %)| =

5 1 30 |Q(0, 0)| %

+ O(%6 ).

(3.22)

Using the relation between affine curvatures of a curve γ and its offset β, given in Lemma 3.5.1, and the fact that the affine curvature of a parabolic arc is zero everywhere, we obtain Q(0, 0) = − 18 k0 + O(%). Substituting this expression into (3.22), we obtain δS (γ, β) =

1 240

|k0 | %5 + O(%6 ).

2. The curve γ has a one parameter family of bitangent conic arcs. Our aim is to give an asymptotic expression for the minimal symmetric difference distance. In our case, the symmetric difference distance between γ and any bitangent conic β is given by the equation (3.23), where σ = c % + O(%2 ), and c ∈ [0, 1], and the bitangent conic β, intersects γ at γ(σ).

72

3. Approximation by conic splines

The symmetric difference distance between a given smooth convex curve γ and a bitangent conic arc β, intersecting γ at γ(σ) is given by δS (β, γ) = |Aβ (0, σ) − Aγ (0, σ)| + |Aγ (σ, %) − Aβ (σ, %)|.

(3.23)

Also see Figure 3.7 (right). Using the Taylor series expansion as given in Lemma 2.2.2 for γ and (3.20) we derive 1 |Aβ (0, σ) − Aγ (0, σ)| = | 60 (5 c4 − 6 c5 + 2 c6 )||Q(0, 0, 0)| %6 + O(%7 ),

and 1 (1 − 2 c + 5 c4 − 6 c5 + 2 c6 )| |Q(σ, σ, %)| %6 + O(%7 ). |Aβ (σ, %) − Aγ (σ, %)| = | 60

Furthermore, Q(σ, σ, %) can be written as Q(σ, σ, %) = Q(0, 0, 0) + c % Qu (0, 0, 0) + O(%2 ), plugging this expression into the expression for Aβ (σ, %) − Aγ (σ, %), we have |Aβ (σ, %) − Aγ (σ, %)| =

1 60 |1

− 2 c + 5 c4 − 6 c5 + 2 c6 | |Q(0, 0, 0)| %6 + O(%7 ).

Using (3.23) we obtain δS (γ, β) =

4 5 6 4 5 6 6 7 1 60 (|1−2 c+5 c −6 c +2 c |+|5 c −6 c +2 c |)|Q(0, 0, 0)| % +O(% ).

(3.24) Since we want to find the asymptotic error bound for the conic minimizing symmetric difference distance, we minimize (3.24) with respect to c. We conclude that δS (γ, β) is minimal for c = 21 . Therefore, equation (3.24) reduces to 1 δS (γ, β) = 192 |Q(0, 0, 0)| %6 + O(%7 ). Referring to the Lemma 3.5.1, relating the affine curvature of offset curves, and the fact that the affine curvatures of conics are constant we have that, 1 0 Q(0, 0, 0) = − 40 k0 + O(%). Plugging in the expression for Q(0, 0, 0) into the last expression for δS (γ, β), we have δS (γ, β) =

0 6 1 7680 |k0 | %

+ O(%7 ).

3.6. Complexity of conic splines

73

3. The asymptotic expansion of the symmetric difference distance between the given arc of γ and its unique equisymmetric conic arc β, is found by equating |Aβ (0, σ) − Aγ (0, σ)| to |Aβ (σ, %) − Aγ (σ, %)|,yielding c = 12 . Further simplifying we see that δS (γ, β) is of the same form as given by (3.19). We therefore conclude that the asymptotic error bounds for a conic minimizing symmetric difference and an equisymmetric conic are the same upto terms of order 6 in %, and therefore we say that the approximation with an equisymmetric conic is near optimal. In the following corollary we give expressions for the symmetric difference between a given convex curve γ and its best approximating parabolic and conic spline. The corollary can be proven using the same techniques as used by McClure and Vitale in [82] and Ludwig in [73]. Corollary 3.6.5 (Symmetric difference distance for an optimal spline). Let γ : I → R2 be a sufficiently smooth convex curve, with strictly increasing or decreasing affine curvature. 1. The symmetric difference between γ and a best approximating parabolic spline Pn with n knots is given by Z %
5 1 1 1 + O( 5 ). δS (γ, Pn ) = 240 |k(r)|1/5 dr 4 n n 0 2. The symmetric difference between γ and a best approximating conic spline Cn with n knots is given by Z %
6 1 1 1 δS (γ, Cn ) = 7680 |k 0 (r)|1/6 dr + O( 6 ). 5 n n 0 Here γ is parametrized by the affine arc length parameter r and the affine curvature of the curve γ is denoted by k. In the following corollary, we give the asymptotic expression for the symmetric difference distance between a given convex curve γ, and its equisymmetric conic spline, with n knots and denoted by ESn . As the name suggests, an equisymmetric conic spline, is a spline such that every element in it is an equisymmetric conic.

74

3. Approximation by conic splines

Corollary 3.6.6 (Symmetric difference distance for an equisymmetric conic spline). The symmetric difference distance between γ and an equisymmetric conic spline with n knots is given by Z %
6 1 1 1 δS (γ, ESn ) = 7680 |k 0 (r)|1/6 dr + O( 6 ). 5 n n 0 Remark 3.6.7. The basic idea behind proving Corollary 3.6.5 or Corollary 3.6.6 is to define functions called parabolic content and conic content. Given a sufficiently smooth strictly convex curve γ : [σ, τ ] → R2 , its parabolic Rτ content is defined by λp = σ |k(r)|1/5 dr. Similarly the conic content of γ Rτ is given by λc = σ |k 0 (r)|1/6 dr. These functions are useful in distributing the knots over the curve γ, in such a way, that the symmetric difference distance of all the segments are equal. Here each segment consists of a region bounded by the arc of γ lying between two knots and the bitangent parabolic or the equisymmetric conic arc approximating it. The aim for this kind of approximation is to distribute the knots uniformly over the curve with respect to the parabolic or the conic content. In fact the methods used by McClure and Vitale in [82] and Ludwig in [73] use this notion of content to show that there exists an optimal spline minimizing the symmetric difference distance for a curve with a given number of knots. The preceding result represents an asymptotic expression for the number of elements of an optimal parabolic or conic spline and also an asymptotic expression for an equisymmetric conic spline in terms of the symmetric difference distance. Corollary 3.6.8 (Complexity of parabolic and conic splines). Let γ : [0, %] → R2 be a sufficiently smooth regular curve with non-vanishing Euclidean curvature of length %, parametrized by affine arc length, and let k(r) be its affine curvature at γ(r). 1. The minimal number of arcs in a tangent continuous parabolic spline approximating γ to within symmetric difference distance ε is Z %
5/4 −1/4 N (ε) = (240)−1/4 ε (1 + O(ε1/4 )). (3.25) |k(r)|1/5 dr 0

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75

2. The minimal number of arcs in a tangent continuous conic spline approximating γ, to within symmetric difference distance ε is Z %
6/5 −1/5 N (ε) = (7680)−1/5 |k 0 (r)|1/6 dr ε (1 + O(ε1/5 )). (3.26) 0

The expression for complexity of an equisymmetric conic spline is of the same form as the expression for complexity of an optimal conic spline. The expressions match in the most significant terms, implying that the minimal number of elements in either case differ by a constant for a given value of ε. For all practical cases this difference turned out to be small.

3.7

Implementation

We implemented an algorithm in C++ using the symbolic computing library GiNaC1 , for the computation of an optimal parabolic or conic spline, based on the monotonicity property with respect to the Hausdorff distance. For computing the optimal parabolic spline, the curve is subdivided into affine spirals. Then for a given maximal Hausdorff distance ε, the algorithm iteratively computes optimal parabolic arcs starting at one endpoint. At each step of this iteration the next breakpoint is computed via a standard bisection procedure, starting from the most recently computed breakpoint. The bisection procedure yields a parabolic spline whose Hausdorff distance to the subtended arc is ε. An optimal conic spline is computed similarly. The bisection step is slightly more complicated, since the algorithm has to select the optimal conic arc from a one-parameter family. Here the equioscillation property gives the criterion for deciding whether the computed conic arc is optimal. We use a similar strategy for computing optimal parabolic or an equisymmetric conic spline, based on the monotonicity property of the symmetric difference distance. Given a local (symmetric difference) stopping condition εl , the algorithm iteratively computes the optimal parabolic arcs starting at one endpoint. We give a local, stopping condition, since from the theory we have that, for a parabolic spline with symmetric difference distance ε and n knots, the local εl is given by εl = nε . In fact our algorithm gives an exact 1

http://www.ginac.de

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3. Approximation by conic splines

match between the theoretical complexity and the experimental complexity, for sufficiently small values of ε. Below we present two examples of computations of optimal parabolic and conic splines for the Hausdorff metric case as well as the symmetric difference distance case. We compare the computed number of elements of these splines with the theoretical asymptotic complexity given in Corollary 3.6.3 and Corollary 3.6.8, thereby neglecting the higher order terms in (3.17), (3.18), (3.25) and (3.26).

3.7.1

A spiral curve

We present the results of our algorithm applied to the spiral curve, parameterized by α(t) = (t cos(t), t sin(t)), with 16 π ≤ t ≤ 2π. Figures 3.8(a) and 3.8(b) depict the result of the algorithm applied to the spiral for different values of the error bound ε, for the approximation by conic arcs and parabolic arcs respectively. For ε ≥ 10−2 , there is no visual difference between the curve and its approximating conic. ε 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8

Parabolic Exp./ Th. 5 9 15 26 46 82 145 257

Conic Exp./ Th. 3 4 6 9 13 21 32 51

Table 3.1: Hausdorff distance. The complexity (number of arcs) of the parabolic spline and the conic spline approximating the Spiral Curve. The theoretical complexity matches exactly with the experimental complexity, for various values of the maximal Hausdorff distance ε.

Table 3.1 gives the number of arcs computed by the algorithm, and the theoretical bounds on the number of arcs for varying values of ε, both for the

3.7. Implementation

77

Conic spline approximation

Parabolic spline approximation

Figure 3.8: Approximation of the to 10−8 .

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3. Approximation by conic splines

Conic spline approximation

Parabolic spline approximation

Figure 3.9: Approximation of the spiral for ε ranging between 10−1 to 10−6 .

parabolic and for the conic spline. Figures 3.9(a) and 3.9(b) depict the result of the algorithm applied to the spiral for different values of the error bound ε, for the approximation by conic arcs and parabolic arcs respectively with respect to the symmetric difference distance. There is no visual difference between the curve and its approximating conic, for the values of ε under consideration. Table 3.2 gives the number of arcs computed by the algorithm,

3.7. Implementation

79 ε 10−1 10−2 10−3 10−4 10−5 10−6

Parabolic Exp./ Th. 9 15 26 46 82 146

Conic Exp./ Th. 4 6 9 13 21 33

Table 3.2: Symmetric difference distance. The complexity (number of arcs) of the parabolic spline and the conic spline approximating the Spiral Curve. The theoretical complexity matches exactly with the experimental complexity, for various values of the symmetric difference distance ε.

and the theoretical bounds on the number of arcs for varying values of ε, both for the parabolic and for the conic spline with respect to the symmetric difference distance.

3.7.2

Cayley’s sextic

We present the results of our algorithm applied to the Cayley’s sextic, the curve parameterized by α(t) = (4 cos( 3t )3 cos(t), 4 cos( 3t )3 sin(t)), with − 34 π ≤ t ≤ 43 π. This curve has a sextactic point at t = 0. For all values of ε we divide the parameter interval into two parts [− 34 π, 0] and [0, 34 π] each containing the sextactic point as an endpoint, and then approximate with conic arcs using the Incremental Algorithm. The pictures in Figure 3.10(a) give the conic spline approximation images for Cayley’s sextic for different values of ε. The first picture in Figure 3.10(b) gives the original curve and its parabolic spline approximation for ε = 10−1 . The rest of the pictures in Figure 3.10(b) gives only the parabolic spline approximation for Cayley’s sextic for different errors, since the original curve and the approximating parabolic spline are not visually distinguishable. Table 3.3 gives the number of arcs computed by the algorithm, and the theoretical bounds on the number of arcs for varying values of ε, both for the parabolic and for the conic spline. The difference in the experimental and

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3. Approximation by conic splines

Conic spline approximation

Parabolic spline approximation

Figure 3.10: Plot of the approximations of a part of Cayley’s sextic for ε ranging from 10−1 to 10−8 .

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81

theoretical bound in the conic case for ε = 10−1 can be explained by the fact that the higher order terms are not taken into consideration for computing the theoretical bound. This causes the anomaly for relatively higher values of ε.

ε 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8

Parabolic Exp./Th. 6 8 14 24 44 76 134 238

Conic Exp./ Th. 4/2 4 6 8 12 18 28 44

Table 3.3: Hausdorff metric. The complexity of the parabolic spline and the conic spline approximating Cayley’s sextic. The theoretical complexity matches exactly with the complexity measured in experiments (except for ε = 10−1 in the conic case), for various values of the maximal Hausdorff distance ε.

The pictures in Figure 3.11(a) give the conic spline approximation images for Cayley’s sextic for different values of ε. The pictures in Figure 3.11(b) gives only the parabolic spline approximation for Cayley’s sextic for different errors, since the original curve and the approximating parabolic spline are not visually distinguishable. The approximations in this case are with respect to the symmetric difference distance. Table 3.4 gives the number of arcs computed by the algorithm, and the theoretical bounds on the number of arcs for varying values of ε, both for the parabolic and for the conic spline with respect to the symmetric difference distance.

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3. Approximation by conic splines

Conic spline approximation

Parabolic spline approximation

Figure 3.11: Plot of the approximations of a part of Cayley’s sextic with respect to symmetric difference distance for ε ranging from 10−1 to 10−6 .

3.8

Conclusion

It would be interesting to determine the constants in the approximation order of some of the existing methods for geometric Hermite interpolation (Floater [44], Schaback [104]), using the methods of this paper. Another open problem is to determine more terms in the asymptotic expansions of

3.8. Conclusion

83 ε 10−1 10−2 10−3 10−4 10−5 10−6

Parabolic Exp./ Th. 6 12 20 34 60 108

Conic Exp./ Th. 4 4 6 10 16 24

Table 3.4: Symmetric difference distance. The complexity of the parabolic spline and the conic spline approximating Cayley’s sextic. The theoretical complexity matches exactly with the complexity measured in experiments, for various values of the symmetric difference distance ε.

the complexity of optimal parabolic and conic splines derived in Section 3.6, like Ludwig [75] extends the complexity bound of the linear spline approximation of Fejes T´ oth [42]. This chapter addresses the issue of approximation with conics with respect to the symmetric difference distance. The problem of finding the unique optimal conic minimizing symmetric difference distance still remains open, our experimental results do show that such a conic exists. To enable certified computation of conic arcs with guaranteed bounds on the Hausdorff distance we would have to derive sharp upper bounds on the Hausdorff distance between a curve and a bitangent conic, extending the asymptotic expression for these error bounds for short curves, as given in Theorem 3.6.1. Such a certified method could lead to robust computation of geometric structures for curved objects, like its Voronoi Diagram. In this approach the curved object would first be approximated by conic splines, after which the Voronoi Diagram of the conic arcs of these splines would be computed. The number of elements of such a conic spline would be orders of magnitude smaller than the number of line segments needed to approximate the curved object with the same accuracy. Deciding whether this feature outweighs the added complexity of the geometric primitives in the computation of the Voronoi Diagram would have to be the goal of extensive experiments.

Chapter 4

Helix spline approximation of space curves

4.1

Introduction

In the area of computer aided geometric design several methods exist to approximate parametric curves with simpler curves, with respect to various metrics, the most commonly used being the Hausdorff metric. Although plane curves have been extensively studied in this field a lot of work still needs to be done for curves in higher dimensions. In this chapter we focus on approximation of curves in space with bihelical arcs with respect to the Hausdorff metric. Our approximation scheme falls within the scope of geometric Hermite interpolation, where we approximate a curve in space with non-zero, monotonically increasing/decreasing curvature by a geometric spline, which in our case is composed of finite number of consecutive segments, where each segment is a bihelical arc. Furthermore, a bihelical arc is made of two helical arcs which join each other at a point in between and are tangent to each other at this point. The point where two helical arcs forming a bihelical arc join is defined as the junction point. Furthermore each bihelical arc is bitangent to the given space curve, implying that a bihelical arc is tangent to a space at the two endpoints. For a detailed overview to the area of geometric Hermite interpolation of curves we refer to Section 1.3.3 in Chapter 1.

4.1.1

Approximation by bihelical splines

We study the complexity of a bihelical spline approximating a sufficiently smooth spiral, i.e., a space curve with non-vanishing monotone curvature, as a function of the Hausdorff distance between the curve and the spline. A bihelical spline is composed of bihelical arcs. A bihelical arc is composed of two helical arcs which are tangent to each other at one of their endpoints and

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4. Helix spline approximation of space curves

tangent to the curve at their other endpoints. The common point is called the junction point, the common tangent at the junction point is the junction tangent. It is of interest to study approximations with biarcs and bihelices as these are easy to use, computationally efficient in shape modeling, and well used as the description of tool path of CNC machines. These kind of applications necessitate a tight hold on the error bound (i.e., the constant of the leading term in the asymptotic expansion) for the Hausdorff metric. Our results generalize the work of Meek and Walton [83] for the approximation of spiral curves in plane with biarcs. The techniques used to find an optimal bihelical spline approximation to a space curve are useful to prove results about biarc approximation of curves in the plane. We discuss the biarc approximation of plane curves in detail in the Chapter 5 of our thesis, also note that the biarcs case becomes an application of the results proved in this chapter. In this chapter we prove that the Hausdorff distance between a bit1 0 3 angent bihelix and the corresponding space curve is at least 324 κ σ + O(σ 4 ), here σ is the arclength of the given space curve.

4.1.2

Results of this chapter

Complexity of bihelix approximants. We show that the complexity the number of elements - of an optimal bihelix spline approximating a space curve to within Hausdorff distance ε is of the form cε−1/3 + O(1), where we express the value of the constant c in terms of derivative of curvature (see Theorem 4.5.1). Similarly, an optimal biarc spline approximating a plane curve to within Hausdorff distance ε has exactly the same form as the expression for the bihelix spline, for a proof we refer to Chapter 5. This bound is obtained by first deriving the expression for the Hausdorff distance of a bihelical arc that is tangent to a (sufficiently short) curve at its endpoints, and minimizing the Hausdorff distance among all such bihelical arcs. Furthermore, we show that the Hausdorff distance is of third order in the length of the curve. We also derive the explicit constant in the asymptotic expansion in terms of the derivative of the curvature of the space curve. Algorithmic issues space curves. For space curves with monotone curvature and non-zero torsion, we consider bihelices tangent to the curve at

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87

its endpoints. On fixing a junction tangent T∗ we get a line of junction points l(T∗ ), where each junction point in l(T∗ ) corresponds to a bihelix tangent to the curve at its endpoints. Furthermore, we show that the locus of all junction tangents corresponding to junction lines parallel to l(T∗ ) is a circle lying on S2 and the union of all these junction lines is a junction cylinder. We conclude that there exists a one parameter family of junction cylinders. Thus for a fixed pair of points p0 , p1 and fixed tangents T0 , T1 at these points we conclude that there exists a three-parameter family of bitangent bihelices. In the case of space curves we find a near optimal bihelix by the following method. 1. Let α : [0, σ] → R3 be an arclength parametrized space curve with monotonic curvature κ and non-zero torsion. Define the junction tangent by 1 0 1 T∗ = T ( 21 σ) − 24 κ ( 2 σ)σ 2 N ( 12 σ), where T is the tangent and N the normal of the curve α. 2. Let the junction point be the point on the junction line l(T∗ ) closest to the point α( 12 σ). 3. Construct the bihelix consisting of the helix segment bitangent to (α(0), T (0)) and (p∗ ), (T∗ ), and the helix segment bitangent to (p∗ ), (T∗ ) and (α(σ), T (σ)). junction tangent we have a unique bihelix. We call this composite curve the near optimal bihelix of α. In our local computations we show that this near optimal bihelix in the limit is that same as the optimal bihelix. Furthermore, we assume the monotonicity property of the Hausdorff distance function, which is supported by various experiments. This gives a bisection based algorithm for the computation of a near optimal interpolating tangent continuous bihelix spline. We implemented such an algorithm, and compare it to the theoretical complexity of the actual number of elements in an optimal approximating bihelix spline. Overview. In Section 4.2 we define circular helices based on the derivation of general helices in Chapter 2 and show that given a point and a tangent there exists a three parameter family of helices.

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4. Helix spline approximation of space curves

In Section 4.3 we derive the necessary and sufficient condition for short helical arc through two points p0 and p1 with tangents T0 and T1 respectively. We conclude this section with an algorithm to find the unique short circular helix through points p0 and p1 with tangents T0 and T1 satisfying the necessary condition (4.3). In Section 4.4 the bihelix problem is considered, to begin with we prove that given a point pL and tangents TL and T0 , we show that locus of all points p0 such that there exists a short helical arc through pL and p0 with tangents TL and T0 is a plane denoted by ΠL (T0 ). Moreover, it is shown that the locus of all junction points p0 which generate a unique bihelix through two points pL and pR with tangents TL and TR respectively and junction tangent T0 is a line of junction points denoted by l(T0 ). Furthermore, we give the construction of the junction cylinder corresponding to all junction tangents which give rise to junction lines parallel to l(T0 ). Finally as T0 varies over S2 we show that there exists a one parameter family of junction cylinders. Section 4.5 presents an Approximation Theorem for space curves, a general result regarding the Hausdorff distance between a regular space curve and an approximating G1 -spline consisting of two smooth arcs, each sharing an endpoint and the corresponding tangent with the curve, and joining in a G1 fashion at their other endpoints (the junction point), whereas the curvature of the spline has zero derivative at the endpoints shared with the curve. A bihelix spline is a special case of such a curve, since it consists of two helical arcs, each with constant curvature (and torsion). The Approximation Theorem gives a lower bound for the Hausdorff distance between the curve and the bitangent spline, and a characterization of bitangent splines that achieve this lower bound (asymptotically). In particular, a bitangent spline is optimal with regard to Hausdorff distance if and only if the distance between its junction point and the midpoint of the curve is of fourth order in the length of the curve, and the junction tangent deviates from the tangent to the curve at its midpoint 1 0 1 by a component in the normal direction of the form − 24 κ ( 2 σ) N ( 12 σ)+O(σ 3 ), where, again, σ is the length of the curve. Furthermore, a bitangent spline is optimal iff the Frenet-Serret frames and the curvatures of the curve and the spline at their common endpoints are equal up to terms quadratic in the length of the curve. We show that there is a bitangent bihelix spline satisfying these conditions for optimality in terms of the junction point and junction

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89

tangent. This leads to an algorithm in Section 4.6 for the construction of an asymptotically optimal bihelix spline. We conclude with examples, in which the experimentally measured complexity of the bihelix spline matches its theoretical complexity almost exactly.

4.2

Circular Helices

In Chapter 2 we consider generalized helices. When the curvature and torsion of a generalized helix are both constant, then the helix that we get lies on a circular cylinder and we define it as the circular helix. Consider a generalized helix with constant curvature κ and constant torsion τ . We call such a curve a circular helix, and refer to it as a helix in the following sections. As before let √ β = κ2 + τ 2 , and κ = κ/β and τ = τ /β. Since β is constant, the functions C and S, defined in (2.18), are give by C(s) =

sin(βs) β

and S(s) =

1 − cos(βs) . β

(4.1)

Substituting these identities into the expression for the helix in Lemma 2.3.4 yields Lemma 4.2.1. The unique helix with constant curvature κ and constant torsion τ through p0 , where its Frenet-Serret frame is equal to the orthonormal frame {T0 , N0 , B0 }, is given by α(s) = p0 + s τ V0 +


κ sin(βs) W0 + (1 − cos(βs) N0 . β

This is the arc-length parametrization of a helix, the axis of which is the line through the point κ N0 , p0 + 2 κ + τ2 with direction V0 . Lemma 4.2.1 also implies that the radius of its supporting cylinder is equal to κ . 2 κ + τ2 If p0 and T0 are fixed there is a one-parameter family of normals N0 . Therefore, the identity in Lemma 4.2.1 defines a three-parameter family of helices, with parameters κ, τ and N0 .

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4. Helix spline approximation of space curves

Corollary 2.3.5 yields a useful expression for the unique helix with given initial point, tangent and normal at that point, and curvature and torsion: Corollary 4.2.2. Let α be the unique helix through p0 , which has FrenetSerret frame {T0 , N0 , B0 } at p0 , and curvature κ and torsion τ . In terms of the frame {T0 , V0 , T0 × V0 }, the helix is given by α(s) = p0 + s τ V0 +


1 sin(βs) (T0 − τ V0 ) + (1 − cos(βs)) T0 × V0 . β

Its unit tangent vector is T (s) = T0 + (cos(βs) − 1) (T0 − τ V0 ) + sin(βs) T0 × V0 . The tangent indicatrix of the helix, i.e., the trace of its unit tangent vector, is a circle on the unit sphere with center τ V0 and radius κ, in the plane perpendicular to V0 . More precisely, using the first identity of Lemma 4.2.1, we get T (s) = τ V0 + κ (cos βs W0 + sin βs N0 ). (4.2) Note that this circle contains the point T0 , for all values of κ (and τ ). In fact, this circle is obtained by intersecting the unit sphere with the plane through T0 with direction N0 . If τ = ±1, and, hence, κ = 0, the tangent indicatrix reduces to its center τ V0 = T0 . If τ = 0, and, hence, κ = 1 and W0 = T0 , we have T (s) = cos βs T0 + sin βs N0 , so the tangent indicatrix is a great circle through T0 , whose tangent vector at this point is parallel to N0 . Obviously, the helix reduces to a planar circle in this case. The identities for α(s) and T (s) in Corollary 4.2.2 yield hα(t) − α(s), T (t) − T (s)i = 0. In other words: Corollary 4.2.3. The unit tangent vectors at two points on a helix make the same angle with the line segment connecting these points.

4.3

Helix segments with constraints

In this section we define a short helical arc. Given two linearly independent vectors T0 and T1 in R3 , we show the existence of a natural basis for R3 .

4.3. Helix segments with constraints

91

Moreover, we show that given two points p0 and p1 and two linearly independent tangents vectors T0 and T1 there exists a unique short helical arc satisfying the condition hp1 − p0 , T1 − T0 i = 0. Furthermore, the techniques used to prove this result give us an algorithm to construct a short helical arc. This algorithm is useful in the construction of the bihelical arc in the implementation for finding a near optimal bihelix spline of a given space curve as discussed in Section 4.6. A natural basis for three-space. The unit vectors T0 and T1 , with T0 6= ±T1 , provide us with a natural basis of R3 . Lemma 4.3.1. Let t0 = hT0 , T1 i, and 1 1 1 p E=p (T1 +T0 ), F = p T ×T , and G = (T1 −T0 ). 1 0 2(1 + t0 ) 2(1 − t0 ) 1 − t20 Then {E, F, G} is an orthonormal basis of R3 with E × F = G. Proof. Since T0 and T1 are unit vectors the vectors E and G are orthogonal unit vectors. Furthermore, a short computation shows E × G = −F . Therefore, E × F = G. Helix segments. Let T (s) denote the parametrized tangent vector to a helix with axis V0 . Furthermore, let T⊥ (s) denote the projection of T (s) onto the plane perpendicular to V0 . A short helical arc is then an arc of a helix whose T⊥ vector traces less than a full circle. Theorem 4.3.2. Let p0 and p1 be given points in three-space, and let T0 and T1 be unit vectors, with T1 6= ±T0 . There is a unique short helical arc with endpoints p0 and p1 and tangent vectors T0 and T1 at these points, respectively, iff hp1 − p0 , T1 − T0 i = 0. (4.3) First a lemma. In view of Corollary 4.2.2, we put ϕ = βs and define T (ϕ, τ , V ) by T (ϕ, τ , V ) = T0 + (cos ϕ − 1) (T0 − τ V ) + (sin ϕ) T0 × V.

(4.4)

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4. Helix spline approximation of space curves

Then T is a map X0 → S2 , where the set X0 ⊂ R5 is defined by X0 = {(ϕ, τ , V ) ∈ R × R × R3 | 0 < ϕ < 2π, |τ | < 1, hV, V i = 1, hT0 , V i = τ , hT1 , V i = τ }. In fact, it straightforward to check that || T (ϕ, τ , V ) || = 1. Lemma 4.3.3. The equation T (ϕ, τ , V ) = T1 , with (ϕ, τ , V ) ∈ X0 , has two one-parameter families of solutions for (τ , V ) in terms of ϕ, given by r t0 − cos ϕ , τ ± (ϕ) = ± 1 − cos ϕ V± (ϕ) =


1 τ ± (ϕ) (T0 + T1 ) + (cot 12 ϕ) (T1 × T0 ) , 1 + t0

where ϕ ranges over the interval [arccos t0 , 2π − arccos t0 ]. Proof. Let us first assume that T (ϕ, τ , V ) = T1 , for (ϕ, τ , V ) ∈ X0 . Then hT (ϕ, τ , V ), T0 i = t0 , so 0 = hT (ϕ, τ , V ), T0 i − t0 = (1 − cos ϕ) τ 2 − (t0 − cos ϕ). From this equality we conclude τ2 =

t0 − cos ϕ , 1 − cos ϕ

(4.5)

which yields the expression for τ . To determine V as a function of ϕ, observe that the decomposition of V with respect to the orthonormal basis {E, F, G}, introduced in Lemma 4.3.1, is V = hV, Ei q E + hV, F i F + hV, Gi G. Since hV, T0 i = hV, T1 i = τ , we have 2 hV, Ei = τ 1+t and hV, Gi = 0. Using the expressions for E, F and G in 0 terms of T0 and T1 we get

V =

τ 1 (T0 + T1 ) + hV, T1 × T0 i (T1 × T0 ). 1 + t0 1 − t20

So it remains to determine hV, T1 × T0 i. To this end, we use hT (ϕ, τ , V ), T1 i = 1, so t0 + (cos ϕ − 1) (t0 − τ 2 ) + sin ϕ hT1 , T0 × V i = 1.

4.3. Helix segments with constraints

93

Using (4.5), a short calculation yields hV, T1 × T0 i = hT1 , T0 × V i = (1 − t0 )

1 + cos ϕ = (1 − t0 ) cot 21 ϕ. sin ϕ

(4.6)

Substituting this identity into (2.14) yields the identity for V . It is straightforward to check that T (ϕ, τ ± (ϕ), V± (ϕ)) is equal to T1 by verifying that its inner product with T1 is equal to one. Remark 4.3.4. If τ satisfies (4.5)), then κ2 = 1 − τ 2 =

1 − t0 . Also note 1 − cos ϕ

that 0 ≤ τ 2 ≤ 21 (1 + t0 ), and 12 (1 − t0 ) ≤ κ2 ≤ 1. Proof of Theorem: 4.3.2 . Corollary 4.2.3 shows that the condition is necessary. To prove sufficiency, we fix the initial point p0 and the unit tangent vector T0 . For a point p1 and unit vector T1 such that hp1 − p0 , T1 − T0 i = 0 we have to find real numbers β, τ , s and a unit vector V such that p1 = α(s) and T1 = T (s), with α(s) and T (s) as in Corollary 4.2.2. In view of Lemma 4.3.3 we consider T± (ϕ) ∈ S2 and S± (β, ϕ) ∈ R3 , defined by T± (ϕ) = T (ϕ, τ ± , V± )
1 S± (β, ϕ) = τ ± ϕ V± + sin ϕ (T0 − τ ± V± ) + (1 − cos ϕ) T0 × V± , β where β > 0 and ϕ ∈ I := [arccos t0 , 2π − arccos t0 ], and τ ± and V± stand for τ ± (ϕ) and V± (ϕ), respectively. Since T± (ϕ) = T1 , by Lemma 4.3.3, we have to show that there is a unique pair (β∗ , ϕ∗ ) such that either S+ (β∗ , ϕ∗ ) = p1 − p0 or S− (β∗ , ϕ∗ ) = p1 − p0 . To this end we determine the inner product of S± (β, ϕ) with the unit vectors E and F . A short computation shows that the inner product with G is zero. See also Corollary 4.2.3. So assume S± (β, ϕ) = p1 − p0 . Then β hp1 − p0 , T1 × T0 i = β hS± (β, ϕ), T1 × T0 i = τ ± (ϕ − sin ϕ) hV± , T1 × T0 i + (1 − cos ϕ) hT0 × V± , T1 × T0 i.

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Since hV± , T1 × T0 i = (1 − t0 ) and

1 + cos ϕ sin ϕ

hT0 , T1 i hT0 , T0 i hT0 × V± , T1 × T0 i = = −τ ± (1 − t0 ), hV± , T1 i hV± , T0 i

a short calculation shows that β hp1 − p0 , T1 × T0 i = τ ± (ϕ) (1 − t0 ) (ϕ cot 21 ϕ − 2).

(4.7)

Since ϕ cot 12 ϕ − 2 < 0, for ϕ ∈ I, it follows that sign τ ± (ϕ) = − signhp1 − p0 , T1 × T0 i.

(4.8)

In other words, (4.8) tells us which of the two signs to choose. To derive an equation for ϕ, we also determine the inner product of p1 −p0 and T0 + T1 : β hp1 − p0 , T0 + T1 i = β hS± (β, ϕ), T0 + T1 i = τ ± (ϕ − sin ϕ) hV± , T0 + T1 i + (1 − cos ϕ) hT0 × V± , T0 + T1 i. Now hV± , T0 + T1 i = 2τ ± , and hT0 × V± , T0 + T1 i = hV± , T1 × T0 i = (1 − t0 )

1 + cos ϕ , sin ϕ

so β hp1 − p0 , T0 + T1 i =


2 ϕ (t0 − cos ϕ) + (1 − t0 ) sin ϕ . 1 − cos ϕ

From (4.7) and (4.9) we obtain √ 2 H(ϕ) = − 1 − t0 where

hp1 − p0 , T1 × T0 i hp1 − p0 , T1 + T0 i ,

√ (ϕ cos 12 ϕ − 2 sin 12 ϕ) t0 − cos ϕ H(ϕ) = , D(ϕ)

with D(ϕ) = ϕ (t0 − cos ϕ) + (1 − t0 ) sin ϕ.

(4.9)

(4.10)

(4.11)

4.3. Helix segments with constraints

95

Algorithm 1 HelixArc(p0 , T0 , p1 , T1 ) Input: p0 , T0 , p1 , T1

// Precondition: hp1 − p0 , T1 − T0 i = 0

t0 ← hT0 , T1 i √ 2 hp1 − p0 , T1 × T0 i c← 1 − t0 hp1 − p0 , T1 + T0 i 2 if c > √ then π 1 + t0 Error: value of c out of range end if Compute ϕ∗ , the unique solution of H(ϕ) = −c on [arccos t0 , π] // Here H(ϕ) is given by (4.11) r t0 − cos ϕ∗ τ ← − signhp1 − p0 , T1 × T0 i 1 − cos ϕ∗
1 τ (T0 + T1 ) + (cot 12 ϕ∗ ) T1 × T0 . V0 ← 1 + t0 p 2(1 − t0 ) + (τ ϕ∗ )2 β← || p1 − p0 || Output: α(s) ← p0 + ϕ∗ and s1 ← β

sin βs
1 − cos βs sin βs T0 + τ s − V0 + T0 × V0 , β β β

Claim: The function H is decreasing on the interval I0 = [arccos t0 , π]. Furthermore, H(arccos t0 ) = 0

2 and H(π) = − √ . π 1 + t0

(4.12)

Appendix B.1 contains the proof of this claim. 2 , there is π 1 + t0 a unique value ϕ ∈ I0 such that H(ϕ) = −c. In particular, the fundamental equation (4.10) has a unique solution ϕ = ϕ∗ (t0 ) ∈ I0 . See Algorithm 1.

This claim implies that for each constant c with 0 < c
n (Proposition 6.2.2). We also show that the germ of the parametrization of the secant variety at the diagonal is never AZ2 -stable for immersions of codimension greater than two (see Proposition 6.2.5). Section 6.3 describes the main result of the present paper, the classification of the germs of secant maps Sˆ at the diagonal of surfaces generically immersed in Rn , n ≥ 2, and for n ≥ 3 any such AZ2 stable germ is the germ of the secant map of some generically immersed surface (for n = 2 there are complex AZ2 -orbits corresponding to several real orbits, whose representatives are distinguished by ± signs, and some of these real forms cannot be the germ of any secant map). For n ≥ 5, the AZ2 classes in this classification of secant maps are in 1:1 correspondence with certain A classes of germs of projections onto hyperplanes of the immersion-germ X at the corresponding points p along a certain bad direction in Tp X. For n = 3 and 4 there are certain AZ2 -orbits of secant germs that can be further stratified by the A-types of such projections. But the A-classes of the projections distinguish the AZ2 -orbits of secants germs for any n ≥ 3. Section 6.4 contains the classification of Z2 stable germs R4 → Rn , for n ≥ 2 (relevant for the secant map-germs Sˆ of generic immersions X : R2 → Rn+1 ).

6.2

Secants and inner projections of immersions X : Rn → Rm+1 , m ≥ n

Let X : Rn → Rm+1 , p = (p1 , . . . , pn ) 7→ X(p) be an immersion, and represent the Pn−1 of lines through 0 ∈ Rn by one unit vector ω on each line. The map β(p, λ, ω) = (p, p + λ · ω) =: (p, q) blows up the diagonal in (Rn )2 , which has codimension n, to the hyperplane β −1 ({p = q}) = {λ = 0} and is one-to-one outside λ = 0. Let [v] denote homogeneous coordinates of the vector v. We want to define a projectivized secant map Rn × R × Pn−1 → Pm given by (p, q) 7→ [X(q) − X(p)], for p 6= q, and on the diagonal by (p, p) 7→ [Dω X(p)] (with Dω the directional derivative). The desired map is given by S˜ : Rn × R × Pn−1 → Pm ,

(p, λ, ω) 7→ [λ−1 (X(p + λ · ω) − X(p))].

˜ 0, ω) = [Dω X(p)], and outside the diagonal we On the diagonal we have S(p, ˜ obtain S(p, λ, ω) = [X(q) − X(p)], for λ 6= 0. (For an immersion X and some neighborhood of λ = 0, the vector λ−1 (X(p+λ·ω)−X(p)) is always non-zero.

134

6. Singularities of secant maps

In Proposition 6.2.4, which describes the off-diagonal behavior of the secant map, we assume that X is an embedding, so that S˜ is defined for all λ.) Next, we want to consider the germ of this secant map at a point of the diagonal. Writing an immersion germ as X(p) = (p, gn+1 (p), . . . , gm+1 (p)),

gi ∈ M2n ,

and by taking an affine chart such that (1, v) = (1, v2 , . . . , vn ), vi = ωi /ω1 we get S˜ = [1, v2 , . . . , vn , λ−1 (gn+1 (q) − gn+1 (p)), . . . , λ−1 (gm+1 (q) − gm+1 (p))]. Composing with the (“symmetry restoring”) linear right coordinate change L(p¯1 , p¯2 , . . . , p¯n , λ, v2 , . . . , vn ) = (p¯1 − λ)/2, (p¯2 − λv2 )/2, . . . , (p¯n − λvn )/2, λ, v2 , . . . , vn




˜ we get a map and omitting the first (constant) component of S, Sˆ : R2n → Rm ,

ˆ p, λ, v). (¯ p, λ, v) 7→ S(¯

The germ of Sˆ at 0 is the germ of the projectivized secant map at the point (p, q) = (0, 0) ∈ (Rn )2 on the source-diagonal and the direction ω = [1 : 0 . . . : 0]. We can (and will) always consider a neighborhood of this direction (in which ω1 6= 0, where ω = [ω1 : . . . : ωn ]) by applying an element of SO(n − 1) to the tangent space of the immersion. We have the following easy lemma Lemma 6.2.1. The germ of the projectivized secant map Sˆ : R2n → Pm at a point on the source-diagonal is even. ˆ p, λ, v) = S(¯ ˆ p, −λ, v), so that, by Whitney’s lemma Proof. We claim that S(¯ ˆ on even map-germs [114], S = f (¯ p, λ2 , v) for some smooth germ f . The term cα pα in gl (where α = (α1 , . . . , αn ) is a multi-index) corresponds in S˜ to Y
λ−1 cα (p1 + λ)α1 (pi + λvi )αi − pα =: λ−1 cα Q, i>1

6.2. Secant and inner projections of immersions

135

where Q is divisible by λ. Now Y Y
Q ◦ L = 2−|α| (¯ p1 + λ)α1 (¯ pi + λvi )αi − (¯ p1 − λ)α1 (¯ pi − λvi )αi i>1

i>1

is odd in λ, hence λ−1 cα Q ◦ L and therefore Sˆ are even as claimed. Hence the Z2 symmetric secant map Sˆ is the composition of a map-germ f : R2n , 0 → Rm , 0, (¯ p, u, v) 7→ f (¯ p, u, v) with a folding map (¯ p, λ, v) 7→ 2 Z 2 (¯ p, λ , v). It is well-known that the A -classification of Z2 symmetric germs corresponds to the A(H)-classification of such germs f (see [2, 18]), where A(H) denotes the geometric subgroup of A in which the diffeomorphisms in the source preserve the hyperplane H := {u = 0} in the source. If R(H) is the subgroup of R of elements preserving H, we set A(H) = L × R(H) and K(H) = C · R(H) (semi-direct product). For Sˆ and f as above, we set ˆ := cod(A(H), f), cod(AZ2 , S) and similarly for the corresponding extended groups AZe 2 and A(H)e of nonorigin preserving diffeomorphisms. The germ Sˆ is (infinitesimally) Z2 stable ˆ := cod(A(H)e , f) = 0, and the AZ2 -codimension of a Z2 stable if cod(AZe 2 , S) germ from R2n to Rm is at most 2n − 1 (see Lemma 4.1). Working with the group of A(H) equivalences of map-germs f — rather than with AZ2 equivalence of equivariant secant-germs Sˆ — not only has technical advantages in the classification but also in transversality arguments relating submanifolds of jet spaces J k (2n, m) of k-jets of maps f to submanifolds in multi-jet spaces of jets of pairs of immersions X (and the results obtained for f then give the desired results for secants maps Sˆ by composing f with the above folding map). The next result shows that, for generic codimension ≥ 2 immersions, the germ of Sˆ at a point of the diagonal is a Z2 stable germ (provided the Z2 stable germs are dense for the relevant pairs of dimensions (2n, p); for other pairs of dimension this statement holds with C 0 -Z2 stable in place of Z2 stable). Proposition 6.2.2. Let (2n, m) be a pair of dimensions for which the K(H)orbits of germs f : R2n , 0 → Rm , 0 of rank at least n and K(H)-codimension at most 2n − 1 are K(H)-simple. Then, for a residual subset of Imm(Rn , Rm+1 ),

136

6. Singularities of secant maps

ˆ p) at the where m > n, the germ of the secant map Sˆ : R2n , (p, p) → Rm , S(p, Z 2 diagonal in the source is a Z2 stable germ of A -codimension at most 2n − 1 and rank at least n − 1. Proof. We will show that, for m > n, the restriction to the blow-up of the diagonal, Γk , of the map (J 2k+1 (n, m + 1))2 → J k (2n, m) sending (2k + 1)-jets of pairs of immersions germs j 2k+1 (X(p), X(q)) to the k-jets of the associated maps f (whose composition with the folding map gives ˆ is transverse to the closure of j k A(H)-orbits of codimension no greater than S) 2n−1. Taking coordinates p and q = p+l·v and restricting to the diagonal λ = 0, we have local coordinates (p, v) ∈ R2n−1 on the (blow-up of the) diagonal, and Γk : J 2k+1 (n, m+1)×Rn−1 → J k (2n, m) maps pairs (cα pα , v), with cα pα a monomial of some component function gl of X, to terms λ−1 cα Q◦L(¯ p, u1/2 , v) of the corresponding component function of f (¯ p, u, v) (recall that λ−1 Q ◦ L, defined in the proof of Lemma 6.2.1, is even in λ). The Γk -preimages of the closures of these j k A(H)-orbits are therefore Whitney stratified subsets of the same codimension or are empty. The image of (j 2k+1 X, v)(R2n−1 ) will therefore, for a residual set of immersions X, miss the Γk -preimages of orbits of A(H)-codimension 2n or greater. Claim 1. Suppose f A(H)-stable, then A(H) · f ⊂ K(H) · f is open. This follows from the following analogue of a result of Mather [81]: suppose f and g are A(H) stable, then they are A(H)-equivalent if and only if they are K(H)-equivalent. In order to prove this, one can adapt a result of Martinet [76] on the relation between Ke -versality and A-stability to the subgroups K(H)e and A(H). (A brief summary of Martinet’s result is given on p. 502 of Wall’s survey [113]: one considers a “regular unfolding” F : Rd × R2n → Rd × Rm of f , i.e., one which is transverse to Rd × {0}, then VF := F −1 (Rd × {0}) is a smooth submanifold of Rd × R2n . Let πF : VF → Rd denote the restriction of the obvious projection onto the first factor. Checking that one can replace the group R by R(H) we get the following variant of Martinet’s result, (3.5) in [113]: F is K(H)e -versal if and only if πF is A(H)-stable. And one deduces the above variant of Mather’s result by taking

6.2. Secant and inner projections of immersions

137

regular unfoldings F (y, x) = (y, f (x) − y) of x 7→ f (x) = y and (similarly) G of g – notice that πF = f , because x 7→ (f (x), x) is a parametrization of VF .) For A(H)-stable germs f , transversality to the K(H)-orbit of f therefore implies transversality to the A(H)-orbit. Claim 2. For m > n the maps Γk are transverse to the Kk (H)-orbits of codimension at most 2n − 1 and rank at least n. The map-germ f is given by f = (v2 , . . . , vn , Gn+1 (¯ p, u, v), . . . , Gm+1 (¯ p, u, v)), where the k-jets of the Gj are the k-jets of functions X

l−1 cα Q ◦ L(¯ p, u1/2 , v)

|α|≤2k+1

(recall the definition of the map Γk ). The map Γ1 is a submersion and the maps R2n → Rm of rank less than n form a submanifold B of J 1 (2n, m) of codimension greater than or equal to (m − n + 1)(n + 1) > 2n − 1, for m > n. 3 The set Γ−1 1 (B) therefore has empty intersection with j X for a residual set of immersions X. Now (avoiding the bad set B of germs of rank less than n) we can choose coordinates p1 , . . . , pn in the source of X and an affine chart (1, v2 , . . . , vn ) for Pn such that either f = (v2 , . . . , vn , p¯1 , Gn+2 (¯ p, u, v), . . . , Gm+1 (¯ p, u, v)), or f = (v2 , . . . , vn , u + Gn+1 (¯ p, 0, 0), Gn+2 (¯ p, u, v), . . . , Gm+1 (¯ p, u, v)) (in the former case Sˆ has rank at least n and in the latter case at least n − 1). Now consider in both cases the restriction f 0 of f to the subspace of the source given by the vanishing of the first n variables, and let Γ0k denote the corresponding restriction of Γk . Again using the functions λ−1 cα Q ◦ L above we see that Γ0k is a submersion, and hence transverse to K(H) · f 0 , and therefore to K(H) · f (f being the suspension of f 0 ).

138

6. Singularities of secant maps

Next, the following relation between secant maps Sˆ and n-parameter families of projections of X(Rn ) ⊂ Rm+1 from centers in X(Rn ) – so called inner projections in the terminology of [4] – will be useful. Recall that the map (p, λ, ω) = (p, p + λ · ω) blows up the diagonal in (Rn )2 , which has codimen¯ sion n, to the hyperplane λ = 0. Its composition β˜ := β ◦L with the symmetry restoring linear right coordinate change L is given by (¯ p, λ, ω) 7→ ((¯ p − λ · ω)/2, (¯ p + λ · ω)/2) =: (p, q). If πc denotes the projection from a center c ∈ Rm+1 , then the projectivized secant map factors as follows: Rn × R × Pn−1 β˜ . Rn × Rn (p, q)

−→ 7−→

& Sˆ Rm+1 × Rm+1 (X(p), X(q))

−→ 7−→

Pm πX(p) (X(q))

Outside the diagonal, where p 6= q, one can use this observation to show that the germ of Sˆ is locally a versal n-parameter deformation of a germ Rn → Rm for a residual set of immersions X, see Proposition 6.2.4 below. On the diagonal, the family of inner projections is only defined after blowing-up the source diagonal and dividing by λ. The following remark implies that the germ of the secant-map Sˆ at (p, λ, v) = (p, 0, 0) determines the projection π1 (X) of the germ of the immersion X at p into a hyperplane in Rm+1 orthogonal to e1 = dXp (e01 ) ∈ Rm+1 , with e01 = (1, 0, . . . , 0) ∈ Tp Rn (notice that e01 = (1, v) for v = (v2 , . . . , vn ) = 0). In Section 6.3 the AZ2 classification of secant-maps Sˆ and the related A-classification of the associated orthogonal projections π1 (X) will be described, the remark is also useful in relating AZ2 -orbit membership conditions to transversality conditions on the immersion. Setting γ(p1 , . . . , pn ) = (p1 , . . . , pn , p1 , p2 /p1 , . . . , pn /p1 ), we have the following:

6.2. Secant and inner projections of immersions

139

Remark 6.2.3. Let Sˆ be a germ of the secant map of an immersion p 7→ X(p) ˆ at (¯ p, λ, v) = (0, 0, 0), then π1 ◦ X(p) = p1 · S(γ(p))). Proof. Let X(p) = (p, gn+1 (p), . . . , gm+1 (p)) be in Monge form. From the diagram above we have Sˆ = [1, v2 , . . . , vn , hn+1 , . . . , hm+1 ], where hi := λ−1 (gi ((¯ p + λ · ω)/2) − gi ((¯ p − λ · ω)/2)) and ω = (1, v2 , . . . , vn ). Composing with γ and multiplying through with p1 gives the desired formula (notice that gi (0) = 0). Next, consider the off-diagonal behavior of the secant map. Notice that we can exchange the roles of X(p) and X(q) in the projection πX(p) (X(q)), this global Z2 -symmetry makes the off-diagonal part of the secant map highly unstable as a bi-germ. But considering Sˆ as a mono-germ we have the following. Proposition 6.2.4. The germ of the secant map ˆ 0 , q0 ) Sˆ : R2n , (p0 , q0 ) → Rm , S(p at any pair of points p0 6= q0 is a versal n-parameter deformation of a germ Rn to Rm for a residual subset of Emb(Rn , Rm+1 ). Proof. Consider disjoint neighborhoods U and V of p0 and q0 , respectively. The n-parameter family of projections πX(U ) : U × Rm+1 → U × Rm , (p, r) → πX(p) (r) is a family of submersions, and hence A-versal. By a transversality theorem for composite maps [90, 46] and a partition of unity argument, we can approximate the embedding germ X(V ) in a neighborhood of X(q0 ) by an embedding germ Y (V ), without changing the embedding germ X(U ) near X(p0 ), such that the composite family πX(U ) : U × V → U × Rm , (p, q) → πX(p) (Y (q)) is an A-versal n-parameter deformation of central projections of the embedding germ Y (V ) ⊂ Rm+1

140

6. Singularities of secant maps

We conclude this section by considering parametrizations of the secant variety of an immersion X : Rn → Rn+c , X(p) = (p, gn+1 (p), . . . , gn+c (p)), gi ∈ M2n . The map F : R2n+1 → Rn+c ,

(p, q, t) 7→ 21 (X(q) + X(p)) + t(X(q) − X(p))

has the symmetry F (q, p, −t) = F (p, q, t). We can again extend F to the “diagonal” p = q by taking q = p + λ(1, v2 , . . . , vn ) and by replacing X(q) − X(p) by λ−1 (X(q) − X(p)). By composing again on the right with the symmetry restoring linear coordinate change L we obtain a map F˜ (¯ p, λ, v, t). And setting w = (1, v) = (1, v2 , . . . , vn ), we see that the first term 1/2(X((¯ p + λw)/2) + X((¯ p − λw)/2) in F˜ is even in λ, and the second term is even by Lemma 6.2.1. Setting u = λ2 , the least degenerate j 1 F˜ is j 1 A(H)-equivalent to h := (t, p¯2 , . . . , p¯n , u, 0, . . . , 0) (if the x2 coefficients of all the gi components of X vanish then we obtain a more degenerate 1-jet). The j 1 A(H)-codimension of h is cn, and Lemma 6.4.1 below implies that the A(H)-stable germs R2n+1 → Rn+c have A(H)-codimension at most 2n. Hence F˜ is A(H)-unstable, and hence AZ2 -unstable, for any c > 2, and we obtain the following Proposition 6.2.5. The germ at the diagonal, F˜ : R2n+1 → Rn+c , of the parametrization of the secant variety of any immersion X : Rn → Rn+c of codimension c > 2 is AZ2 -unstable. Remark 6.2.6. The restriction of the secant variety to the diagonal is the tangent variety, and the latter has already for the simplest case of space curves in R3 non-isolated singularities of infinite A-codimension. The map F˜ might therefore be more degenerate than just AZ2 -unstable. (In fact, one checks, that for curves in 3-space the map F˜ has infinite AZ2 -codimension.)

6.3

Secant maps of generically immersed surfaces in Rn , n ≥ 3

In this section the germs of the secant map at a point in the source diagonal will be classified for generically immersed surfaces in Rn , n ≥ 3. Locally such

6.3. Secant maps of generically immersed surfaces

141

an immersion is given by an embedding-germ X(x, y) = (x, y,

X

(3)

aij xi y j , . . . ,

i+j>1

X

(n)

aij xi y j )

i+j>1

at (x, y) = (0, 0). The projectivized secant map (in the affine chart (1, v) for ω, see Section 6.2) is given by S˜ = [1 : v : S3 : . . . : Sn ], hence we have a mapˆ x, y¯, l, v) := (v, S3 , . . . , Sn ), where x germ into (n − 1)-space S(¯ ¯, y¯ correspond to the coordinates p¯1 , p¯2 in Section 6.2 (after applying the coordinate change ˆ Using the map Γk (defined in the L that restores the Z2 -symmetry of S). proof of Proposition 6.2.2), one obtains (i)

(i)

(i)

(i)

Si = a20 x ¯ + 12 a11 (¯ y + v¯ x) + a02 v y¯ + 41 a30 (u + 3¯ x2 ) + . . . with u = λ2 . Let πx ◦ X be the projection-germ at x = y = 0 of the immersed surface X(R2 ) along the x-direction. By Remark 6.2.3, the projection-germ πx ◦ X is determined by the secant map-germ Sˆ at x ¯ = y¯ = λ = v = 0 (the 2 direction (1, v) = (1, 0) in TX(0,0) X(R ) corresponds to the x-direction), but AZ2 -equivalence of Sˆ in general does not preserve the A-class of πx ◦ X. In the classification of secant-germs of generically immersed surfaces we therefore decompose certain AZ2 -orbits of secant-maps into strata on which the A-type of the corresponding projection πx ◦ X is invariant. In the normal forms Sˆ certain monomials are therefore present which affect the A-type of πx ◦ X but which could be removed by an AZ2 -change. On the other hand, redundant terms in the corresponding projection-germs πx ◦ X have been eliminated ˆ denotes the AZ2 (by suitable A-changes). (Also recall that if cod(AZ2 , S) codimension of Sˆ in the space of Z2 -symmetric germs and Sˆ = f ◦ (¯ x, y¯, λ2 , v) ˆ = cod(A(H), f).) The main result of the present section is then cod(AZ2 , S) then the following. Theorem 6.3.1. For a residual subset of Imm(R2 , Rn ), n ≥ 3 the germ of the secant map Sˆ : R4 → Rn−1 at a point on the diagonal is equivalent to one of the Z2 stable germs in Table 6.1. The projection-germs πx ◦ X associated with the Sˆ are also listed (up to A-equivalence).

142

6. Singularities of secant maps

n

ˆ x, y¯, λ, v) S(¯

AZ2 -cod

πx ◦ X ∼A

A-cod

3

(v, x ¯) ∼AZ2 (v, y¯ + x ¯2 ) ∼AZ2 (v, y¯ + x ¯3 ) ∼AZ2 (v, y¯ + x ¯4 ± x ¯6 )

0

(y, x2 ) (y, xy + x3 ) (y, xy + x4 ) (y, xy + x5 ± x7 )

1 2 3 4

(v, λ2 + x ¯2 ± y¯2 )

2

(y, x3 ± xy 2 )

3

(v, λ2 + v y¯ + x ¯2 + y¯3 )

3

(y, x3 + xy 3 )

4

(v, λ4

vλ2

+

4

+

x ¯3

+x ¯y¯)

(v, x ¯, y¯) s±

s+

0

(v, x ¯, ±λ2

y¯2 )

= + 4 ∼AZ2 = (v, y¯ + x ¯ , λ2 + x ¯2 ) ∼AZ2 (v, y¯ + x ¯7 , λ2 + x ¯2 ) (v, x ¯, λ2 + v y¯ + y¯3 ) (v, x ¯, λ4

± (y, xy + x5 , x3 ) (y, xy + x8 , x3 )

3 4 5

2

(y, x2 , x3 + xy 3 )

4

±

xy 2 )

4

±

xy 4 )

5

(v, x ¯, v y¯ + x ¯y¯2 + y¯3 ± y¯λ)2

3

(y, x2 , xy 3 ± x3 y)

5

3

(y, x2 , xy 2

5

+

vλ2

±

x ¯y¯2 λ6

±

x ¯ λ4 )

+

(v, x ¯, y¯, λ2 )

±

x7 )

0

(y, x2 , xy, x3 )

3

1

(y, x2 , xy, x5 )

4

2

(y, x2 , xy, x7 )

5

2

(y, x2 , x3 , xy 2 )

5

3

(y, x2 , xy, x9 )

6

3

(y, x2 , x3 ± xy 2 , xy 3 )

6

(v, x ¯, y¯, λ2 , 0)

0

(y, x2 , xy, x3 , 0)

4

(v, x ¯, y¯, λ4 + x ¯λ2 , vλ2 )

2

(y, x2 , xy, x5 , 0)

6

3

(y, x2 , xy 2 , x3 , 0)

7

0

(y, x2 , xy, x3 , 0, 0)

5

3

(y, x2 , xy, x5 , 0, 0)

8

0

(y, x2 , xy, x3 , 0, . . . , 0)

n−2

(v, x ¯, λ2 (v, x ¯, y¯, λ8

+

vλ2 )

+

+

vλ6

v y¯, y¯2 ) +

x ¯ λ4

+

y¯λ2 )

(v, x ¯, λ2 ± y¯2 , v y¯ + y¯3 + x ¯y¯2 )

(v, x ¯, y¯2 , λ2

+ v y¯, x ¯y¯)

(v, x ¯, y¯, λ2 , 0, 0) (v, x ¯, y¯, λ4

≥8

1

xy 2 )

3

+ v y¯ +

±

(v, x ¯, y¯, λ6 + vλ4 + x ¯ λ2 )

7

2

(y, x2 , x3

(y, x2 , x3

(v, x ¯, y¯, λ4

6

(y, x2 , xy)

4

2

(v, x ¯, y¯2 5

+

x5 )

y¯4 )

+

y¯2 )

+

x4

(y, x2 , x5

(v, x ¯, λ2

vλ2

3

(y, x2 y

+

vλ2 , x ¯λ2 , y¯λ2 )

(v, x ¯, y¯, λ2 , 0, . . . , 0)

Table 6.1: Generic secant germs for immersions in Rn

6.3. Secant maps of generically immersed surfaces

143

Remark 6.3.2. 1. Projecting an immersion-germ X : R2 , p → Rn , X(p) along a direction w ∈ TX(p) X(R2 ) always gives rise to a singular projection-germ of A-codimension at least n − 2. Furthermore, in a 3-parameter family of such projection-germs, varying with (p, w) ∈ R2 × P1 , we generically expect that n − 2 ≤ cod(A, πx ◦ X) ≤ n + 1. Comparing the A-classes of projections πx ◦ X in the theorem (which correspond to secant-germs Sˆ of generic immersions) with the existing classifications of A-orbits of maps R2 → Rn−1 we find that for n 6= 5 all A-classes [f ] with n − 2 ≤ cod(A, f) ≤ n + 1 arise as the A-class ˆ for n = 3 we get the of a projection πx ◦ X associated with some S: A-classes 2, 3, 4k (k = 2, 3), 5, 6 and 115 from [101] and for n = 4 we get Sk (k = 0, . . . , 4), Bk (k = 2, 3), C3 and Hk (k = 2, 3) from [89]. Finally, one checks that in higher dimensions n ≥ 6 this is also true (the normal forms for πx ◦ X in the table represent the only A-orbits of A-codimension between n − 2 and n + 1). For n = 5 we get Ik (k = 1, . . . , 4), II2 , and III2,3 from [63], but there is no Z2 -stable secant germ Sˆ whose associated projection πx ◦ X is A-equivalent to VII1 (and cod(A, VII1 ) = 6, see [63]). 2. For n = 3 there are AZ2 -stable germs that cannot be equivalent to ˆ The AZ2 -stable germs (v, λ2 − x any germ of a secant-map S. ¯2 − y¯2 ), (v, λ2 + v y¯ − x ¯2 + y¯3 ) and (v, λ4 + vλ2 + ¯ x2 + ¯ y 2 ) ( = ±1) which, over C, are AZ2 -equivalent to the representatives Sˆ of the third, fourth and fifth AZ2 -orbit in the table, respectively, are not equivalent to the germ ˆ The reason in all three cases is that an ax3 term in the last of any S. component of the immersion-germ X yields an a(λ2 + 3¯ x2 )/4 term in ˆ the last component of S. Proof of Theorem 6.3.1. We consider map-germs f (¯ x, y¯, u, v) (up to A(H)equivalence) whose composition with (¯ x, y¯, u, v) 7→ (¯ x, y¯, λ2 , v) yield Z2 symmetric map-germs. Theorem 6.4.2 in Section 6.4 contains the classification of A(H)-stable germs f : R4 , 0 → Rn−1 , 0, n ≥ 3. Proposition 6.2.2 implies that for a residual set of immersion-germs X : R2 → Rn , n ≥ 4, the associated germs f are A(H)-stable. Furthermore, for n = 3 (codimension-1 immersions) the proof of Proposition 6.2.2 implies that for a residual set of immersions f is an A(H)-stable germ of rank 2 or a germ of lower rank. From the classific-

144

6. Singularities of secant maps

ation in Theorem 6.4.2 we have that for n − 1 = 2 there are three real stable A(H)-orbits of rank 1 with representatives fi = (v, u2 + vu + 1 x ¯2 + 2 y¯2 ), where (1 , 2 ) = (+1, +1) for i = 1, (−1, −1) for i = 2 and (+1, −1) for i = 3, and all germs of rank less than two lie in the closure of these orbits. Now one checks that the jet-map Γ2 (defined in the proof of Proposition 6.2.2) has empty intersection with the closures of j 2 A(H) · fi , i = 1, 2 and that 2 3 Γ−1 2 (A), where A is the closure of j A(H) · f3 , is a submanifold of J (2, 3) × R (3) (3) (3) 2 defined by a20 = a11 = a30 = 0 (notice that f3 ∼A(H) (v, u + vu + x ¯y¯)). 2 Furthermore, the Γ2 -preimages of the orbits of j A(H)-codimension greater than three in the closure of j 2 A(H) · f3 have codimension greater than three (or are empty). Hence we conclude that in all cases (i.e. the ones covered by Proposition 6.2.2 – or its proof – and also for n = 3 and rank ≤ 1) we ˆ for obtain A(H)-stable map-germs f (and hence AZ2 -stable secant maps S) a residual set of immersion-germs. Finally, we have to consider the following two points: (i) we want to see ˆ which A(H)-stable germ R4 → Rn−1 can be realized as a germ f (or S) of an immersion and (ii) we want to decompose the A(H)-orbits of f into A-invariant strata of πx ◦ X. For (i) we consider the list of A(H)-stable map-germs f in Theorem 6.4.2 and check whether the Γk -preimage of a ksufficient orbit A(H) · f is non-empty. In the normal forms of Theorem 6.4.2 the variables x1 and u always correspond to v and u in f (or to v and λ2 in ˆ One finds that for n ≥ 4 all these Γk -preimages are non-empty, and for S). n = 3 the Γk -preimages are empty in the cases mentioned in Remark 6.3.2 (2). For (2) we simultaneously consider two jet-maps: the map Γk : × R1 → J k (4, n − 1), sending (j 2k+1 X, v) to j k f , and a second ˜ 2k+1 : J 2k+1 (2, n) × R1 → J 2k+1 (2, n − 1), sending (j 2k+1 X, v) to map Γ j 2k+1 (π(1,v) ◦ X). Here π(1,v) denotes the projection onto a hyperplane in Rn orthogonal to the (1, v) in the tangent plane TX(p) X(R2 ). (Recall that we actually fix the direction (1, v) = (1, 0), the x-direction, in the tangent plane of X(x, y) = (x, y, g3 , . . . , gn ) at (0, 0) and instead rotate the surface about an axis orthogonal to the (x, y)-plane, but the transversality arguments are perhaps clearer if we consider varying directions (1, v).) Using the correspondence of pairs consisting of germs of secant maps Sˆ = f (¯ p, λ2 , v) at (¯ p, v) J 2k+1 (2, n)

6.3. Secant maps of generically immersed surfaces

145

and projection germs π(1,v) ◦ X from Remark 6.2.3, we obtain submanifolds k 2k+1 ˜ −1 Γ−1 A · (π(1,v) ◦ X)) ⊂ J 2k+1 (2, n) × R1 , k (j A(H) · f ) ∩ Γ2k+1 (j

and such submanifolds of codimension greater than three will generically have ˆ πx ◦ X in Table 6.1 empty intersection with (j 2k+1 X, v)(R3 ). For the pairs S, we find that the codimensions of these submanifolds are given by the sum of the AZ2 -codimension of Sˆ and the difference of the A-codimensions of πx ◦ X and of the least degenerate projection-germ corresponding to some secant ˆ map-germ in the AZ2 -orbit of S. The projection-germs πx ◦ X distinguish all the AZ2 -orbits of secant mapgerms in the classification above. In some cases there is also a relation between Sˆ and curvature properties of X(R2 ). Remark 6.3.3. For n = 3 the last three AZ2 -orbits of secant map-germs Sˆ occur at parabolic points of the surface X(R2 ) and the direction (1, v) = (1, 0) is an asymptotic direction. The last orbit occurs, in fact, at points of tangency between the parabolic and the flecnodal curve (such points of tangency are called godron- or gutter-points and correspond to cusps of the Gauss map, see e.g. [62]). For the first AZ2 -orbit there is no restriction on the Gaussian curvature (here the degenerations of the corresponding projection-germs occur for asymptotic directions (1, v) = (1, 0) at a hyperbolic point). For n = 4 the AZ2 -type of Sˆ imposes no restriction on the second fundamental form of X(R2 ). Only for the degenerations of the projection-germs corresponding to the secant map-germ s+ we obtain such a restriction. These occur at parabolic points p that are not inflection points (the invariant ∆ introduced by Little [72] vanishes at p, but the 3 × 2 matrix αX of the second fundamental form has rank 2) and such p have negative Gaussian curvature. For n = 5, consider the 3×3 matrix αX of the second fundamental form of a surface X(R2 ) and the sets Mi := {p : rankαX (p) = i}. Mochida et al. [88] have shown that generic surfaces consist of regions of M3 points and curves of M2 points, but the AZ2 -types of Sˆ in our classification can occur at M3 as well as at M2 points.

146

6.4

6. Singularities of secant maps

Classification of Z2 stable map-germs R4 → Rn

Let f : Rn , 0 → Rp , 0 be a smooth Z2 -equivariant germ. We wish to classify the AZ2 -stable germs, where AZ2 = L×RZ2 (i.e., Z2 -equivariant diffeomorphisms in the source and arbitrary diffeomorphisms in the target). By an observation of Arnol’d [2], the AZ2 classification of Z2 -symmetric map-germs f (x1 , . . . , xn−1 , y) = f (x1 , . . . , xn−1 , −y) over C coincides with the A(H)classification of germs f (x1 , . . . , xn−1 , u), where A(H) = L × R(H) and where R(H) is the group of diffeomorphisms preserving the hyperplane H = {u = 0} in the source (which are of the form k = (k1 (x, u), . . . , kn−1 (x, u), u · kn (x, u)). Substituting u = y 2 into the normal forms of the latter classification gives the desired Z2 -symmetric germs. Over R the last component of the source diffeomorphism k has to satisfy the extra condition kn (0, 0) > 0 (so that k preserves the set {(x1 , . . . , xn−1 , u) : u > 0}). We denote the local rings of smooth source and target functions by Cn and Cp , respectively, and Mn and Mp are the corresponding maximal ideals. Let θf denote the Cn -module of vector fields over f (i.e., sections of f ∗ T Rp ). Set θn = θ(1Rn ) and θp = θ(1Rp ), and define homomorphisms tf and wf : tf : θn → θf ,

tf (ξ) = df · ξ,

(where df is the differential of f ), and wf : θp → θf ,

wf (φ) = φ ◦ f.

The tangent spaces to the groups G = A, K, R and C are then defined in the usual way (see [113]). Hence we shall only indicate the required modifications for the subgroup A(H) of A (where we have to restrict tf to θn (H), see below). Apart from A(H) we need the the group K(H) = C · R(H) (semi-direct product), both groups are so-called geometric subgroups of A and K so that the usual unfolding and determinacy results hold (see Damon [22]). For any such group G we denote, as usual, by Ge and G1 the extended pseudo-group (of non-origin preserving diffeomorphisms) and the subgroup of diffeomorphisms with 1-jet the identity. For calculations of complete transversals (see [19]) and of determinacy degrees we use the notation H for a unipotent subgroup

6.4. Classification of Z2 stable map-germs R4 → Rn

147

of A(H) (H can contain certain elements of A(H) \ A(H)1 ). Since A(H)equivalence is much finer than A-equivalence, we often have to work with bigger unipotent groups H than A(H)1 , even for stable germs f – frequently one also has to use the whole group A(H) (which is not unipotent) and apply Mather’s lemma [81]. Combining the determinacy results in [17, 35] we get k+1 the following useful criterion: if Mln θf ⊂ T K(H)e ·f +Ml+1 n θf and Mn θf ⊂ T H · f + Mk+l+1 θf then f is k-H- and hence k-A(H)-determined. Recall that n T R(H)e · f = tf (θn (H)), where θn (H) is the Cn -module of source vectorfields P tangent to H (i.e. θn (H) 3 ξ = n−1 i=1 ai (x, u)∂/∂xi + u · b(x, u)∂/∂u, where ai , b ∈ Cn ). The following relation is also useful in the classifications below. Lemma 6.4.1. For an A(H)-finite map-germ f : Rn , 0 → Rp , 0 we have the following relation between codimensions of orbits: cod(A(H)e , f) = max[0, cod(A(H), f) − n + 1]. Proof (Sketch). The argument is almost the same as that for the analogous formula for ordinary A-equivalence, see for example Proposition 4.5.2 (ii) of [113]. For A(H)-stable germs f the formula holds trivially, hence suppose f unstable. In that case the formula is equivalent to
dim T A(H)e · f /T A(H) · f = n + p − 1, which in turn is equivalent to: if α ∈ θn (H) and β ∈ θp are such that tf (α) + wf (β) ∈ T A(H) · f then β ∈ Mp · θp and π(α) ∈ Mn · θn−1 , where π is the projection onto the first n − 1 components (this makes the difference to the usual formula for A). The proof now concludes as in the A-case. The classification of A(H)-stable map-germs with source dimension four is given by the following Theorem 6.4.2. Any A(H)-stable map-germ f : R4 , 0 → Rn , 0, where n ≥ 2, is equivalent to one of the germs in Table 6.2. (Here i := ±1 and the cases (1 , 2 ) = (+1, −1), (−1, +1) are equivalent.) Remark on Proof. By Lemma 6.4.1 one obtains the A(H)-stable germs (of A(H)e -codimension 0) by classifying the A(H)-orbits of germs R4 , 0 → Rn , 0

148

6. Singularities of secant maps n= 2

f (x1 , x2 , x3 , u) = (x1 , x2 ) (x1 , u + 1 x22 + 2 x23 ) (x1 , u + x1 x2 ± x23 + x32 ) (x1 , u2 + x1 u + 1 x22 + 2 x23 )

cod(A(H), f) 0 2 3 3

3

(x1 , x2 , x2 ) (x1 , x2 , u ± x23 ) (x1 , x2 , u + x1 x3 + x33 ) (x1 , x2 , u2 + x1 u ± x23 ) (x1 , x2 , u + x1 x3 + x2 x23 ± x43 ) (x1 , x2 , x1 x3 + x2 x23 + x33 + x3 u) (x1 , x2 , x23 + x1 u ± u2 )

0 1 2 2 3 3 3

4

(x1 , x2 , x3 , u) (x1 , x2 , x3 , u2 + x1 u) (x1 , x2 , x3 , u3 + x1 u + x2 u2 ) (x1 , x2 , u + x1 x3 , x23 ) (x1 , x2 , x3 , u4 + x1 u + x2 u2 + x3 u3 ) (x1 , x2 , u ± x23 , x1 x3 + x33 + x2 x23 )

0 1 2 2 3 3

5

(x1 , x2 , x3 , u, 0) (x1 , x2 , x3 , u2 + x2 u, x1 u) (x1 , x2 , x23 , u + x1 x3 , x2 x3 )

0 2 3

6

(x1 , x2 , x3 , u, 0, 0) (x1 , x2 , x3 , u2 + x1 u, x2 u, x3 u)

0 3

(x1 , x2 , x3 , u, 0, . . . , 0)

0

≥7

Table 6.2: A(H)-stable germs R4 → Rn

of A(H)-codimension at most three. The group A(H) is a geometric subgroup of the group A (as defined by Damon [22]), the classification techniques are therefore the same as for A. Below we give an outline of this classification for the case n = 4, p = 3 (indicating the structure of the classification pattern of the A(H)-stable orbits, but omitting all calculations). For the other p ≥ 2 the classifications are similar (and often less extensive).

6.4. Classification of Z2 stable map-germs R4 → Rn

149

So for n = 4, p = 3 we have the following classification. Proposition 6.4.3. Any A(H)-stable map-germ f : R4 , 0 → R3 , 0 is equivalent to one of the following germs: (x1 , x2 , x3 ), (x1 , x2 , u ± x23 ), (x1 , x2 , u + x1 x3 + x33 ), (x1 , x2 , u + x1 x3 + x2 x23 ± x43 ), (x1 , x2 , x23 + x1 u ± u2 ), (x1 , x2 , x23 + x1 u + u3 + x2 u2 ) and (x1 , x2 , x3 u + x1 x3 + x33 + x2 x23 ). Outline of Proof. The elements of R(H) are germs of diffeomorphisms k of R4 preserving the hyperplane H = {u = 0}, which — by Hadamard’s lemma — have the form k = (k1 , k2 , k3 , uk4 ) with ki = ki (x1 , x2 , x3 , u) and k4 (0, 0, 0, 0) > 0. By Lemma 6.4.1 the A(H)-stable germs f : R4 , 0 → Rn , 0 are those with A(H)-codimension at most three. Let A(H)k := j k A(H) denote the Lie group of k-jets of elements of A(H). For n = 3, one finds (by direct A(H) coordinate changes) A(H)1 -orbits of codimension 0, 1 and 2, respectively, with the following representatives: (x1 , x2 , x3 ),

(x1 , x2 , u),

(x1 , x2 , 0).

(6.1)

The remaining A(H)1 -orbits lie in the closure of A1 · (x1 , u, 0) and have codimension at least 4 and can therefore be discarded. The first germ in (6.1) is 1-A(H)-determined, and a complete 2-transversal for the second germ σ is spanned by x23 , x1 x3 and x2 x3 in the last component. From the “general” 2-jet (x1 , x2 , u + ax23 + bx1 x3 + cx2 x3 ) we obtain over the 1-jet σ the A(H)2 -orbits represented by (x1 , x2 , u ± x23 ), f = (x1 , x2 , u + x1 x3 ) and (x1 , x2 , u). The first of these has codimension 1 and is 2-determined, the third has codimension 4, and the second has codimension 2 and has to be considered further. Over f we find three A(H)3 -orbits given by (x1 , x2 , u + x1 x3 + x33 ),

(x1 , x2 , u + x1 x3 + x2 x23 ),

(x1 , x2 , u + x1 x3 ).

The first has codimension 2 and is 3-determined, the third has codimension 4, and over the second (which has codimension 3) there is one A(H)4 -orbit that has codimension 3 (the others have higher codimension), which is 4determined and given by (x1 , x2 , u + x1 x3 + x2 x23 ± x43 ). This completes the classification of the A(H)-stable germs over the second 1-jet in (6.1). Finally, consider the third 1-jet σ = (x1 , x2 , 0) in (6.1). A complete 2transversal consists of all degree 2 monomials in the third component involving x3 or u. We consider two cases: the x23 coefficient is non-zero or zero.

150

6. Singularities of secant maps

In the first case we can reduce, up to A(H)2 -equivalence, to (x1 , x2 , x23 + au2 +bx3 u+cx1 u+dx2 y). Assuming that c and d are not both zero (otherwise the A(H)2 -codimension is greater than 3) we can reduce to the case where c = 1 and d = 0, and letting x3 7→ x2 − bu/2 we get: (x1 , x2 , x3 + x1 u + (a − b2 /4)u2 ). Hence we get the following three A(H)2 -orbits: (x1 , x2 , x3 ± u2 ) (these are 2-determined and have A(H)-codimension 2) and σ = (x1 , x2 , x23 + x1 u) (of A(H)2 -codimension 3). A complete 3-transversal for σ is given by u3 and x2 u2 in the third component, and when the coefficients of these two terms are both non-zero we can reduce to f = (x1 , x2 , x23 + x1 u + u3 + x2 u2 ). Some more substantial calculations then show that f is 3-determined and cod(A(H), f) = 3. The other A(H)3 -orbits over σ (for which the product of the two coefficients vanishes) have codimension at least four. In the second case (zero x23 coefficient) the A(H)2 -codimension is at least three, and in the best possible case the coefficients of x3 u and x1 x3 are nonzero (any degeneration would lead to A(H)2 -orbits of codimension at least four) so that we can reduce to the 2-jet σ = (x1 , x2 , x3 u + x1 x3 ). A complete 3-transversal for σ is given by x33 and x2 x23 in the third component, and using Mather’s lemma one shows one can reduce to f = (x1 , x2 , x3 u + x1 x3 + x33 + x2 x23 ) provided that the coefficients of both terms are non-zero. Finally one shows that f is 3-determined and cod(A(H), f) = 3. Acknowledgements . The work in this chapter has been carried out in part during the authors’ stay at the Banach Center in Warsaw (with the financial support of the EU Center of Excellence programme, IMPAN – Banach Center, ICAI-CT-2000-70024) and at the University of Sao Paulo at Sao Carlos (with financial support of a CAPES-DAAD project, 415-br-probral/poD/04/40407). We thank Maria Ruas for valuable discussions.

Appendix A

Appendix to Chapter 3

A.1

Divided differences and the Division Property

Recall that, for a real-valued function f defined on an interval I and points x0 , x1 , . . . , xn ∈ I, the n-th divided difference [x0 , . . . , xn ]f is defined as the coefficient of xn in the polynomial of degree n that interpolates f at x0 , x1 , . . . , xn . This definition is equivalent to the well-known recursive definition; see [29, Chapter 4] or [99, Chapter 5]. The interpolating polynomial can be written in the Newton form p(x) = f (x0 ) + (x − x0 ) [x0 , x1 ]f + · · · + (x − x0 ) · · · (x − xn−1 ) [x0 , . . . , xn ]f. (A.1) The n-th divided difference is well defined if the points are distinct. However, if f is sufficiently differentiable on I, then the n-th divided difference is also defined if some of the points coincide. More precisely, if f is a C n -function, then the n-th divided difference has the following integral representation, known as the Hermite-Genocchi identity: Z [x0 , x1 , · · · , xn ] f = f (n) (t0 x0 + t1 x1 + · · · + tn xn ) dt1 · · · dtn , Σn

Pn

where t0 = 1 − i=1 ti , and the domain of integration is the standard Σn = {(t1 , . . . , tn ) | t1 + · · · + tn ≤ 1, ti ≥ 0, for i = 0, 1, . . . , n}. For a proof we refer to [14, Chapter 1], [86] or [91]. The Hermite-Genocchi identity implies that [x0 , x1 , · · · , xn ] f is symmetric and continuous in (x0 , x1 , . . . , xn ). If f is a C m -function, with m ≥ n, this divided difference is a C m−n -function of (x0 , x1 , . . . , xn ). Furthermore, if xi = ξ for i = 0, . . . , n, then 1 (n) [ξ, . . . , ξ ] f = f (ξ). | {z } n! n+1

(A.2)

152

A. Appendix to Chapter 3

Furthermore, taking x0 = · · · = xn−1 = ξ, and xn = x, we see that 1 [ξ, . . . , ξ , x] f = | {z } (n − 1)! n

Z

1


(1 − u)n−1 f (n) (1 − u)ξ + ux du.

(A.3)

u=0

The key result used in this paper is the following ‘Newton development’ of a function f , akin to the Taylor series expansion. Lemma A.1.1. Let f : I → R be a C m -function defined on an interval I ⊂ R, and let x0 , . . . , xn−1 ∈ I. Then f (x) = f (x0 )+

n−1 X k−1 Y

(x−xi ) [x0 , . . . , xk ] f +

k=1 i=0

n−1 Y

(x−xi ) [x0 , x1 , · · · , xn−1 , x] f.

i=0

If m ≥ n, then [x0 , x1 , · · · , xn−1 , x] f is a C n−m -function of x. Furthermore, if x0 = . . . = xn−1 = ξ, then the preceding identity reduces to the Taylor expansion with integral remainder: n R1 P (x−ξ)k (k) n−1 f (n) (ux + (1 − f (x) = f (ξ) + n−1 f (ξ) + (x−ξ) k=1 k! (n−1)! u=0 (1 − u) u)ξ) du. The result follows from the observation that the polynomial p, defined by (A.1), interpolates f at x0 , . . . , xn , so in particular f (xn ) = p(xn ). Taking xn = x yields the first identity. The Taylor expansion follows using identities (A.2) and (A.3). Since [x1 , . . . , xk ] f = 0 if f (xi ) = 0, 1 ≤ i ≤ k, a straightforward consequence of Newton’s expansion (Lemma A.1.1) is the following. Lemma A.1.2 (Division Property). Let I ⊂ R be an interval containing points x1 , . . . , xn , not necessarily distinct, and let f : I → R be a C m -function, m ≥ n, having a zero at xi , for 1 ≤ i ≤ n. Then f (x) =

n Y (x − xi ) [x1 , . . . , xn , x] f. i=1

where the divided difference [x1 , . . . , xn , x] f is a C m−n -function of x.

A.2. Approximation of n-flat functions

A.2

153

Approximation of n-flat functions

In this section we derive error bounds for univariate real functions with multiple zeros at the endpoints of some small interval [0, r]. To stress that the error also depends on the size of the interval we consider a one-parameter family of functions (u, r) 7→ f (u, r), where r is a small positive parameter. We look for a bound of the error max0≤u≤r |f (u, r)|. To obtain asymptotic bounds for this error as r goes to zero, we assume that the function f is defined on a neighborhood of (0, 0) in R × R. Lemma A.2.1. Let I ⊂ R be an interval which is a neighborhood of 0 ∈ R. 1. Let f : I × I → R be a C m -function such that the function u 7→ f (u, r) has an n-fold zero at u = 0 and at u = r, with 2n + 2 ≤ m. Then 1 max0≤u≤r |f (u, r)| = 2n 2 (2n)!

2n 2n ∂ f 2n+1 ). ∂u2n (0, 0) r + O(r

2. Let f : I ×I ×I → R be a C m -function such that the function u 7→ f (u, s, r) has an n-fold zero at u = 0 and at u = r, and an additional single zero at u = s, with 2n + 3 ≤ m. Let δ(s, r) = max0≤u≤r |f (u, s, r)|. Then δ is a continuous function, and cn min0≤s≤r δ(s, r) = (2n + 1)! where cn =

2n+1 ∂ 2n+1 f r (0, 0, 0) + O(r2n+2 ), ∂u2n+1 nn 1

2n+1 (2n + 1)n+ 2

(A.4)

.

Moreover, the minimum in (A.4) is attained at s = s0 (r), where s0 is a C m−2n+1 -function, with s0 (0) = 21 .

154

A. Appendix to Chapter 3

Proof. 1. We prove that, for r > 0 sufficiently small, the function u 7→ f (u, r) has a unique extremum in the interior of the interval (0, r). According to the Division Property (see Appendix A.1, Lemma A.1.2), there is a C m−2n function F : I × I → R such that f (u, r) = un (r − u)n F (u, r). Observe that ∂ 2n f (0, 0) = (−1)n (2n)! F (0, 0). ∂u2n Note that the ‘model function’ g(u) = un (r − u)n F (0, 0) has its extreme 1 value 22n F (0, 0) on 0 ≤ u ≤ r at u = 12 r. We shall prove that the function f (u, r) has its extreme value at u = 21 r + O(r2 ). To this end we apply the ∂f Implicit Function Theorem to solve the equation (u, r) = 0. ∂u Since 0 ≤ u ≤ r, we scale the variable u by introducing the variable x such that u = rx, with 0 ≤ x ≤ 1, and observe that f (rx, r) = r2n f˜(x, r), with f˜(x, r) = xn (1 − x)n F (rx, r). Therefore, ∂ f˜ (x, r) = nxn−1 (1 − x)n−1 E(x, r), ∂x where E(x, r) = (1 − 2x) F (0, 0) + O(r), uniformly in 0 ≤ x ≤ 1. Since x 7→ ∂ f˜ (x, r) has an (n − 1)-fold zero at x = 0 and x = r, the Division Property ∂x allows us to conclude that E is a C m−2n+1 -function. Since E( 12 , 0) = 0, and ∂E ( 1 , 0) = −2F (0, 0) 6= 0, the Implicit Function Theorem tells us that there ∂x 2 ∂ f˜ (x(r), r) = 0. is a unique C m−2n+1 -function r 7→ x(r) with x(0) = 21 and ∂x ˜ Therefore, f (·, r) has a unique extremum at x = x(r). Hence, max0≤u≤r |f (u, r)| = |f˜(x(r), r)| r2n = |f˜( 1 , 0)| r2n + O(r2n+1 ) 2

|F (0, 0)| 2n r + O(r2n+1 ) 22n 2n ∂ f 2n 1 r + O(r2n+1 ). = 2n (0, 0) 2 (2n)! ∂u2n =

2. The proof of the second part goes along the same lines, but is slightly more complicated due to the occurrence of two critical points of the function f (·, s, r) in the interior of the interval (0, r). Again, the Division Property guarantees the existence of a C m−2n−1 -function F : I × I × I → R such that f (u, s, r) = un (r − u)n (s − u) F (u, s, r).

A.2. Approximation of n-flat functions

155

The ‘model function’ g(u) = un (r − u)n (s − u) F (0, 0, 0) has two critical points for 0 ≤ u ≤ r: one on the interval [0, s] and one on the interval [s, r]. The derivative of this function is of the form
g 0 (u) = un−1 (r − u)n−1 −(2n + 1) u2 + (2ns + n + 1) u − ns F (0, 0, 0). A straightforward calculation shows that g 0 has two zeros u± (s), and that the critical values of g at these zeros are equal iff s = 21 . In the remaining part of the proof we show that the function f (·, s, r) has its extreme values at u = u± (s) + O(r2 ), again by applying the Implicit Function Theorem to ∂f (u, s, r) = 0. solve the equation ∂u The critical values of f (·, s, r). Putting u = rx and s = ry, with 0 ≤ x, y ≤ 1, we obtain f (rx, ry, r) = r2n+1 f˜(x, y, r), with f˜(x, y, r) = xn (1 − x)n (x − y) F (rx, ry, r). To determine the critical points of x 7→ f˜(x, y, r) on the interval (0, 1), we observe that ∂ f˜ (x, y, r) = xn−1 (1 − x)n−1 Q(x, y, r), ∂x

(A.5)

where Q is a function of the form
Q(x, y, r) = −(2n + 1) x2 + (2ny + n + 1) x − ny F (0, 0, 0) + O(r), ∂ f˜ ∂ f˜ uniformly in x, y ∈ [0, 1]. Since is a C m−1 -function such that x 7→ (x, r) ∂x ∂x has (n − 1)-fold zeros at x = 0 and x = 1, the Division Property allows us to conclude that Q, determined by (A.5), is a C m−2n+1 -function. Assume F (0, 0, 0) > 0 (the case F (0, 0, 0) < 0 goes accordingly). Then, if 0 < y < 1, the function x 7→ f˜(x, y, 0) has one minimum at x = x0− (y) and one maximum at x = x0+ (y), where x0± are the zeros of the quadratic ∂Q 0 function x 7→ Q(x, y, 0). Since (x (y), y, 0) 6= 0, the Implicit Function ∂x ± Theorem guarantees the existence of C m−2n+1 -functions x± : I × I → R, with x− (y, r) < x+ (y, r), such that x± (y, 0) = x0± (y), and Q(x± (y, r), y, r) = 0. So, in view of (A.5), the function x 7→ f˜(x, y, r) has one minimum at x = x− (y, r), and one maximum at x = x+ (y, r). Putting ˜ r) = max0≤x≤1 |f˜(x, y, r)|, δ(y,

(A.6)

156

A. Appendix to Chapter 3

we see that
˜ r) = max |f˜(x− (y, r), y, r)|, |f˜(x+ (y, r), y, r)| . δ(y,

The minimax norm of the family {f (·, s, r) | s ∈ [0, r]}. For fixed x and r, with 0 < x < 1 and r > 0 sufficiently small, the function y 7→ f˜(x, y, r) is decreasing. See Figure A.1. This follows from the observation that ∂ f˜ (x, y, r) = −xn (1 − x)n E(x, y, r), ∂y with E(x, y, r) = F (0, 0, 0) + O(r), uniformly in x, y ∈ [0, 1]. Therefore, there is a %0 > 0 such that, for 0 ≤ r ≤ %0 , we have E(x, y, r) > 0 for 0 ≤ x, y ≤ 1, ∂ f˜ and hence (x, y, r) < 0. ∂y

Figure A.1: Graph of the function x 7→ f˜(x, y, r), for r fixed and y = y0 (solid), y = y1 (dashed), and y = y2 (dotted), with y0 < y1 < y2 .

From this observation it follows that, for fixed r and y ranging from 0 to 1, the graphs of the functions x 7→ f (x, y, r) are disjoint, except at their ˜ r) attains endpoints. See again Figure A.1. Therefore, the function y 7→ δ(y, its minimum iff ∆(y, r) = 0, where ∆(y, r) = f˜(x− (y, r), y, r) + f˜(x+ (y, r), y, r). Claim: There is a C m−2n+1 -function y0 , such that, for 0 ≤ r ≤ %0 : ∆(y, r) = 0 iff y = y0 (r).

A.2. Approximation of n-flat functions and y0 (r) =

1 2

157

+ O(r).

To prove this claim, we first prove that ∆( 21 , 0) = 0. To see this, observe that f˜(x, 12 , 0) = −f˜(1 − x, 12 , 0), so

∂ f˜ ∂ f˜ (x, 21 , 0) = (1 − x, 12 , 0). ∂x ∂x

Therefore, x+ ( 12 , 0) = 1 − x− ( 21 , 0), and hence ∆( 21 , 0) = 0. Since ∂∆ ∂ f˜ ∂ f˜ (y, 0) = (x− (y, 0), y, 0) + (x+ (y, 0), y, 0) < 0, ∂y ∂y ∂y the function y 7→ ∆(y, 0) has a unique zero at y = 21 . Furthermore, the Implicit Function Theorem guarantees the existence of a C m−2n+1 -function y0 with ∆(y0 (r), r) = 0, and y0 (0) = 12 . In view of (A.6) we have ˜ r) = |f˜(x± (y0 (r), r), y0 (r), r)| min0≤y≤1 δ(y, = |f˜(x± ( 1 , 0), 1 , 0)| + O(r) 2

2 n

= max0≤x≤1 |x (1 − x)n (x − 21 )| + O(r) = cn + O(r). ˜ r) = cn r2n+1 +O(r2n+2 ). The Finally, min0≤s≤r δ(s, r) = r2n+1 min0≤y≤1 δ(y, minimum is attained at s = s0 (r) = r y0 (r). Obviously, s0 is a C m−2n+1 function. This concludes the proof of the second part of the Lemma.

Appendix B

Appendix to Chapter 4

B.1

Claim in the proof of Theorem 4.3.2

A straightforward calculation shows that H 0 (ϕ) =


(1 + t0 ) sin ϕ2 √ t0 K0 (ϕ) + K1 (ϕ) , 2 2 D(ϕ) t0 − cos ϕ

where K0 (ϕ) = 4 − ϕ2 − 4 cos ϕ − ϕ sin ϕ,

and

K1 (ϕ) = −1 + cos 2ϕ + ϕ sin ϕ + ϕ2 cos ϕ. We now prove that both K0 and K1 are negative on the interval (0, π). (k)

To prove that K0 is negative on this interval, observe that K0 (0) = 0, (4) (4) for 1 ≤ k ≤ 4. Furthermore, K0 (ϕ) = −ϕ sin ϕ, so K0 is negative on this interval. Therefore, K0000 is decreasing, and hence also negative on the whole interval. Continuing this way we arrive at the conclusion that K0 is negative on (0, π). It is more complicated to prove that also K1 is negative on (0, π). Since 1 K1 (ϕ) = − 360 ϕ6 + O(ϕ8 ), we rewrite K1 as K1 (ϕ) = g1 (ϕ) + g2 (ϕ) + g3 (ϕ) + p(ϕ), where g1 , g2 and g3 are the deviations of cos 2ϕ, ϕ sin ϕ and ϕ2 cos ϕ from the lower order parts of their Taylor expansion at ϕ = 0 that include terms of order six, and such that these deviations are negative on the interval (0, π).

160

B. Appendix to Chapter 4

More precisely, g1 (ϕ) = cos 2ϕ −

4 X k=0

g2 (ϕ) = ϕ (sin ϕ −

1 (2ϕ)2k , (2k)!

2 X k=0

2

g3 (ϕ) = ϕ (cos ϕ −

1 ϕ2k+1 ), (2k + 1)!

2 X k=0

7 ϕ6 + p(ϕ) = − 180

2 315

1 (2ϕ)2k ), (2k)!

ϕ8 .

Note that g1 (ϕ) is the difference between cos 2ϕ and the terms of its Taylor (k) expansion at ϕ = 0 up to and including terms of order eight. Since g1 (0) = 0, (8) for 1 ≤ k ≤ 8, and g1 (ϕ) = 28 (cos(2ϕ) − 1) < 0, for 0 < ϕ < π, it follows that g1 is negative on (0, π). (k) (6) Since g2 (0) = 0, for 0 ≤ k ≤ 6, and g2 (ϕ) = −1 + cos ϕ − ϕ sin ϕ < 0, a similar argument shows that g2 is negative on the whole interval. (k) To prove that g3 is negative, first observe that g3 (0) = 0, for 0 ≤ k ≤ 6. (6) Furthermore, g3 (ϕ) = −30 (1−cos ϕ)−ϕ2 cos ϕ−12ϕ sin ϕ, which is certainly (6) negative for ϕ ∈ (0, π/2]. Rewriting g3 (ϕ) = −30+(30−ϕ2 ) cos ϕ−12ϕ sin ϕ, (6) we see that g3 is also negative on (π/2, π). Again the conclusion is that g3 is negative on the whole interval. √ Finally, the polynomial p is negative for 0 < ϕ < 47 2 = 2.47 . . . Since √ 3 7 3 4 π < 4 2, we already conclude that K1 is negative on the interval (0, 4 π]. 3 To prove that K1 is also negative on ( 4 π, π), note that K1 (ϕ) ≤ ϕ h(ϕ), where h(ϕ) = ϕ cos ϕ + sin ϕ. It is not hard to show that h is negative on ( 43 π, π), since on that interval h0 (ϕ) = 2 cos ϕ − ϕ sin ϕ < 0, so h is decreasing. √ Furthermore, h( 43 π) = 4−3π < 0, so h is negative. Therefore, K1 is also 4 2 negative on ( 43 π, π).

B.2

Intersecting planes in three-space

We determine a parametrization of the line of intersection of the planes Π0 and Π1 through the points p0 and p1 in R3 , respectively, with normals U0 and

B.2. Intersecting planes in three-space

161

U1 , in case these normals are independent vectors in R3 . Lemma B.2.1. Planes Π0 and Π1 intersect in the line p = p + λ U0 × U1 , where λ ranges over the real numbers, and where p is the point given by p = p0 +

hp1 − p0 , U1 i (−hU0 , U1 i U0 + || U0 ||2 U1 ). || U0 × U1 ||2

Proof. The planes intersect in a line consisting of the points p satisfying the system of linear equations hp − p0 , U0 i = 0,

(B.1)

hp − p1 , U1 i = 0.

Note that the vectors U0 , U1 and U0 × U1 form a basis of R3 . The direction of the line defined by (B.1) is U0 × U1 . The solution of (B.1) is of the form p = p(λ) = p0 + a U0 + b U1 + λ U0 × U1 ,

(B.2)

where λ is a real parameter, and a and b satisfy the following system of equations, obtained by substituting (B.2) into (B.1): ! ! ! hU0 , U0 i hU0 , U1 i a 0 = , hU0 , U1 i hU1 , U1 i b γ where γ = hp1 − p0 , U1 i. Therefore, ! ! 1 a || U1 ||2 −hU0 , U1 i = ∆ −hU0 , U1 i b || U0 ||2

0 γ

!

γ = ∆

! −hU0 , U1 i , || U0 ||2

(B.3)

where ∆ = || U0 ||2 || U1 ||2 −hU0 , U1 i2 = || U0 ×U1 ||2 . Substituting these values for a and b into (B.2) gives the parametrization of the line of intersection as stated in the lemma.

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Summary

In this thesis we study: (i) geometric approximation of curves in the plane and in space, and (ii) singularities of secant maps of immersed surfaces from a geometric perspective.

Geometric Curve Approximation. Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied are mostly two or three dimensional, although many of the tools and principles can be applied to sets in any finite dimension. Today geometric modeling is done with computers and for computer-based applications. Curve modeling is a part of geometric modeling and in the current thesis our focus is on problems related to approximation of parametric curves in the plane with conic arcs and biarcs, and in space with bihelical arcs. Biarcs are curves formed by joining two circular arcs in a tangent continuous fashion. Similarly, a bihelical arc is formed by joining two circular helices in a tangent continuous manner. Two curves are said to join in a tangent continuous fashion at a point if they meet at a point and the derivative of both curves at that point are parallel to each other. Approximating a parametric curve with some spline curve is always based on some metric with which we measure how close the original curve and its approximation are to each other. A spline curve in general is defined piecewise by some special class of functions. In this thesis we consider tangent continuous splines, e.g., a conic spline is made up of piecewise conic arcs, where two consecutive conic arcs join in a tangent continuous manner. In our study we primarily consider the Hausdorff distance and, in case of parabolic and conic arcs, we also consider approximation with respect to the symmetric

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difference distance. The Hausdorff distance in case of plane curves is defined as the maximum of the distance function between two curves: For every point on a given parametric curve we consider the Frenet-Serret frame and consider the distance between the given point and the point of intersection of the normal line at that point with the approximating curve. The symmetric difference distance between two curves sharing the endpoints is the area enclosed by the curves.

Example of a biarc

One-parameter family of biarcs

Figure 6.1: Example of a biarc formed by two circular arcs and a one-parameter family of bitangent biarcs.

Complexity. Our goal is to compute the complexity (minimum number of elements) of approximation of a sufficiently smooth curve, with non-vanishing curvature, in the plane with biarc, parabolic and conic splines. We also determine the complexity of approximation of space curves with bihelix splines. Circles are the only curves in plane with non-zero constant curvature, conics are the only curves in plane with constant affine curvature, where affine curvature is a differential invariant arising in affine differential geometry and is a notion similar to curvature. Furthermore, circular helices are curves in space with constant curvature and constant torsion. We exploit these properties of the curves in our computation of the Hausdorff distance and show that the approximation error improves by one order of magnitude in going from

Summary

177

biarc splines (third order) to parabolic splines (fourth order) to conic splines (fifth order). We also compute the leading term in the approximation error, and, hence, in the complexity of the approximating spline. Consequently the complexity can be expressed in terms of differential geometric quantities like curvature and affine curvature and for: • biarc splines depends on the derivative of the curvature, • parabolic splines depends on the affine curvature, and for conic splines depends on the derivative of the affine curvature, and for • bihelical splines depends on the derivative of the curvature.

One-parameter family of conics

Family of displacement functions

Figure 6.2: One-parameter family of bitangent conics and their corresponding displacement functions.

Geometry of Approximation. We consider the approximation of spiral arcs with bitangent biarcs. Spirals are curves in the plane with monotonically increasing/decreasing curvature. A bitangent biarc to a spiral arc is tangent to a spiral arc at its endpoints. We prove that there exists a one-parameter family of biarcs bitangent to a spiral arc. Furthermore, we show that in this one-parameter family of biarcs there exists a unique biarc which is closest to the curve with respect to the Hausdorff distance. Similar to the notion of spirals, there is the notion of affine spirals in the plane. Affine spirals are

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curves with monotonically increasing/decreasing affine curvature. We show that there exists a one-parameter family of bitangent conic arcs to a given affine spiral arc. Moreover, the distance function of this one-parameter family is bimodal as shown in Figure 6.2.

We prove that there exists a unique bitangent conic which is closest to the given affine spiral with respect to the Hausdorff distance. In this case the bimodal displacement function is equioscillatory in nature as shown by the red curve in Figure 6.2. In case of biarcs as well as conics we also prove that the displacement function and hence the Hausdorff distance function is monotonic, i.e., if we reduce the length of the given spiral or affine spiral arc the Hausdorff distance of its best approximating biarc or conic arc keeps decreasing respectively. Due to these properties of the approximating biarc and conic spline we are able to design an algorithm for computing a best approximating biarc or conic spline of a given spiral or affine spiral curve. In case of approximation of regular space curves, we prove a general result in Chapter 4 regarding the Hausdorff distance between a regular space curve and an approximating G1 -spline consisting of two smooth arcs which are tangent to each other at one of their endpoints (junction point) and the spline formed by these two smooth arcs is bitangent to the given space curve. Imposing the condition that the derivative of curvature of this kind of spline curve is zero at the endpoints shared with the curve, we obtain a lower bound on the Hausdorff distance between a curve and such a bitangent spline. Furthermore, we obtain an asymptotic characterization of the junction point and tangent at the junction point (junction tangent) for a bihelix spline where the lower bound of the Hausdorff distance is obtained. We present an algorithm using these characterizations of the junction point and the junction tangent and conclude that the experimentally measured complexity of the bihelix spline matches its theoretical complexity almost exactly. Nonetheless, in this case the problem of finding a bitangent bihelix which is closest to the curve is complex and still remains open.

Summary

179

Singularities of Secant Maps Singularity theory is a far-reaching generalization of the study of functions at its critical points. It provide some useful tools for the study of the geometry of curves and surfaces locally i.e., properties around a small neighborhood of a point lying on a curve or surface. Given a function f : I → R, x0 is called its critical point if f 0 (x0 ) = 0. A critical point or a singular point gives crucial information about the shape of the graph f (x). The maximal or minimal points of a function are its nondegenerate critical points. Moreover x0 is a degenerate critical point if f 0 (x0 ) = f 00 (x0 ) = 0. In Whitney’s theory functions are replaced by mappings, i.e., collections of several functions of several variables. Generically all critical points are non-degenerate, but degenerate critical points may occur in generic families of functions. The problem considered in this thesis is an extension of Bruce’s work where he studies the singularities of secant maps of curves in space. Given a space curve γ, which is a regular, smooth embedding from R to R3 , the projectivized secant map Sˆ : R × R → P2 maps a pair of distinct points t1 and t2 in R to an unoriented line onto a real projective space P2 . Furthermore, it maps two non-distinct points onto the corresponding unoriented tangent line. The goal is then to study the local behavior of the secant map for points on the diagonal ∆ = {(t, t) | t ∈ R}. In this thesis we study the secant maps and projectivized secant maps of a surface immersed in Rn , where n ≥ 3. A secant map acting on two points p, q on a surface, maps them to the line p − q. We discuss in detail the singularities of such maps and its projectivized counterpart in this thesis along the points on the diagonal.

Samenvatting

In dit proefschrift bestuderen we: (i) de meetkundige benadering van krommen in het vlak en de ruimte en (ii) singulariteiten van secant-afbeeldingen van ingedompelde oppervlakken vanuit een meetkundig oogpunt.

Meetkundige Benadering van Krommen Meetkundig modelleren is een vakgebied binnen de toegepaste wiskunde en computationele meetkunde waarin methoden en algoritmes worden bestudeerd voor de wiskundige beschrijving van vormen. De bestudeerde vormen zijn meestal twee- of drie-dimensionaal, maar veel van de concepten en technieken kunnen worden toegepast op verzamelingen met een willekeurige, maar eindige, dimensie. Tegenwoordig wordt meetkundig modelleren gedaan met behulp van computers en voor computertoepassingen. Een specifiek onderwerp binnen meetkundig modelleren is het benaderen van een gegeven kromme. In dit proefschrift bestuderen we problemen gerelateerd aan het benaderen van geparametriseerde krommen. In het vlak benaderen we krommen met twee-bogen en kegelbogen; in de ruimte benaderen we krommen met bihelische bogen. Twee-bogen worden gevormd door het verbinden van twee cirkelbogen zodanig dat de raaklijnen in het verbindingspunt samenvallen (geometrische continu¨ıteit). Bihelische bogen worden gevormd door het verbinden van twee cirkelvormige helices met geometrische continu¨ıteit. Een spline is een kromme die wordt gevormd door de aaneenschakeling van krommen uit een speciale klasse. In dit proefschrift beschouwen we geometrisch-continue splines. Een conische spline wordt bijvoorbeeld gevormd door een aaneenschakeling van bogen van kegelsneden (ellipsen, parabolen of hyperbolen) waarbij twee opeenvolgende stukken geometrisch-continu zijn verbonden. Met behulp van een metriek meten we hoe goed een spline een gegeven

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Samenvatting

geparametriseerde kromme benadert. In dit proefschrift beschouwen we voornamelijk de Hausdorff-afstand, maar in het geval van paraboolbogen en meer algemene bogen van kegelsneden bekijken we ook de afstand gedefini¨eerd door het symmetrische verschil. De Hausdorff-afstand tussen twee krommen in het vlak is gedefini¨eerd als het maximum van de afstand tussen de twee krommen: voor ieder punt op een gegeven geparametriseerde kromme beschouwen we het Frenet-Serret-driebeen en berekenen we de afstand tussen het gegeven punt en het snijpunt van de normaallijn met de benaderende kromme. De afstand gedefini¨eerd door het symmetrische verschil van twee krommen met samenvallende eindpunten is gedefini¨eerd als de oppervlakte ingesloten door de twee krommen.

Voorbeeld van een twee-boog

E´en-parameter familie van tweebogen

Figure 6.3: Voorbeeld van een twee-boog gevormd door twee cirkelbogen en een ´e´en-parameter familie van bitangenti¨ele twee-bogen.

Complexiteit. We benaderen een voldoende gladde kromme in het vlak, waarvan de kromming nergens nul is, met behulp van twee-bogen, paraboolen kegelsplines. Voor ruimtelijke krommen gebruiken we bihelische splines. In beide gevallen is het doel om de complexiteit van de benadering, dat wil zeggen het minimale aantal elementen, te berekenen. Cirkels zijn de enige vlakke krommen met een constante kromming ongelijk

Samenvatting

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aan nul; bogen van kegelsneden zijn de enige vlakke krommen met een constante affiene kromming. (Affiene kromming is het analogon van kromming in de affiene differentiaalmeetkunde.) Bovendien zijn cirkelvormige helices ruimtelijke krommen met een constante kromming en torsie. Deze eigenschappen worden gebruikt bij het berekenen van de Hausdorff-afstand tussen twee krommen. We tonen aan dat de benaderingsfout verbetert met ´e´en orde van grootte bij de overstap van twee-boog splines (derde orde) naar paraboolsplines (vierde orde) en naar kegelsplines (vijfde orde). We berekenen de leidende term van de benaderingsfout en daarmee ook van de complexiteit van de benaderende spline. Hiermee kan de complexiteit worden uitgedrukt in termen van (affiene) kromming. De complexiteit hangt af van: • de afgeleide van de kromming in het geval van twee-boog splines en bihelische splines; • de affiene kromming en de afgeleide daarvan in het geval van respectievelijk paraboolsplines en kegelsplines.

E´en-parameter familie van kegelbogen

Familie van verplaatsingsfuncties

Figure 6.4: E´en-parameter familie van bitangenti¨ele kegelbogen en hun bijbehorende verplaatsingsfuncties.

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Samenvatting

Meetkunde van benaderingen. We beschouwen de benadering van spiraalbogen met bitangenti¨ele twee-bogen. Spiralen zijn vlakke krommen met een monotoon dalende of stijgende kromming. Een twee-boog benadering van een spiraalboog heet bitangenti¨eel als de twee-boog de spiraalboog in de twee eindpunten raakt. We bewijzen dat er een ´e´en-parameter familie van tweebogen bestaat waarvan elke twee-boog bitangenti¨eel is aan de spiraalboog. Verder bewijzen we dat binnen deze familie er een unieke twee-boog is die de Hausdorff-afstand tot de gegeven spiraalboog minimaliseert. Analoog aan spiralen zijn affiene spiralen vlakke krommen met een monotoon dalende of stijgende affiene kromming. We bewijzen het bestaan van een ´e´en-parameter familie van kegelbogen waarvan elke kegelboog bitangenti¨eel is aan de gegeven affiene spiraalboog. Bovendien is de afstandsfunctie van deze familie bimodaal zoals weergegeven in Figuur 6.4. We bewijzen dat er een eenduidige bitangenti¨ele kegelspline bestaat die de Hausdorff-afstand tot de gegeven affiene spiraalboog minimaliseert. In dit geval heeft de bimodale verplaatsingsfunctie een equioscillatie-eigenschap zoals weergegeven door de rode kromme in Figuur 6.4. In het geval van zowel twee-bogen als kegelsplines bewijzen we ook dat de offestfunctie, en dus de Hausdorff-afstandsfunctie, monotoon is: als de lengte van de gegeven spiraalboog (affiene spiraalboog) afneemt, dan neemt de Hausdorff-afstand tot de best benaderende twee-boog (kegelspline) af. Deze bijzondere eigenschappen stellen ons in staat om een algoritme te ontwerpen voor het berekenen van de beste benadering van een gegeven (affiene) spiraalboog. In Hoofdstuk 4 bewijzen we een algemeen resultaat voor de Hausdorffafstand tussen een ruimtelijke kromme en een benaderende G1 -spline. De G1 -spline bestaat uit twee gladde bogen die elkaar raken in het verbindingspunt, en is bitangenti¨eel aan de gegeven kromme. Onder de voorwaarde dat de afgeleide van de kromming van de G1 -spline in de eindpunten gelijk is aan nul, krijgen we een ondergrens voor de Hausdorff-afstand tussen de gegeven kromme en de benaderende spline. Voor een bihelische spline waarvoor de ondergrens van de Hausdorff-afstand wordt bereikt, verkrijgen we een asymptotische karakterisering van het verbindingspunt en de raaklijn in het verbindingspunt. We presenteren een algoritme dat gebruik maakt van deze

Samenvatting

185

karakteriseringen en concluderen dat de experimenteel gemeten complexiteit van de bihelische spline en de theoretische complexiteit bijna overeenstemmen. Het vinden van de beste benadering van een gegeven kromme met een bitangenti¨ele bihelische spline is echter nog een open probleem.

Singulariteiten van Secant-afbeeldingen Singulariteitentheorie is een een verregaande generalisatie van de studie van functies in hun kritieke punten. Deze theorie geeft bruikbare technieken voor het bestuderen van de lokale meetkunde van krommen en oppervlakken, dat wil zeggen: de eigenschappen in een omgeving van een punt op de kromme of het oppervlak. Een punt x0 heet een kritiek punt van een functie f : I → R als f 0 (x0 ) = 0; een kritiek punt heet ontaard als ook f 00 (x0 ) = 0. Een kritiek punt geeft belangrijke informatie over de vorm van de grafiek van de functie f . In de theorie van Whitney worden functies vervangen door afbeeldingen, dat wil zeggen collecties van functies van meerdere variabelen. In het algemeen zijn kritieke punten niet-ontaard, maar ontaarde kritieke punten kunnen wel in generieke families van functies voorkomen. Het probleem dat we bestuderen in dit proefschrift is een uitbreiding van het werk van Bruce waarin hij de singulariteiten van secant-afbeeldingen van ruimtelijke krommen bestudeert. Bij een ruimtelijke kromme γ, welke een reguliere, gladde inbedding van R naar R3 geeft, beeldt de geprojectiviseerde secant-afbeelding Sˆ : R × R → P2 een paar punten t1 6= t2 in R af op een niet-geori¨enteerde lijn in het projectieve vlak P2 . Twee samenvallende punten worden afgebeeld op de niet-geori¨enteerde raaklijn. Het doel is het bestuderen van het lokale gedrag van de secant-afbeelding voor punten op de diagonaal ∆ = {(t, t) | t ∈ R}. We bestuderen (geprojectiviseerde) secant-afbeeldingen van een oppervlak dat ge¨ımmerseerd is in Rn voor n ≥ 3. Een secant-afbeelding beeldt twee punten p en q op een oppervlak af op de lijn p − q. Van dergelijke afbeeldingen en hun geprojectiviseerde tegenhangers bespreken we in detail de singulariteiten op de diagonaal.