Applications in Computer-Aided Ship Design

NATIONAL TECHNICAL UNIVERSITY OF ATHENS

DEPARTMENT OF NAVAL ARCHITECTURE AND MARINE ENGINEERING

Fairing Methods of Planar and Space Curves under Design Constraints Applications in Computer-Aided Ship Design by

Konstantinos G. Pigounakis Submitted in Partial Ful lment of the Requirements for the Degree:

Doctor of Engineering

Thesis Supervisor: Panagiotis D. Kaklis, Associate Professor

Athens, February 1997

Fairing Methods of Planar and Space Curves under Design Conditions Applications in Computer-Aided Ship Design by

Konstantinos G. Pigounakis Submitted in Partial Ful lment of the Requirements for the Degree:

Doctor of Engineering Advisory Committee Panagiotis Kaklis, Associate Professor (Supervisor) Apostolos Papanikolaou, Professor Gerasimos Politis, Assistant Professor

Examination Committee Vasilios Dougalis, Professor { National and Capodistrean University of Athens Panagiotis Kaklis, Associate Professor { National Technical University of Athens Theodoros Loukakis, Professor { National Technical University of Athens Apostolos Papanikolaou, Professor { National Technical University of Athens Gerasimos Politis, Assistant Professor { National Technical University of Athens Panagiotis Sakkalis, Assistant Professor { Agricultural University of Athens Stylianos Stamatakis, Assistant Professor { Aristotle University of Thessaloniki Athens, February 1997

To my parents, George and Ee.

Abbriviations 2D / 3D CAD CAGD CAM DW FEM Ff FOB / FOs GM LCB / LCF NLP NTUA NURBS OK PDE QP SAC SLC SQL St Wl

Two-/Three-Dimensional Computer-Aided Design Computer-Aided Geometric Design Computer-Aided Manufacturing Design Waterline Finite Element Method Frenet frame Flat of Bottom / Side Metacentre, Metacentric height Longitudinal Centre of Buoyancy /Flotation Non-Linear Programming National Technical University of Athens Non-Uniform Rational B-Splines O ending Knot Partial Di erential Equations Quadratic Programming Stational Area Curve Stational Leverarm Curve Sequential Quadratic Programming Station Waterline

List of Symbols 6

(a b)

a?b ab ab a or da=ds a_ or da=du (da=dt) jaj ja b cj (a b c) (a b) kak r 0

A

Anj k

AW AX j B=BW

b

Cr CB CP CW CX

D

Smaller angle between two vectors, a and b a is vertical to b Inner product of a and b Outer or cross product of a and b Di erentiation of a with respect to arc length Di erentiation with respect to the parameter Absolute value of a Determinant of three vectors, a, b, and c Plane determined by the point vectors, a, b, and c Plane determined by the free vectors, a and b Euclidean norm of vector a Displacement Area de ned by a curve and the x;axis Parameter-dependent terms of L for a B ;spline of degree n Waterplane area Maximum sectional area Parameter-dependent coecients of Q_ within a segment Breadth of ship (moulted) / Breadth at the waterline Unit binormal vector (or binormal vector) Order of continuity r Block coecient Prismatic coecient Waterplane coecient Maximum section coecient Control polygon of a B ;spline curve

D

;

di = (dxi dyi dzi )T hi Ixx Iyy J , J 0 , Jr

K k

k kmax

L

LOA=LW LCB LCF j limx x0 Mxx Myy Mj (t) Njn (u) Njn (r) (u) !

n

Depth of ship Di erence symbol (e.g., dj 1 = dj ; dj 1) Vertex i of the control polygon Global tolerance Absolute torsion discontinuity at ui Sum of hi for a cubic spline curve Maximum permissible value for Inertias of area with respect to the x; and y;axis Fairing functionals Curvature vector Signed curvature k;th iteration, maximum number of iterations Curvature (absolute value) Numerator of the signed curvature Overall length of ship / Length at the waterline Longitudinal centre of buoyancy Longitudinal centre of otation Parameter-dependent coecients of L for a cubic B ;spline Limit of variable x going to the value x0 Moments of area with respect to the x; and y;axis j ;th transformed basis function of degree 3 j ;th B ;spline basis function of degree n

o() P Q(t) Q0 (t)

;

r;th parametric derivative of Njn (u) Unit principal normal vector (or normal vector) Sum of nodal sign changes of torsion, wi Lower-case Landau order symbol Problem reference Spline (B ;spline) curve with components Qx(t) Qy (t) Qz (t) Initial spline curve

Q (t)

IR

R

ri i = (1 0 0) S(t) s(t) Pk1 j =k0 R x1 x0

T

T

t

max U = fu0 u1 ::: ung ui wi zi zik ziK

Changed (faired) spline curve The set of real numbers Ratio of discontinuities and fairness metrics Tolerance radius, corresponding to the i;th control vertice Desired direction of signed curvature in i;th segment Parametric curve in IR3 with components x(t) y(t) z(t) Arc length of a parametric curve Sum of the index j between the values k0 and k1 Integral between the values x0 and x1 Local parametrization Draught of ship Unit tangent vector Torsion Absolute maximum value of torsion Parametrization or knot vector i;th knot Torsion sign changes at nodal points, ui Magnitude of discontinuity at ui for , or k , or K , respectively Sum of zi Maximum permissible value for 0

0

0

Abstract The present work can be classi ed in the areas of Computer Aided Geometric and Ship Design (CAGD - CAShipD). New automatic methods for fairing planar and spatial curves, which can be subjected to geometric and/or general design constraints, are examined. The methods handle low degree polynomial spline curves of at least second order derivative continuity (C 2): The curves are given through the B ;spline representation, which has been chosen for its exibility in geometric applications. Along with the fairing methods a process is proposed for developing ship lines when integrated and shape characteristics of a ship-hull are available. The dissertation starts with a general introduction to the Di erential Geometry of parametric curves and the B ;spline representation, so that the reader becomes acquainted with the notions, the mathematical tools and the notation of the work. A local and iterative fairing method for three dimensional cubic B ;splines follows, which can be considered an extension and generalization of Sapidis & Farin '90]. A new fairness criterion for spatial C 2 cubic splines is introduced, focusing on the quality of the curvature and torsion plots as well as on the shape characteristics of the curve. New sucient conditions derive for satisfying the new criterion and the proposed algorithm improves the curve locally at every step by raising di erent kinds of nodal discontinuities. The second method is also automatic and handles spatial cubic B ;splines. Shape constraints are set for the control polygon of every segment of the curve, which secure the i

ii smooth movement of the Frenet frame vectors along the curve. The solution of the resulting Non-Linear Programming problem exhibits substantial improvement of the torsion plot of the curve, while the curvature plot is less a ected. The third method is appropriate for fairing planar curves under geometric constraints, such as end conditions, shape, and deviation of existing data poins, and/or integral constraints, e.g. area, rst and second moments of area. The method is eciently used for solving a number of problems, and it is mainly applicable to Ship Design. Based on the latter method, a process is proposed for developing ship lines, when hydrostatic characteristics of the desirable hull are available. Although the process is similar to the traditional method that naval architects follow, it assures that the produced lines satisfy all requirements within a small tolerance throughout the process. All fairing methods incorporate tolerance constraints and bound the di erence between the initial and the faired curve. This fact secures that the resulting faired curve stands close to the initial one, which is important for design applications.

Acknowledgements The present thesis consummates my postgraduate studies at the National Technical University of Athens, which have been partially funded by the Greek Scholarship Foundation, the Commission of the European Union and the Committee of Research of NTUA, to which I owe special thanks. First, I would like to express my sincere gratitude to my supervisor, Professor Panagiotis Kaklis, not only for his continual guidance and help, but also for his friendship and his moral support all these years I have known him. It has been a great fortune for me to work under his supervision and be taught by his way of thinking and acting. He has given me the opportunity to start materializing my scienti c ambitions, though some times this may have turned out to be an èncumbrance' for his personal life. I would like to thank Professor Apostolos Papanikolaou for his valuable and always willing assistance and support. His deep knowledge and ideas for Ship Design helped me many times to keep in touch with the essence of my thesis, and his remarks were always to the target. I am indebted to Dr. Nickolas Sapidis for his constant and valuable support, his helpful comments, his ideas and his friendship. Thanks to him I got in touch with the CAGD world. I must thank Assistant Professor Gerasimos Politis for his interest in my work and his practical advice, which saved me from aimless loss of time. iii

iv I am also indebted to the members of the examination committee, who honoured me by accepting participating in that. I was also fortunate to collaborate and be a friend with Melkon Isirikian, Spyridon Kapniaris and Polyvios Kehagioglou, who had worked for their diploma theses under the supervision of Prof. Kaklis. Especially to Melkon and Spyridon, I am indebted for their help. It would have been a great omission, had I not mention my trusted friends and colleagues, Elias Karyampas and Christodoulos Koskinas, who encouraged me in the rst steps of postgraduate studies and have been of great help throughout the years. Special thanks go to all my colleagues of the Ship Design Laboratory, who have always stood by me and honoured me with their trust and friendship. Along with them, I would like to thank the members of the laboratories of Ship Design and of Shipbuilding Technology, and especially Professor Vassilios Papazoglou for his care and support and Dr. Nicolas Tsouvalis for his advice and friendship. Also, I feel that the accomplishment of this dissertation is undoubtly due to the Department of Naval Architecture and Marine Engineering, to which I am grateful. I am grateful to Professor Joseph Hoschek of the Technische Hochschule Darmstadt (THD), Germany, for his support and experienced advice all the time I spent with him. I also thank the postgraduate students and the personnel of the Zentrum fur Praktishe Mathematik for their spontaneous help. They made my stay there an exceptional experience. Last but not least, I would like to express my deep gratitude to my father, George, my mother, Eustathia, and my sister Garyfalia. This work would not have even started without their consent and constant support, moral and nancial. Along with them, I would like to thank Despena for her love and patience.

Contents Chapter 1. Introduction

1

1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Organisation of the dissertation . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1 Fairing methods for spatial curves . . . . . . . . . . . . . . . . . .

4

1.3.2 Planar fairing under constrains - Ship lines development . . . . . .

6

Chapter 2. Basic concepts and denitions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9

2.2 Elements from Di erential Geometry . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Arc length of a regular curve . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Frenet frame (Ff) - Curvature and torsion . . . . . . . . . . . . . . 10 2.2.3 Signed curvature - Inections . . . . . . . . . . . . . . . . . . . . . 13 2.3 B ;spline curves: de nitions and properties . . . . . . . . . . . . . . . . . 15 2.3.1 De nition of integral B ;splines . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Di erentiation and integration formulae . . . . . . . . . . . . . . . 17 v

Contents

vi

Chapter 3. Local fairing for 3D curves

18

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 Mathematical descriptions for fairness . . . . . . . . . . . . . . . . 18 3.2 Fairness indicators and discrete fairness metrics for cubic curves . . . . . . 20 3.3 Local continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1 Torsion continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.2 Continuity related to curvature . . . . . . . . . . . . . . . . . . . . 23 3.3.3 Continuity of third order (C 3) . . . . . . . . . . . . . . . . . . . . 25 3.4 Automatic method for local fairing . . . . . . . . . . . . . . . . . . . . . . 26 3.4.1 Determination of the o ending knot . . . . . . . . . . . . . . . . . 27 3.4.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.6 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 4. Fairing of 3D curves under shape constraints

41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Frenet frame and inections . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 More local properties of C 2 cubic B ;splines . . . . . . . . . . . . . . . . . 43 4.4 Acuteness and local convexity . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Contents

vii

4.7 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 5. Fairing of 2D curves under design constraints

62

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.1 End conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.2 Calculation of integrated measurs . . . . . . . . . . . . . . . . . . . 63 5.2.3 Local convexity of planar cubic B ;splines . . . . . . . . . . . . . . 66 5.3 Fairing of B ;spline curves under constraints . . . . . . . . . . . . . . . . 67 5.3.1 The problem (Pdesign) . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3.2 Solution of the problem (Pdesign) - An example . . . . . . . . . . . 69 5.4 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter 6. Applications in Computer Aided Ship Design

76

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 Ship design and hull geometry . . . . . . . . . . . . . . . . . . . . . . . . 77 6.3 Development of ship lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.3.1 Main characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3.2 Process for solving the problem (Pmesh) . . . . . . . . . . . . . . . 81 6.4 A CAShipD example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.5 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

List of Figures 2.1 The Frenet frame vectors when a planar curve turns left (right). . . . . . . 13 3.1 The conditions for K and C 3 continuity, expressed as geometric locii . 24 0

3.2 Orthogonal projections of the initial (dashed) and nal (solid) chine curve. 33 3.3 Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Binormal vector distribution of the initial (upper) and nal (lower) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Curvature vector distribution of the initial (upper) and nal (lower) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Orthogonal projections of the initial (dashed) and nal (solid) GM curve.

37

3.7 Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 Binormal vector distribution of the initial (upper) and nal (lower) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.9 Curvature vector distribution of the initial (upper) and nal (lower) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 viii

List of Figures

ix

4.1 Comparison of two curves with the same sign of torsion, but di erent movement of Frenet frame. The lower curve exhibits a rapid change of shape but no inection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Acute polygons. Left: Planar. Right: Spatial. . . . . . . . . . . . . . . . . 46 4.3 Local coordinate system. Left: The initial position of the coordinate system, P1xy. Right: The relative position of the two coordinate systems, P1xy and P2x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 0

0

4.4 Subspaces that the two coordinate systems can create, and possible positions of the components of Q_ (u) and L(u). . . . . . . . . . . . . . . . . . . 49 4.5 Orthogonal projections of the initial (dashed) and nal (solid) chine curve. 54 4.6 Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.7 Binormal vector distribution of the initial (upper) and nal (lower) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.8 Curvature vector distribution of the initial (upper) and nal (lower) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.9 Orthogonal projections of the initial (dashed) and nal (solid) GM curve.

58

4.10 Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.11 Binormal vector distribution of the initial (upper) and nal (lower) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.12 Curvature vector distribution of the initial (upper) and nal (lower) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Planar symmetrical section of a symmetric body. Axis of symmetry: Ox. . 70

List of Figures

x

5.2 Control polygons: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Curves: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid). . . . 74 5.4 Curvature Plot: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.1 Initial Hull: Body plan St0.1 of Hull0.1. . . . . . . . . . . . . . . . . . . . 87 6.2 Initial Hull: Body plan St0.4 of Hull0.2. . . . . . . . . . . . . . . . . . . . 87 6.3 Sets of waterlines of the initial hull: Wl0.1 (upper) and Wl0.2 (lower). . . 88 6.4 Aft (left) and fore (right) of Wl0.1 (upper) and Wl0.2 (lower).

. . . . . . 88

6.5 Initial (dashed-I) and modi ed (solid-II) distributions, SAC and SLC, of the initial hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.6 Comparison between SAC-1/SLC-1 (dashed) and SAC-F/SLC-F (solid). . 89 6.7 Initial Hull: Body plan St1.1 of Hull1.1. . . . . . . . . . . . . . . . . . . . 90 6.8 Final Hull: Body plan St1.5 of Hull1.3. . . . . . . . . . . . . . . . . . . . 90 6.9 Distributions of Aw , LCF , Ixx and Iyy with respect to the draught of the ship (G-distributions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.10 Sets of waterlines: Wl1.1 (upper) and Wl1.2 (lower). . . . . . . . . . . . . 92 6.11 Waterlines: Wl1.4 (upper and lower). . . . . . . . . . . . . . . . . . . . . 92

List of Tables 3.1 The fairness metrics for the initial and nal chine curve. . . . . . . . . . . 30 3.2 The fairness metrics for the initial and nal GM curve. . . . . . . . . . . . 31 5.1 Characteristics of the boundary curves of the example section.

. . . . . . 72

6.1 Initial and required characteristics of the hull. . . . . . . . . . . . . . . . . 84 6.2 The characteristics of hulls calculated through the development process. . 84

xi

Chapter 1 Introduction 1.1 Preface The recent developments in hardware and software tools have stimulated the transition of design from traditional methods to integrated computer systems. The increasing computing power that has been achieved, leads even the most reluctant to bene t of Computer Aided Design (CAD) and Manufacturing (CAM), which o er improvement of productivity, consistently high quality and reduced development time for new products. All these advantages enable a rm to keep up with the modern, customer-driven market conditions, which require capability to design and release frequently new and improved products, so that this rm gains strategic bene ts Roy '95]. That explains the apparent increase of CAD companies with innovated systems, targeting even not-so-ourishing markets and industry sectors, like the marine market and Shipbuilding, which are relatively nite Wake '93] { if not shrinking. The need for a wide variety of types, morphology and size of products has boosted Computer Aided Geometric Design (CAGD), the branch of CAD that focuses on free form curve-, surface- and solid-modelling. Since the appearance of a product is important for potential buyers Burchard et al. '94], the construction of visually pleasing objects has become vital for industrial design and styling, although this boost was initially not due to 1

2

Chapter 1. Introduction

market reasons, but to practical de ciencies. Pierre Bezier, one of the pioneers in CAGD, explains why Renault adopted his ideas in the early sixties by pointing out the necessity of \a complete, distortion-free and unquestionable denition" for their designs, particularly when they had to \exchange information" Bezier '90]. From 1960 until today, research and development have resulted in quite a few methods for constructing curves, surfaces and solids. Now industry research is focused in shape quality and fairness Burchard et al. '94], while vendors aim at an even cooperation between solid and surface modellers Smith '96]. For a non-connoisseur of the principles of CAGD it may seem oxymoron that, in a state-of-the-art system { like those based on Non-Uniform Rational B-Splines (NURBS) { one often faces undesirable shape features (inections/oscillations). Unfortunately, this occurs when any spline representation { NURBS included { is used to interpolate/approximate lower dimension entities, or to ll a gap between entities, or even when two or more entities must be joined together Jensen et al. '91], Massabo '96]. Also, there is a certain need for tools to repair imported geometry from `foreign' systems Smith '96]. In order to develop a CAGD system able to produce entities as good as a draftsman can draw, a `complete set' of aesthetic constraints must be available, like the ones used by draftsmen in judging the quality of a curve or a surface, which should be followed by their translation into mathematical conditions Burchard et al. '94]. Though that may sound complicated, the principle of the simplest shape is widely accepted { originating mainly in the ideas of Birkho '33] { and has been followed by the majority of researchers. The process of eliminating undesirable features of geometric entities is called fairing, and it is by no means inuenced only by styling. A number of practical constraints, like manufacturability, functionality and performance according to hydrodynamics/aerodynamics should be also taken into account, so that such a process can be successful. Materialising a CAGD system with robust techniques for automatic fairing is not an easy

1.2. Organisation of the dissertation

3

task and is still a challenge Burchard et al. '94], Jensen et al. '91]. That is justi ed by the plethora of existing fairing tools Burchard et al. '94], Catley '89], CETENA '85], Hohenberger & Reuding '95], Hottel et al. '91], Klass et al. '91], KCS '94], which, however, often stand far from satisfying the end-users Hays '92] and lack of `simple' utilities, like fairing of truly spatial curves, or automatic fairing under design requirements along with the styling ones. Even in the area of Computer Aided Ship Design (CAShipD), spatial curves are ùnavoidable evils', since they can be found on every vessel, from a high-speed craft to an ultra-large tanker Clement '63], Clement & Blount '63], Comstock-PNA '67], Serter '92], Serter '94], so fairing tools are quite necessary for ship hull generation. Nevertheless, for that process, spatial curve fairing is not the target. As in most engineering applications, geometry should satisfy a number of requirements and consist the optimal solution. This approach starts to become a trend in CAD, and the incorporation of constraints in modelling processes is a fact Anderl & Mendgen '96], Feng & Kusiak '95], Zou et al. '96]. For these reasons, the dissertation investigates possible ways of automating local fairing for 3D curves, based on two di erent approaches! one that aims at àberrations' of curvature and torsion, and one that controls the shape through the Frenet frame. Additionally, it gives attention to a fairing method for planar curves, which incorporates several design constraints, derived mainly from the area of Ship Design. Based on that method, a process for the development of hull lines is proposed, which can be automated and consist a tool for CAShipD.

1.2 Organisation of the dissertation The rst chapter of the thesis continues with a general literature review related to the work areas, while the second chapter focuses on the mathematical background needed

4


for comprehending the contents of the following chapters. It gives a brief review of the notions of Di erential Geometry related with planar and spatial parametric curves, and some elements for the B ;spline representation. Chapters 3 and 4 deal with fairing methods for three-dimensional (3D) curves. New fairing criteria are introduced and algorithms are developed in order to ful l them. In Chapter 3 di erent kinds of continuity are discussed, and an iterative local fairing algorithm is proposed. In Chapter 4 the behaviour of the Frenet frame is controlled in any segment, and this results to the quality improvement of the curve. Chapter 5 proposes a method for fairing planar curves under constraints. Some constraints are set for the curve itself (end conditions, curvature sign, tolerances), while others are set for integral characteristics (area, rst and second order moments of area) of the planar domain bounded by the curve and one of the cartesian axes, assuming that the curve can be described by a function of one variable. The well known energy functional is the fairness criterion, consisting of two integrated derivative magnitudes of the curve. Mainly based on the method of Chapter 5, a design process is proposed in Chapter 6, which develops ship lines under a number of design requirements and can be automated.

1.3 Literature Review 1.3.1 Fairing methods for spatial curves Though a number of fairness criteria for 3D curves have been proposed (see Farin '93], ch. 23, Hoschek & Lasser '93], ch. 13, and references therein), the majority of them refer to integral functionals, which, in most cases, involve one or more squared parametric derivatives up to third order, and in few cases are directly related to curvature and torsion. Using integral functionals of the former kind, many algorithms have been proposed using

1.3. Literature Review

5

polynomial splines Eck & Hadenfeld '95a], Fang & Gossard '95], Hohenberger & Reuding '95], Hagen & Bonneau '91], Huanzong et al. '92], Hu & Schumaker '86], Reinsch '67], Reinsch '71], most of which solve the interpolation/ approximation (or fairing) problem analytically and quickly, but the results are not always satisfactory, partially because of the assumption that the parametrization of the curve is close to the natural one (see Pigounakis et al. '95]). Proposed improvements, such as in Liu et al. '92] and Vassilev '96], have not been tested for truly 3D curves, so this kind of algorithms should be considered supplements of the planar case. The second category of integral functional criteria are mainly covered by Roulier et al. '91], Roulier & Rando '94], and by Moreton & Sequin '93], Moreton & Sequin '95]. Though more sophisticated, these criteria cannot be implemented with analytic methods and the resulting algorithms are either simple and iterative or materialized by Non-Linear Programming (NLP) techniques1 . Also, in Moreton's work the employed criteria deal only with curvature and the variation of curvature, while they ignore torsion. On the other hand, Roulier & Rando '94] seems to focus more on the integrated study of integral functional methods, since it explains the e ect of a variety of fairness criteria and proposes iterative algorithms for implementing them. The criteria mentioned above exhibit two inherent disadvantages, namely, (i) they assume that the fairness of curve is almost constant along its length, so they act globally, and (ii) they cannot cope up with the shape of the curve, given by the sign of torsion. An exception for (i) is Eck & Hadenfeld '95a], where a global fairness measure is used for a local criterion. A more detailed analysis for global and local criteria is given in the introduction of Chapter 3. A completely di erent approach is found in Wagner et al. '95], where a new kind of polynomial splines, called spring splines, is introduced, which incorporate constraints for NLP techniques are also iterative. Nonetheless, this distinction aims to underline that NPL techniques are much more time consuming and their iterations may require the solution of large linear systems. 1

6


curvature and torsion, based on mechanics. Nevertheless, the whole problem is set in a rather simpli ed way and no results for 3D curves are given. The discussion above points out the necessity of local fairing criteria and methods which are mainly intended for spatial curves, i.e., they take into consideration both curvature and torsion. Such criteria and a method for C 2 cubic B ;splines are presented in Chapter 3. Experience has shown that fairing of 3D curves means, among others, endeavour for `right' shape, which is not always evident. Presumably, that is why the rst papers on shape preservation in IR3 with polynomial splines are very recent Goodman & Ong '91], Kaklis & Karavelas '96], Labenski & Piper '96]. Along with them, a number of methods for detecting shape failures, and particularly inections can be used for or extented to 3D space Goodman '91], Hoschek '84], Manocha & Canny '91], Hansen & Nielsen '95a], Hansen & Nielsen '95b]. However, the problem of hailing any kind of shape failures, like sign-changes of torsion, has not been faced yet. The only exception is met in Jones '87], where a method is proposed for shape manipulation of spatial curves, though the conditions are rather application oriented and not tested for 3D curves. Also, a constructive method has been introduced in Higashi & Kaneko '88] for Bezier curves, but the imposed condition seems too stringent, since it guarantees monotonic curvature variation. In the fourth chapter of the dissertation, a method for elimination of shape failures is proposed, which controls the Frenet frame vectors along a cubic C 2 B ;spline. The control is achieved by setting constraints on the control polygon of the curve.

1.3.2 Planar fairing under constrains - Ship lines development Industrial design lacks of methods that incorporate design requirements along with curve/ surface generation, and that gives birth to design spirals until geometry comes in accordance with functionality. Especially in ship design, such conditions are in the every-day

1.3. Literature Review

7

practice, because a hull must be a compromise of numerous and sometimes controversial constraints. As a result, the elimination of such spirals is still considered unavoidable, while CAD systems simply \speed the trip around the classic design spiral" Hays '92]. So, the majority of ship designers change the geometry of a ship in a more or less arbitrary way in order to achieve the appropriate characteristics Firth '95], while the more progressive take advantage of the mathematical representation of their system and estimate the result of the modi cations Birmingham & Smith '95]. At the antipode of this trend, there exist a few methods for generating/ fairing curves and surfaces under general constraints. The Institut fur Schi s- und Meerstechnik of the Technische Universitat Berlin have proposed procedures for developing fair curves and surfaces under constraints Nowacki & Lu '94], Nowacki et al. '90], Nowacki et al. '95], Standerski '88]. More speci cally, Nowacki et al. '90] and Nowacki & Lu '94] solve the problem of fairing a planar curve under end and area constraints. The former refers to an interpolation scheme using a single-segment Bezier curve, while the latter approximates given data with quintic Hermite polynomial splines. In Standerski '88] a ship hull is modelled by a B ;spline surface with uniform parametrization and rectangular grid for the control points. For speci c isoparametric lines (sections) area moments of rst and second order are required, but the restrictive setting of the problem leads to results, which can be used only as guidelines and not for lines development. In none of the afore mentioned works exist constraints for shape, and all of them employ an energy functional minimization as a fairness criterion. Along with them, Ganos & Papanikolaou '95] proposes a method for generation of hull forms of traditional greek shing boats, which employs non-parametric cubic form splines and a rectangular grid. Though the framework seems rather restrictive, the results are quite satisfactory, at least for the particular, simple-shaped hulls. Another related work is Lowe et al. '94], where hydrodynamic criteria are incorporated in a Partial Dierential


8 Equations (PDE)2 method for yacht hull generation.

For di erent purposes, Bercovirer & Jacobi '94] suggests a constrained minimization algorithm for o set curve construction with composite Bezier curves employing a Finite Element Method (FEM). Also, Gopalsamy & Reddy '93] presents an algorithm for determining the shape of a curve along with the minimization of a combined functional for error and fairness. The fth chapter of this thesis develops a method for fairing planar B ;spline curves of arbitrary ;practically up to seventh; degree, by minimizing a combined error and energy functional under constraints for the ends of the curve, the sign of curvature, the area, the rst and second moment of area, as well as tolerance constraints for the nodal points. In the last chapter of the dissertation, a process is proposed for the development of ship hull lines under constraints, taking advantage of the method of Chapter 5. Though design constraints are employed only in Standerski '88] and Ganos & Papanikolaou '95], the hull generation problem has been faced by a few researchers Bedi & Vickers '89], Kouh & Chau '93], Liu et al. '92], Rogers & Fog '89], Rossier '90], and nowadays numerous systems provide such methods AUTOSHIP '96], KCS '94], Wake '93]. The novelty of the proposed process is that the design constraints are present throughout the calculations and the designer can de ne the desirable shape of any line or local area of the hull. Furthermore, the variety of constraints is the widest among the existing methods, and the process can be automated.

PDE methods are introduced and widely used by M.I.G. Bloor, M. Wilson and their collaborators at the Department of Mathematical Studies of the University of Leeds. 2

Chapter 2 Basic concepts and denitions 2.1 Introduction Di erential Geometry provides the necessary mathematical background for interrogation and study of the behaviour of a curve, which can be seen as a geometric set of points, i.e. a locus, and, at the same time, as a path traced out by a particle in IR3 Millman & Parker '77], Kreyszig '91], Stoker '88], Struik '88] etc. The latter consideration is compatible with the parametric expression of a curve, which o ers information not only for the successive positions of the particle, but also for the time that this particle passes through a speci c position. One can understand that changing the correspondence between positions and times, the kinematic condition of the particle is altered while the path remains the same. Though this piece of information is valuable, there are certain unchanged geometrical properties due to the fact that the curve is the same, no matter how the particle moves along the path. These properties enable one to estimate the quality of the curve. Apart from notions of Di erential Geometry, this chapter presents properties and intrinsic characteristics of polynomial spline parametric curves expressed with the so-called B ;spline representation. The choice of B ;splines is based on the CAD/CAM practice, where NURBS have become a standard, since their properties o er many advantages 9

10

Chapter 2. Basic concepts and denitions

for changing and manipulating such curves eciently Farin '91], Farin '93], Farin '94], Hoschek & Lasser '93], Piegl & Tiller '95], Rogers & Adams '90].

2.2 Elements from Dierential Geometry 2.2.1 Arc length of a regular curve Let S(t) = (x(t) y(t) z(t))T be a real vector function of a real variable t, representing a parametric curve in IR3. As stated in many books (e.g., Stoker '88], pp. 12-13) and in order to avoid ambiquities, it is practical to work with a speci c family of parametric curves, of which all three components are at least one time continuously di erentiable and, for any t the rst derivative of at least one of the components is non-zero. Such curves are called regular. Since IR3 is a Euclidean space, the length, s(t), of the curve S(t) from a xed point (say, from S(t = t0 )) to a variable point equals to

s(t) =

Zt t0

jjS_ (l)jjdl

(2.1)

and is called the arc length of the curve. Dot denotes di erentiation with respect to the parameter t and "jj jj" is the Euclidean norm. It can be shown that the arc length is an invariant property of the curve (see, e.g., Willmore '93]). If a curve S(t) is de ned as a function of its arc length, its parametrization is called natural and holds jjS jj = jjS_ (s)jj = 1. From now on, di erentiation with respect to arc length, s, will be denoted by prime and the resulting derivatives will be also called natural. 0

2.2.2 Frenet frame (Ff) - Curvature and torsion No matter what the parametrization of the curve is, the rst derivative of a parametric curve is a vector, tangent to the curve, which points in the direction that the parameter

2.2. Elements from Di erential Geometry

11

increases. In other words, this vector gives the velocity of a moving particle for the particular path. If such a vector, t, is of magnitude 1, namely _ t = S_ (t) = S (s) (2.2) jjS(t)jj is called the unit tangent vector or, simply, the tangent vector of the curve. 0

Assuming that the curve is of continuity order C r , r 2, we can di erentiate the relation t t = 1 with respect to arc length and obtain t t = 0. Hence, if the vector t := S is not the null vector, it is orthogonal to t. The vectors K = t = S and n = jjSS ((ss))jj (2.3) are the curvature vector and the unit principal normal vector or, simply, the normal vector, respectively. The direction of these vectors does not depend on the orientation of the curve, i.e., their direction does not change if the parameter s is replaced by s = ;s or any other allowable parameter (Kreyszig '91], p.34). For an arbitrary parametrization of the curve, the expression of the curvature vector is h i K(t) = _ 1 2 S(t) ; t(t) t(t) S(t) (2.4) jjS(t)jj where dot " " denotes inner product. 0

0

00

00

0

00

00

The magnitude of the curvature vector, which is known as curvature, , measures the rate of change of the tangent vector Millman & Parker '77], Stoker '88] Kreyszig '91], Porteous '94]. It can be shown (see, e.g., Kreyszig '91], Struik '88]) that = jjS (s) S (s)jj = jj_L(t)jj3 (2.5) jjS(t)jj where " " denotes outer (or cross) product,and 0

00

L(t) = S_ (t) S(t):

(2.6)

The equation (2.5) is commonly used as a de nition for curvature, which is assumed always positive.


12

The normal vector and the tangent vector are orthogonal and de ne the osculating plane. For a planar curve the osculating plane is identical to the plane of the curve, while a spatial curve belongs locally to its osculating plane. For di erent ways of de ning the osculating plane, as well as its properties, one can refer Kreyszig '91], Stoker '88], Willmore '93]. Another vector, orthogonal to t and to n, can be introduced for any regular curve with non-vanishing second natural derivative:

b = t n = jjLL((tt))jj :

(2.7)

This vector is called the unit binormal vector or, simply, the binormal vector. From the above de nition, one can observe that the vectors t n b in this order, have the righthanded direction. The triplet they form is called the moving trihedron or the Frenet frame (Ff). Apart from the osculating plane, there are two more planes de ned by the Ff vectors: t de nes the rectifying plane, and n de nes the normal plane Struik '88], Kreyszig '91], Willmore '93]. Apart from measuring the change of tangent through curvature, it is of high interest to measure the vector of deviation of a curve from the osculating plane, or, in other words, the rate and direction of change of the osculating plane. Since b is the vector which de nes the plane, we study the derivative b for a C r curve S, r 3, and non-vanishing curvature. It can be shown (see, e.g. Kreyszig '91], Stoker '88], Willmore '93]) that b = ; n, where = (s) is the torsion of the curve at S(s): 0

0

_ S S...j j S S S j j S = ;n b = = jjS jj2 = _ 2 jjS Sjj where "jabcj" denotes the determinant of the three vectors a, b and c. 0

00

0

00

000

(2.8)

Unlike curvature, the sign of torsion has a geometric signi cance: it shows whether a curve 'curves' right (positive) or left (negative). The sign of torsion is orientation invariant, while for planar curves torsion vanishes identically (see Kreyszig '91], p. 39).

2.2. Elements from Di erential Geometry

13

t 11 00

n b

b n

t

Figure 2.1: The Frenet frame vectors when a planar curve turns left (right). The relation between the Ff vectors and their natural derivatives are given through the Serret-Frenet formulae:

2 3 2 t 64 n 75 = 64 ;0 0

0

b

0

0 0 ;

3 2

3

0 t7 7 6 5 4 n 5: 0 b

(2.9)

2.2.3 Signed curvature - Inections If 0 holds along an entire curve, then it follows from (2.5) that the curve is a straight line. In case that 6= 0 and the curve is planar, b is constantly orthogonal to the plane of the curve, whereas n lies on the plane. If the planar curve turns to the left (right) with respect to the tangent vector, then at any position along the curve the normal vector points to the left (right) of the tangent vector, and the binormal vector points up (down)! see Figure 2.1. In that particular case we can de ne a positive/negative direction with reference to the osculating plane and introduce the signed curvature:

_ k = b = S(t)_ S3(t) : jjS(t)jj

(2.10)

(A detailed analysis for planar curvature can be found in Stoker '88], pp. 23{27.) Until now, it has been assumed that neither S nor S S equal to null vector, but this is not an unusual case, especially in IR2. These two conditions have the same interpretation, 00

0

00


14

namely the curve does not change in direction, so its tangent is constant. For an arbitrary parametrization the tangent vector could remain constant (S = 0) or change only in magnitude (S_ S = 0), but both result to vanishing curvature. If ((s ) = (t ) = 0) holds, then this position, S(s ) = S(t ), is called a point of inection. Excluding the case that the curve is a straight line itself, at the neighbourhood of an inection the curve exhibits a straight line behaviour, since its tangent does not change. Then the osculating plane and the Ff vectors can be de ned using the rst non-vanishing derivative of the curve, though at a point of inection, even a C -curve need not possess a unique osculating plane (an example is given in Willmore '93]). Apart from the tangent vector, which is determined for any regular curve, the other two vectors can be expressed through Taylor expansion using higher derivatives of the curve at the neighbourhood of s . Assuming that only the second natural derivative is zero at s and the curve is at least C 4;continuous in the neighbourhood of s , while its fth-order natural derivative exhists ahd is nite there, we get: 2 3 S(s h) = S (s ) hS (s ) + h2 S (s ) h6 S (s ) + o(h4) (2.11) 2 h (2.12) S (s h) = S (s ) hS (s ) + 2 S (s ) + o(h3) S (s h) = S (s ) hS (s ) + o(h2 ): (2.13)

1

0

00

00

000

000

00

000

000

0000

0000

0000

where o() is the lower-case Landau order symbol. From the de nitions of the Ff vectors and the equations (2.11)-(2.13) we have: S (s ) S (s ) S (s ) : lim n ( s h ) = lim b ( s h ) = (2.14) h 0 h 0 jjS (s )jj jjS (s )jj 000

0

000

000

!

000

!

Following the same procedure as above, one can calculate the torsion at an inection, which is non-vanishing and continuous: 1 jS (s )S (s )S (s )j : (2.15) lim ( s h ) = h 0 2 jjS (s )jj2 For the special case that the curve is a cubic polynomial, the value of torsion at the inection equals to zero according to (2.15). 0

000

000

!

0000

2.3. B ;spline curves: denitions and properties

15

If the curve is determined through an arbitrary parametrization, the following expressions for the Ff vectors can be used:

_ t(t) = S_ (t) jjS(t)jj

(r) b(t) = jjLL(r)((tt))jj

n(t) = b(t) t(t)

(2.16)

where L(r) (t) = S_ (t) S(r) (t) and r 2 is the order of the rst derivative that L(r) is non-vanishing at t. Looking at (2.14) one can realize that n and b are discontinuous at an inection point. Nevertheless the osculating plane remains the same, though its orientation changes. In order that this discontinuity is raised, the following convention is usually adopted in Di erential Geometry: at the neighbourhood of an inection b is considered constant in direction and is taken to be positive or negative according to the sign of the inner product t n (Brand '84], p. 310, Struik '88], p. 15). This is in accordance with the formulae of (2.9) and o ers an explanation for the signed curvature in plane. For a deeper analysis on the Serret{Frenet equations, one can trace back in Nomizu '59]. Nevertheless, in this work the convention for the sign of at an inection will not be followed, since Ff is going to be the principal mathematical tool for interrogating 3D curves. 0

2.3 B ;spline curves: denitions and properties The most widely used method for de ning curves, in computer-aided hull design and other CAD applications, is the B ;spline method. Integral B ;splines de ne parametric piecewise-polynomial curves, while rational B ;splines de ne parametric curves that consist of rational-polynomial segments. The fundamental properties of integral B ;splines are briey reviewed below! detailed expositions for both representations are available in, e.g., Farin '91], Farin '93], Hoschek & Lasser '93], Piegl & Tiller '95].


16

2.3.1 Denition of integral B ;splines Integral B ;splines (in short, B ;splines) are classical polynomial splines represented as linear combinations of B ;spline basis functions. The coecients of this family of splines have a geometrical interpretetion, as they form a polygonal line, referred to as the control polygon, which reveals the shape of the corresponding curve. The fundamentals of B ;splines are included in most of CAGD textbooks! see, for instance, de Boor '78], Farin '93], Hoschek & Lasser '93] e.t.c. Let D = fdi = (dxi dyi dzi )T i = 0 1 : : : M g be a set of points in IR3, and U = fu0 u1 : : : uN g N = M + n + 1, a non-decreasing sequence of real numbers (knots). The B ;spline, Q(u) = (Qx(u) Qy (u) Qz (u))T , of order n corresponding to the control polygon D and the knot sequence (parametrization) U is given by: M X

Q(u) = diNin (u) u 2 un uM +1]

(2.17)

i=0

where Njn(u) j = 0 1 : : : M are the B ;spline basis functions de Boor '78], Farin '93], Schumaker '81], which are de ned by the recursive formula

Nin(u) = u u ;;uiu Nin 1 (u) + uui+n+1;;uu Nin+11(u) u 2 (;1 +1) ;

i+n

;

i

with

i+n+1

Ni0 (u) =

(

i+1

1 0

(2.18)

if u 2 ui ui+1) elsewhere:

Assuming that all control points are distinct, the degree of continuity between the polynomial segments of the spline is dictated by the knot vector: if ui has multiplicity r (i.e., ui = ui+1 = ui+2 = : : : = ui+r 1), then Q(u) is at least C n r at that knot. Finally, making the multiplicity of the rst and the last knot equal to the degree of the spline plus one (: n + 1) forces the spline to interpolate the rst and the last control point, a property highly desirable to designers. The particular structure of these basis functions produces a plethora of properties for the curve Q(u), making it a powerful tool for CAD ;

;

2.3. B ;spline curves: denitions and properties

17

applications. Among others, a B ;spline has the `variation diminishing' and the `convex hull' properties, it can be eciently evaluated using the de Boor algorithm, and it provides the ability of `local control'. Furthermore, the derivatives of a B ;spline can be also expressed as B ;splines of lower degree! see Hoschek & Lasser '93], Farin '93].

2.3.2 Di erentiation and integration formulae The restriction of a B ;spline curve, Q(u), on ui ui+1], i = n(1)M , with multiplicity (n + 1) at both ends, can be written as:

Q(u) =

Xi

dj Njn(u)

(2.19)

j =i n ;

and its derivative of order r equals to:

Q (u) =

Xi

(r )

j =i n

dj Njn(r)

(2.20)

;

where the derivatives of the basis functions are calculated recursively: Nin (r) (u) = u n; u Nin 1 (r 1) (u) u n; u Nin+11 (r 1) (u): i+n i i+n+1 i+1 Especially the rst and second derivatives are written in the form: i X dj 1 N n 1 (u) Q_ (u) = n j j =i n+1 uj +n ; uj ;

;

;

;

;

;

(2.21)

(2.22)

;

# n1 i " d X Nj (u) d j 1 j n +1 Q (u) = n(n ; 1) ; j =i n+1 uj +n ; uj uj +2 ; uj 1 uj +2 ; uj ;

;

;

;

;

(2.23)

where dj 1 = dj ; dj 1. ;

;

If the knots at the ends of a curve, Q(u), uk1 and uk2, exhibit multiplicity n + 1 (open curve), then it holds that: Z uk2 k2 kX 1 n+2 I = u Q(u)du = n +1 1 di(ui+n+1 ; ui): (2.24) k1 i=k1 ;

;

Chapter 3 Local fairing for 3D curves 3.1 Introduction Farin et al. '87] introduced the concept of knot removal { knot reinsertion1 for a planar cubic B ;spline, which was subsequently applied in the development of local fairing algorithms for curves Eck & Hadenfeld '95a], Farin & Sapidis '89], Hottel et al. '91], Pigounakis & Kaklis '94], Poliako '96], Sapidis & Farin '90], and for point sets Eck & Jaspert '94] (discrete version). Starting from the same idea, in this chapter criteria for evaluating the fairness of C 2 spatial curves are introduced and methods for local fairing of C 2 cubic B ;splines are proposed. The basic tools for interrogation are the curvature vector, the curvature plot and the torsion plot of the curve, i.e. the magnitudes of curvature and torsion versus parameter, since the proposed solutions aim to correct locally undesirable behaviours in those plots.

3.1.1 Mathematical descriptions for fairness Curvature plots are extensively used by CAD researchers, developers and users for inspecting curves (and surfaces) on a computer screen Catley '89], Sapidis & Farin '90], More about knot removal can be found in Hoschek & Lasser '93], Farin '93], as well as in Eck & Hadenfeld '95b], while for knot insertion in Boehm '80] among others. 1

18

3.1. Introduction

19

Farin & Sapidis '89], Eck & Hadenfeld '95a]. The basic idea can be found already in Birkho '33], and its applications have been evaluated in related studies on planar curves, which have concluded that a fair curvature plot is free of any unnecessary variation, i.e., the distribution of curvature on a fair curve must be as uniform as possible Burchard et al. '94], Farin '94], Sapidis '94]. A number of researchers have proposed mathematical criteria aiming at materializing the above idea, and these criteria can be divided into two categories:

Criterion A. (Direct Criterion for C 2 Planar Curves) A curve is characterized as fair if the corresponding curvature plot is comprised of as few as possible monotone (strictlier: almost linear) segments Farin et al. '87], Farin & Sapidis '89], Sapidis '94].

Criterion B. (Indirect or Ènergy' Criterion for C 2 Planar Curves) A curve is considered fair if it minimizes the integral of the squared curvature or the squared slope of curvature (or an approximation to one those) with respect to arc length (see review of papers in Roulier & Rando '94]). Indirect criteria, and the associated curve design and curve fairing methods, have been investigated since the early days of CAD, with mixed results. Their major disadvantages are: (i) their e ectiveness, in terms of consistently producing fair curves, has been marginal, (ii) they disallow user involvement as they always function as a `numerical black box', and (iii) they are global, i.e., they a ect the whole curve, while in practice designers are interested in removing local imperfections from a curve without altering the correct parts of it. On the other hand, the direct criterion A and the corresponding fairing methods Farin et al. '87], Farin & Sapidis '89], Sapidis & Farin '90], are free of the limitations (ii) and (iii). This fact, combined with the empirical observation that these fairing algorithms do remove local undulations from curves, has created a considerable interest in them among researchers and practitioners Hottel et al. '91], Klass et al. '91], Eck & Jaspert '94], Eck & Hadenfeld '95a].


20

In order to state a direct fairness criterion for spatial curves, analogous to the one stated above for planar curves, the elimination of unnecessary variations in curvature and torsion is required, since this is the reason that causes a viewer's eye to stop when inspecting the curve. Especially for a C 2 curve, e.g., a cubic B ;spline with simple knots, the torsion and the derivative of curvature are discontinuous functions, namely they exhibit a variation at least at the knots. Based on the above, the following can be proposed:

Criterion C. (Direct Fairness Criterion for C 2 Spatial Curves) A C 2 curve is characterized as fair if (a) its curvature plot is comprised of as few as possible monotone (strictlier: almost linear) segments, (b) its torsion plot is as close as possible to being continuous also with the fewest possible number of monotone (strictlier: almost linear) pieces, (c) the sign changes in the torsion plot are as few as possible, and (d) the value of the torsion, at each point of the curve, is as small as possible. Trying to adopt the proper indicators in order to judge the smoothness of a cubic spline, the notion of the curvature vector can be very helpful. Manning '74] proposed that a measure of smoothness for curves is the curvature vector, and this idea has been exploited in a number of works dealing with geometric continuity! see, e.g., Boehm '88].

3.2 Fairness indicators and discrete fairness metrics for cubic curves The third derivative of a C 2 cubic spline curve, Q(u), is, in general, discontinuous at the nodal points, and this fact a ects the derivative of curvature as well as the torsion of the curve. For curvature, one can investigate the discontinuity of the slope of curvature, or of the slope of the signed curvature, or of the curvature vector with respect to arc length. The derivative of curvature with respect to arc length equals to ... n Q d = ds = _ 3 ; 3 t _ Q2 kQk kQk 0

(3.1)

3.2. Fairness indicators and discrete fairness metrics for cubic curves

21

while the derivative of the signed curvature, as de ned by (2.10), equals to

...

k = ddsk = t _ Q3 ; 3k t _ Q2 kQk kQk

(3.2)

0

and the derivative of the curvature vector with respect to arc length, is equal to:

...

...

K = ddsK = ;2 t + n + b = n(n Q) +_ b3 (b Q) ; 3K t _ Q2 ; 2 t : kQk kQk

(3.3)

0

0

The absolute discontinuity of at a nodal point (knot), ui, equals to 0

... k n ( u i ) Q(ui )k (3.4) = j (ui ) ; (ui )j = kQ_ (ui)k3 ... ... ... where Q(ui) = Q(u+i ) ; Q(ui ), and, analogously, for the discontinuity of k at ui we zi

0

+

0

;

;

0

have the expression for the magnitude:

... k t ( u i) Q(ui )k zi = kk (ui ) ; k (ui )k = : kQ_ (ui)k3 k

0

+

0

;

(3.5)

The discontinuity of the curvature vector at ui equals to

... ... k n ( u ) n ( u ) Q ( u ) + b ( u ) b ( u ) Q (ui) k i i i i i : (3.6) ziK = kK (u+i ) ; K (ui )k = kQ_ (ui)k3 0

0

;

All these types of discontinuity that are presented above will be analysed in the next section, where conditions for di erent kinds of continuity at ui are set. On the other hand, torsion is discontinuous at any nodal point, ui, and its absolute discontinuity is ... b ( u ) Q (ui) : i hi = j (u+i ) ; (ui )j = _ (3.7) kQ(ui) Q(ui)k ;

Part (a) of Criterion C (x3.1.1) implies the requirement that the derivative of curvature with respect to arc length is continuous, which is not true, in general, at the knots of a cubic spline. Similarly, from part (b) of Criterion C derives the requirement for


22

torsion continuity. The corresponding discrete fairness metrics for the whole curve are, respectively: M M X X := zi and := hi: (3.8) i=4

i=4

The third part of Criterion C is expressed through the sum

=

M X i=4

wi

(

(u+i ) (ui ) < 0 where wi = 10 ifotherwise ;

(3.9)

while part (d) is checked by introducing the absolute maximum value of torsion, max = maxfj (u)jg 2, as another metric.

3.3 Local continuity Let ui 1 ui] and ui ui+1] be two consecutive parameter intervals of a cubic B ;spline. These two segments are a ected by the control points dj j = i ; 4 ::: i. We will see how a speci c kind of continuity can be achieved by moving anyone of those control points. Since we are interested in the movement of one point only, we denote that point by D = (Dx Dy Dz )T := dj , and we split the curve in two parts, namely the one that is a ected by D and the residual of the curve, Qres. The corresponding basis function is denoted by N := Nj3 (ui) and the curve at ui can be written as ;

Q(ui) = Qres(ui) + DN = (Qxres Qyres Qzres)T + (Dx Dy Dz )T N

(3.10)

For reasons of simplicity, the variable u is omitted and all functions are calculated at u = ui, unless it is stated otherwise. The derivatives of the curve are written, respectively,

Q_ = Q_ res + DN_ = (Q_ xres Q = Q res + DN = (Q xres

Q_ yres Q yres

Q_ zres)T + (Dx Dy Dz )T N_ Q zres)T + (Dx Dy Dz )T N

(3.11) (3.12)

For a cubic polynomial the absolute maximum value of torsion can be found analytically, since the numerator of torsion equals to a constant value (Kaklis & Karavelas '96], x3), and the denominator is a polynomial of fourth degree. 2

3.3. Local continuity

23

whereas for the third derivative we have

... ... ... ... ... Q(ui ) = Q(ui ) + DN (ui ) = Q res + DN ... ... ... ... ... Q(u+i ) = Q(u+i ) + DN (u+i ) = Q +res + DN + ;

;

;

;

;

(3.13) (3.14)

and the di erence of the third derivative from the left and from the right is:

... ... ... ... ... ... ... Q = Q(u+) ; Q(u ) = (Qxres Qyres Qzres)T + (Dx Dy Dz )T N: ;

(3.15)

3.3.1 Torsion continuity In order to achieve torsion continuity at ui moving only one point, we set the following ... equation (see also (3.7)), assuming that Q(ui) 6= 0 and (ui) 6= 0:

... ... hi = 0 () b(ui) Q(ui) = 0 () Q_ Q Qu=ui = 0:

(3.16)

The locus of the points, D, that satisfy (3.16) is a plane (see Fig. 3.1) and can be expressed as

A1 Dx + A2Dy + A3 Dz + A4 = 0 where

A1 = A2 = A3 = A4 =

...

x N...

y N...

z N...

x Qxres

...

;z Q...yres ;x Q...zres ;y Q...xres + y Qyres

... +y Q...zres +z Q...xres +x Q...yres + z Qzres

(3.17)

(3.18)

and

x = Q_ yresQ zres ; Q yresQ_ zres x = N_ Q xres ; N Q_ xres

y = Q_ zresQ xres ; Q zres Q_ xres y = N_ Q yres ; N Q_ yres

y (3.19)

z = Q_ xresQ yres ; Q xresQ_ res

z = N_ Q zres ; N Q_ zres:

(3.20)


24 τ continuity

C 3 continuity

1111111111111111111111111111111111111 0000000000000000000000000000000000000 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 0000000000000000000000000000000000000 1111111111111111111111111111111111111 b 0000000000000000000000000000000000000 1111111111111111111111111111111111111

t

K continuity

Q(ui )

n

Figure 3.1: The conditions for K and C 3 continuity, expressed as geometric locii 0

3.3.2 Continuity related to curvature ...

Assuming again that Q(ui) 6= 0 and (ui) 6= 0, then, according to (3.6), the continuity of the derivative of the curvature vector with respect to arc length holds when

... ... ziK = 0 () kn n Q + b b Q ku=ui = 0:

(3.21)

The two vectors of the sum in (3.21) lie along n and b respectively, thus they are linearly independent and belong to the normal plane. Therefore, the condition holds when both ... ... are of zero magnitude, namely Q(ui) ? (n b) () t(ui) Q(ui) = 0. Looking at (3.5), one observes that

...

...

ziK = 0 () zik = 0 () t(ui) Q(ui) = 0 () Q_ (ui) Q(ui) = 0:

(3.22)

With respect to the variable point, D, equation (3.22) is written as a system of three linear equations of rank at most two, which represents a straight line (see Fig. 3.1):

0 1 0 z ;y B@ ;z 0 x C A (Dx Dy Dz )T + (x y z )T = 0 where y ;x 0 ... ... x = N_ Qxres ; N Q_ xres

... ... y = N_ Qyres ; N Q_ yres

(3.23)

... ... z = N_ Qzres ; N Q_ zres (3.24)

3.3. Local continuity

25

... ... ... ... ... ... x = Q_ yresQzres ; Q_ zresQyres y = Q_ zresQxres ; Q_ xresQzres z = Q_ xresQyres ; Q_ yresQxres: (3.25) For planar curves, the condition for continuity coincides with the condition (3.22). Nevertheless, (3.22) is sucient but not necessary for continuity in 3D space. In that case the condition is set as follows: 0

0

... ... zi = 0 () n(ui) Q(ui) = 0 () b t Qu=ui = 0:

(3.26)

The same condition with respect to D is:

( y y + z z )(Dx)2 + (z z + xx)(Dy )2 + (xx + y y )(Dz )2

; ( xy + y x)DxDy ; (y z + z y )Dy Dz ; (xz + z x)Dz Dx + ( z y ; y z + y z ; z y )Dx + ( xz ; z x + z x ; xz )Dy + ( y x ; xy + xy ; y x)Dz + ( xx + y y + z z ) = 0

(3.27)

Eq. (3.27) represents a quadric surface. The classi cation of that surface is not obvious and depends on the relative positions of the other four control points that a ect the curve at ui.

3.3.3 Continuity of third order (C 3) The continuity of the third derivative is achieved when ...x ...x ...z T ... ( Q x y z T ... Qres) : Q(ui) = 0 () (D D D ) = ; res Qres N

(3.28)

It is obvious that the position of the control point which satis es the C 3 continuity at ui, satis es also continuity of torsion and continuity of the derivative of curvature Boehm '88], Lee et al. '92]! see also Fig. 3.1. In case that D = di 2 then the position for C 3 continuity coincides with the result of the 22 algorithm in Sapidis & Farin '90]. ;


26

3.4 Automatic method for local fairing Aiming at minimizing the various previously mentioned discrete fairness metrics, di erent methods can be developed. The herein local fairing method is iterative, i.e., in every step the position of a speci c control point is changed so that a particular kind of discontinuity is eliminated. After a number of iterations both curvature and torsion plots become smoother, so the corresponding curve is fairer than the initial one. Such methods are of heuristic nature, since convergence cannot be guaranteed. Nevertheless, testing a similar one with practical examples has shown that they are most reliable Pigounakis et al. '95], Ives-Smith '96]. According to the ideas of Sapidis & Farin '90] and the pertinent bibliography for 2D curves, the obvious solution for reducing all kinds of discontinuity is to achieve C 3 continuity in every step. Practical experience has shown that C 3 continuity is a rather stringent condition and after a few iterations the resulting curve deviates from the initial one. Thus in every step the òptimal' continuity should be established. Two questions arise about the location that a discontinuity should be removed, and the control point that should be changed for that purpose. For the rst question the answer is based on the idea of the oending knot Sapidis & Farin '90], though with an extended sense (see below). For the second one, the answer is the same with Sapidis & Farin '90], i.e., the control point that a ects most the o ending knot is chosen. This decision is based on the fact that, for uniform parametrization, the basis function Ni3 2(u) exhibits the highest value among all basis functions at the knot ui, therefore the movement of the control point di 2 a ects the curve at Q(ui) more than any other control point. It is not far from true to say that in most practical cases the choice of di 2 is the right one. Even if this were not exact, the theoretical background would not fail, since any control point a ecting ui can be chosen for erasing a discontinuity. ;

;

;

3.4. Automatic method for local fairing

27

3.4.1 Determination of the o ending knot Let Q(k) (u) be a cubic B ;spline curve at the k;th iteration1 of a local fairing procedure, which does not change the initial knot vector, U . For Q(k)(u) the following knots can be found:

up so that zp = maxfzi (k) i = 4(1)M g uq so that hq = maxfh(ik) i = 4(1)M g (k) (k) ur so that max 2 ur 1 ur ] or max 2 ur ur+1] ;

(k) (k) where max = maxfj (k) (u)jg and (k) (u) is the torsion of Q(k) (u). Since max appears in general in the interior of a segment, two knots can be chosen. So, ur is determined as the knot where the curve also exhibits greater discontinuity of torsion.

Next, the following ratios are calculated with respect to the discontinuities and metrics of the current iteration, k:

Rz = zp Rh = hq and R = max : (0) max

(3.29)

Denition 3.1 The oending knot (OK) is that knot among fup uq ur g, where the maximum value of the corresponding ratios Rz Rh or R is found3, unless wh = 1, when the oending knot is uq .

De nition 3.1 can be interpreted as follows: in every iteration, we look for any kind of discontinuity, which, if eliminated, improves most the corresponding fairness metric of the curve. In order to incorporate part (c) of Criterion C, the de nition `gives bounty' to knots sign changes of torsion. The initial curve is denoted as the curve of zero iteration, Q(0) . It is not expected that two or more of the ratios exhibit exactly the same value, so the maximum can be determined 1 3


28

3.4.2 The algorithm The input data for the fairing procedure are, apart from the curve, Q(0) (u), the following values, which are used as termination criteria: a global tolerance, , which should be an upper bound for the sum P M := i=4 jjQ(k)(ui) ; Q(0) (ui)jj

the maximum number of iterations, kmax values and corresponding to and respectively, which determine whether the curve can be considered fair enough1 . The proposed algorithm can be described with the following steps: 0

Calculate the fairness metrics,

1

IF

k kmax

]

THEN find the knots

up uq ur

(0) (0) (0)

and

k) u(OK

and

(0) max

]

]

ELSE STOP 2

IF

up = uq = ur

]

THEN find the point, 3

IF

k) u u(OK p

k) d(OK

2,

;

which guarantees

]

THEN find the closest point2 , 4

IF

k) u u(OK q

] OR

k) u u(OK r

] 2

THEN find the closest point , 5

IF

k) d(OK k) d(OK

C3

continuity ]

2

for which

zK (k) = 0

2

for which

h(k) = 0

;

;

(k) ] OR (k) ] ] Q(k 1) (u) with Q(k) (u) ] AND k = k + 1

]

]

] AND

THEN replace

;

]

ELSE STOP 6

GOTO 1

It is quite dicult for the designer to set values for and , since the sense of fairness cannot be easily interpreted in terms of discontinuity sums. In most cases these values are set to zero. 2 Closest control point to the existing control point, d(k;1) . OK ;2 1

3.5. Examples

29

In Step 2 of the algorithm, the position for C 3 continuity is unique, whereas in Steps 3 and 4 one should calculate the orthogonal projection of the existing control point, corresponding to the o ending knot, to a geometric locus (plane for torsion continuity, straight line for curvature vector slope continuity) in order to nd the new position. The choice of the closest point guarantees that the curve changes as less as possible.

3.5 Examples The performance of the algorithm is tested in two practical examples. The rst one originates in the Parent Model 4667-I of the TMB Series 62 for hard-chine planning hulls Clement '63], Clement & Blount '63]. The second data have been provided by General Motors Design Center, (Warren, MI). In all tests the initial curve is depicted with dashed line, while the nal one with solid line. The same convention stands for curvature and torsion plots. In both examples, the rst and the last control points remain unchanged. First the three projections on the cartesian planes are given (Figs. 3.2, 3.6), where one can see the deviation between the initial and the nal curve. Then the plots of curvature and of torsion vs. parameter are presented (Figs. 3.3, 3.7). Since the values of torsion vary strongly 1, these gures also depict the torsion plots in two di erent scalings. Finally, 3D-views of the scaled distributions of the binormal vector { Figs. 3.4, 3.8 { and the curvature vector { Figs. 3.5, 3.9 { along the initial and the nal curve are given, respectively. An e ort is made so that the best view is chosen for every 3D plot, though that leads sometimes to confusion of the orientation of the curve! thus, an `S' is printed near the starting point of the curves in 3D plots.

Example 1: The initial curve is an interpolant of thirteen (13) points, obtained by digitizing the chine curve of the model 4667-I. The control polygon consists of fteen The torsion plot may not show the exact value of the local maxima of torsion, especially if these maxima correspond to peaks nonetheless, the local maxima are analytically found, so the algorithm is accurate. 1

30


(15) points and the parametrization of the curve corresponds to the chord-length of the interpolated data. Because of digitizing, the quality of the initial curve is rather poor. One can notice three local maxima of torsion in the intervals u11 u12] u12 u13] and u14 u15], which decrease to one, in u13 u14] after fairing! see Fig. 3.3. Also, the algorithm improves the curvature and torsion plot, the latter of which becomes of constant sign (negative). This improvement is obvious in Figs. 3.4 and 3.5: the strong variation of the binormal vector along the initial curve almost disappears in the nal curve, while the distribution of the curvature vector becomes very smooth. The global tolerance, , is set to 0:200m (reference length: 10:000m), and the maximum number of iterations kmax = 100. The values of and are set to zero. After 81 iterations, the maximum total deviation between the initial and the nal nodal points is found = 0:197m and the maximum nodal deviation 0:027m. The fairness metrics of the initial and nal curve are presented in Table 3.1. Fairness Metrics Q0 (u) maxfzi g14i=4 0.502139 maxfhi g14i=4 2.577463 1.694275 14.407156 8. max 136.741117

Q(u)

0.059400 0.222267 0.157899 0.567479 0. 6.726018

Reduction 88.17 % 91.37 % 90.68 % 96.06 % 100.00 % 95.08 %

Table 3.1: The fairness metrics for the initial and nal chine curve. Example 2: The initial curve is a feature line of the hood of a car. The control polygon contains twenty-two (22) points, and the parametrization is equidistant (uniform). The quality of the curve is good. No serious problems can be detected from the curvature plot, except for the intervals u20 u21] u21 u22] where the values of curvature vary. On the other hand, in the torsion plot there exist peaks near the two ends (Fig. 3.7). These problematic areas are quite obvious in the distributions of binormal and the curvature vector (Figs. 3.8, 3.9). The faired curve stays close to the initial one, while its curvature

3.6. Conclusions - Future work

31

and torsion plots are quite smooth and the maximum values are signi cantly reduced (Fig. 3.7), which can be also noticed in the gures of vector distributions along the curves. Here, the global tolerance is = 10mm (reference length: 700mm), and the maximum number of iterations kmax = 500. The values of and are also set to zero. The maximum total deviation is found = 9:879mm, the maximum nodal deviation 2:624mm and the number of iterations k = 114. The fairness metrics of the initial and nal curve are presented in Table 3.2. Fairness Metrics Q0 (u) maxfzi g22i=4 5.07206310 22 maxfhi gi=4 7.28103410 1.39746510 1.40067410 11. max 23.157029

4

;

2

;

3

;

1

;

Q(u) 2.16110210 9.20743910 1.37056810 7.08790210 3. 0.117412

5

;

4

;

4

;

3

;

Reduction 95.74 % 98.74 % 90.19 % 94.95 % 72.72 % 99.49 %

Table 3.2: The fairness metrics for the initial and nal GM curve.

3.6 Conclusions - Future work The fairness criteria of this chapter are adequate for improving curves with local problems. Furthermore, the iterative nature of the presented method, as well as other possible methods based on these criteria, leads to an overall correction of curvature and torsion, so they can be compared with global methods that minimize integral functional (see Ives-Smith '96], Pigounakis et al. '95]). Such methods exhibit comparative advantages, namely they are easily implemented, automatic, fast, with low required information in input, which make them considerably important for industrial use. Apart from developing variations of the algorithm presented here in order to eliminate speci c kinds of discontinuity, the interest should be focused on the investigation of the necessary and sucient condition for continuity of the derivative of curvature. The clas-

32


si cation of the quadric surface of Eq. (3.27) and the determination of the closest point to the control point to be moved could result to more ecient algorithms.

0

0

Figure 3.2: Orthogonal projections of the initial (dashed) and nal (solid) chine curve.

0

1.6

0

0

1.6

1.6

0

1.6

10

10

Figures of examples 33


34 0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 u3

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

20

0

-20

-40

-60

-80

-100

-120

u3

2

0

-2

-4

-6

-8

u3

Figure 3.3: Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) chine curve.

Figures of examples

35

2 S 0

-2 6 0

10

0

2 S 0

-2 6 0

10

0

Figure 3.4: Binormal vector distribution of the initial (upper) and nal (lower) chine curve.


36

S 2 0 -2 0

10

6

S 2 0 -2 0

10

6

Figure 3.5: Curvature vector distribution of the initial (upper) and nal (lower) chine curve.

0

0

Figure 3.6: Orthogonal projections of the initial (dashed) and nal (solid) GM curve.

4650

4650 950 1050

5050

5050

950

1050

700

700

Figures of examples 37


38 0.012

0.01

0.008

0.006

0.004

0.002

0 u3

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

2

1.5

1

0.5

0

-0.5

-1

u3

0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025

u3

Figure 3.7: Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) GM curve.

Figures of examples

39

1050 S

650

5100

700

4600

0

1050 S

650

5100 700 4600

0

Figure 3.8: Binormal vector distribution of the initial (upper) and nal (lower) GM curve.


40

S 1050

5100

650

700

0

4600

S 1050

5100

650

700

0

4600

Figure 3.9: Curvature vector distribution of the initial (upper) and nal (lower) GM curve.

Chapter 4 Fairing of 3D curves under shape constraints 4.1 Introduction According to Di erential Geometry, a spatial curve is {up to a rigid motion{ fully determined by its curvature and torsion Kreyszig '91], Stoker '88], Struik '88], Willmore '93]. Since curvature is sign indi erent, it is not suitable for shape interrogation, while torsion is useful for extracting basic shape information from its sign (see Chapter 2). For a more detailed analysis, a good background in Di eretial Geometry is needed in order to properly interpret the behaviour of the curve and propose methods for shape improvement. On the other hand, the Ff can also be employed for studying spatial curves Stoker '88], Struik '88], and that approach is becoming popular in the CAGD community! see, for example, Moreton '95], Greiner et al. '95]. As shown in Chapter 2, the notion of inection can be introduced in exactly the same way for 2D and 3D curves, because at such a point two of the Ff vectors change their direction! see Eqs. (2.14). In this chapter the combined use of the Ff and the sign of torsion is examined for the case of a C 2 cubic B ;spline, so that the curve can exhibit the proper shape and its behaviour become predictable. The proposed method is based on a new notion, the latent inection 41


42

(Def. 4.2), which is a stronger notion of inection. The choice of cubic splines, which exhibit discontinuous torsion at the nodal points, does not a ect the principal shape features of the method.

4.2 Frenet frame and inections When the curvature of a regular 3D curve vanishes at u , namely L(u ) = 0, the curve exhibits a straight-line behaviour there. In such cases, if the tangent straight-line at u intersects the curve, then L(u ) L(u+)=jjL(u)jj2 = ;1. Since planar curves do not di er from spatial ones in that, the direction of the inection point can be uni ed for both 2D and 3D curves:

;

Denition 4.1 A curve Q(u) exhibits an inection at u , i the binormal vector is discontinuous there, i.e., b(u ) b(u+) = ;1 or, equivalently, (b(u ) b(u+)) = :

6

;

;

The symbol ( ) denotes the smaller angle between two vectors. The requirement for an angle of always holds for inections in plane and in space, because the osculating plane of the curve must change orientation therein, though the osculating plane itself remains the same. 6

In 3D space the binormal vector of a curve can change its direction enormously, but not so abruptly as it does at an inection. That causes a substantial change of the osculating plane, though the sign of torsion may remain constant (Fig. 4.1). In order to classify this undesirable case, which is similar to an inection, we give the following de nition:

Denition 4.2 The curve Q(u) exhibits a latent inection, i there exists an interval (ua ub), so that b(ua) b(ub) 0 or, equivalently, (b(ua ) b(ub)) =2. 6

4.3. More local properties of C 2 cubic B ;splines

43

b

t

n

b

n

b

n

t

t

b

b

t

n

t

n b t

n

Figure 4.1: Comparison of two curves with the same sign of torsion, but di erent move-

ment of Frenet frame. The lower curve exhibits a rapid change of shape but no inection. De nition 4.2 di ers from De nition 4.1, in two ways. First, the positions Q(ua) and Q(ub) can be close to, but not necessarily in the same neighbourhood of the curve. Second, the angle (b(ua) b(ub)) though greater than =2, does not necessarily equal to : As a result, De nition 4.2 is more general than De nition 4.1, and one can easily show that: 6

Lemma 4.1 If a curve exhibits no latent inections in an interval, then there exist no inections therein.

4.3 More local properties of C 2 cubic B ;splines Expanding Eq. (2.22), the rst derivative of a cubic B ;spline, Q(u), in ui ui+1] can be written as: 2 i _Q(u) = 3 X dj 1 Nj (u) = 1 (u)di 3+2(u)di 2 +3(u)di 1witha1 2 3(u) 0: uj+3 ; uj j =i 2 ;

;

;

;

;

(4.1)


44

Taking into consideration (4.1) and (2.23), L(u) equals to:

L(u) = 18f1(u)(di 3 di 2 )+ 2(u)(di 3 di 1 )+ 3(u)(di 2 di 1 )g (4.2) ;

;

;

where

;

;

;

"

2 1 1 1 1 (u) = (u ; u )(1 u ; u ) (u(iu+2 ;;uiu+1)()Nu i (u;)Nui 1()u) + (u(Ni ;1(uu)) ) i+1 i 2 i+2 i 1 i+2 i i+1 i 1 i+1 i 1 ;

;

;

;

1 Ni1 1(u) 2 (u) = (u ; u 1)(u ; u ) ((uui+1 ;;uui))(Nui (u); i+1 i 2 i+3 i i+2 i i+1 ui 1 ) " 2 # 1 ( u ( N ( u )) 1 i ; ui 1 )Ni1 (u)Ni1 1 (u) i 3 (u) = (u ; u )(u ; u ) (u ; u )(u ; u ) + (u ; u ) ;

;

;

;

i+2

i

1

;

i+3

i

i+2

#

;

;

(4.3)

;

i

i+1

i

;

1

i+2

i

One can easily realize that the coecients j (u) j = 1 2 3 are non-negative, and for uniform parametrization 1(u) and 3(u) are always greater than 2(u). (An extended discussion can be found in Pigounakis & Kaklis '94]). The normal vector lies in the direction of the quantity L(u) Q_ (u), which, with the aid of (4.1) and (4.2), is written as a sum of nine factors: i 2 X i 1 X i 1 X ;

j =i

;

;

3 k=j +1 r=i;1

6+j+k 2i(u) 4+r i(u)(dj dk ) dr ]: ;

;

(4.4)

;

For u 2 ui ui+1], the above sum is a linear combination of the double cross products of the vectors dj j = i ; 2(1)i, and its coecients are non-negative.

...

(u) Q(u)j, since it Another quantity of high interest is the numerator of torsion, jQ_ (u)Q controls its sign. After some algebra the numerator can be expressed as: Ci jdj 3dj 2dj 1j ;

;

(4.5)

;

where Ci is a positive constant number, depending on the the knots uj j = i ; 2(1)i + 3.

Remark: It is well known that each cross product of the type dk 1 dk , divided by ;

the lengths of the two vectors involved, expresses a vector, the magnitude of which equals

4.4. Acuteness and local convexity

45

to the sine of the angle between dk 1 and dk . This sine is also the numerator of the discrete curvature of the points dk 1dk dk+1! see Sauer '70], x2.3. Additionally, the scalar product in (4.5), divided by the lengths of the involved vectors, expresses the sine of the bihedral angle between the planes (dj 3dj 2dj 1) and (dj 2dj 1dj ). That sine is the numerator of the discrete torsion{ again in the sense that Sauer '70] introduced{ of the polygon dj 3dj 2dj 1dj . So, one can observe how the discrete properties of the control polygon a ect the continuous ones of the corresponding curve. ;

;

;

;

;

;

;

;

;

;

4.4 Acuteness and local convexity Taking advantage of the simple formulae derived in the previous section, we can predict the local behaviour of the Ff of a spatial C 2 cubic B ;spline, when its control polygon possesses some attributes. But rst, some new de nitions are necessary.

Denition 4.3 A planar polygon p0 p1 p2 is called acute, i it is convex and 0 !1 !2 < =2, where !1 = (p0 p1 ) and !2 = (p1 p2)! see Fig. 4.2. 6

6

A planar acute polygon is also regular according to Goodman '91].

Denition 4.4 A spatial polygon p0 p1p2 is called acute, i the orthogonal projection of p2 on the plane of (p0 p1 ) forms an acute polygon with three legs or, equivalently, i 0 !1 !2 < =2, where is the smaller bihedral angle of the planes (p0 p1 ) and (p1 p2 )! see Fig. 4.2. It can be shown that the condition of De nition 4.4 implies that the orthogonal projection of p0 on the plane of (p1 p2) forms an acute polygon, too. So, the requirement for !1 2 2 0 =2) is equivalent to that for `three-legs acute polygon'.


46 P0

P0

p

P3

0

p

2

p

1

2

P1

P3 1111111111111111111111111 0000000000000000000000000 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 p 0000000000000000000000000 0000000000000000000000000 1111111111111111111111111 p 1111111111111111111111111 2 0000000000000000000000000 1111111111111111111111111 0 1111111111111111111111111 0000000000000000000000000 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 2 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 P 1111111111111111111111111 p

1

1

1

1

P2

P2

Figure 4.2: Acute polygons. Left: Planar. Right: Spatial. z z

z’

e y

P1

-a

1 (0,0,0)

P0

-d

y

P3 P2

2

P2

P1

x

c

P3

x

(1,0,0)

-b

P0

xy’-plane y’

Figure 4.3: Local coordinate system. Left: The initial position of the coordinate system, P1xy. Right: The relative position of the two coordinate systems, P1xy and P2x y . 0

0

Using the above de nitions, we can show how the change of the vectors of the Ff can be bounded and, thus, predictable.

Proposition 4.1 If di 3 di 2 di 1 is a spatial acute polygon, corresponding to a segment of Q(u) u 2 ui ui+1], then the change of direction of the binormal vector is always ;

;

;

less than =2 therein, i.e., there is no latent inection in ui ui+1 ].

Proof: Without loss of generality, we assume that the knots that a ect the segment Q(u) u 2 ui ui+1], de ne the set T = ftj = (ui+j 2 ; ui)=(ui+1 ; ui) j = 0(1)5g and the curve is expressed as bfQ = Q(t) through the transformation t = (u ; ui)=(ui+1 ; ui) 2 0 1]: ;

4.4. Acuteness and local convexity

47

Furthermore it is assumed kp1 k = 1: The coordinate system is set as shown in Figure 4.3, namely the xy-plane is identical to the plane of (p0 p1) with x-axis starting at the second vertex, P1, of the polygon and going along p1 . Due to acuteness (Def. 4.4), the coordinates of the control vertices are: P0 = (;a ;b 0)T P1 = (0 0 0)T P2 = (1 0 0)T and P3 = (c ;d e)T where a b c d e 2 IR+ a 6= 0 c > 1: The segment Q(t) t 2 0 1] can be written as:

2 3 ; a 1 c P0 M0(t) + P2M2 (t) + P3M3 (t) = 64 ;b 0 ;d 75 M0 (t) M2(t) M3(t)]T 0 0 e

(4.6)

where Mk (t) k = 0(1)3 are the transformed basis functions, corresponding to Nj3 (u) j = i ; 3(1)i: The cross product of the rst and second derivative of the curve can be written

2 3 2 3 l ( t ) Lx(t) x 64 ly (t) 75 L(t) = 64 Ly (t) 75 = (1 ; t )(1 ; 18 t1)(t4 ; t1 )t4 t5 lz (t) 0 Lz (t)

(4.7)

where

lx(t) = be(t4 ; t1 )(1 ; t)t ly (t) = ;ea(t4 ; t1 )(1 ; t) + (1 ; t0 )(t ; t1)]t

(4.8)

lz (t) = ;d(1 ; t0 )(t ; t1 )t ; ad + b(c ; 1)](t4 ; t1 )(1 ; t)t ; bt5 (t4 ; t)(1 ; t) Furthermore 4.8imply that, for t 2 (0 1), all three components of L(t) are of constant sign, namely Lx(t) is positive, and Ly (t) Lz (t) are negative. For t = 0 1 the vector L(t) is vertical to the planes of (p0 p1 ) and (p1 p2 ) respectively. That implies that the angle between the binormal vectors of any two positions within the segment is less than or equal to =2. In order to secure that this angle is always strictly less than =2, the coordinate system is rotated around the x-axis. In the new system, P1xy z , the xy -plane is identical to 0

0

0


48

the plane of (p1 p2) (Fig. 4.3). If we repeat the calculation of L(t), we can notice that Lx(t) and Ly (t) are positive, while Lz (t) is negative. From this second result and the rst one, we conclude that L(t) lies within the subspace that is de ned by the positive x-axis, the negative z-axis and the negative z -axis. Then, all possible directions of L(t) form among them angles smaller than =2, since none of them can be parallel to x-axis, as the algebraic expressions imply. So, the possible directions of binormal vector in the segment lie within the convex spherical sector formed be the unit sphere at P1 and the positive x;axis, the negative z;axis and the negative z ;axis. 0

0

0

0

A similar result holds true for the normal vector:

Proposition 4.2 If di 3 di 2 di 1 is a 3D acute polygon, corresponding to a segment of Q(u) u 2 ui ui+1 ], then the change of direction of the normal vector is always less ;

;

;

than therein.

Proof: If one employs the same technique of rotating the coordinate system P1xyz, one can express the rst derivative and nd out that the x; and z;components of Q_ (t) are positive in both coordinate systems. For the initial coordinate system, the rst derivative can be written as:

2 3 2 Q_ (t) 3 q ( t ) x x Q_ (t) = 64 Q_ y (t) 75 = (1 ; t )(1 ; t3 )(t ; t )t t 64 qy (t) 75 0 1 4 1 4 5 qz (t) Q_ z (t)

(4.9)

where

qx(t) = a(t4 ; t1 )t4t5 (1 ; t)2 + (1 ; t0)(1 ; t1)t5 (t4 ; t)t + (1 ; t0 )t4t5 (t ; t1)(1 ; t) + (c ; 1)(1 ; t0 )(1 ; t1 )(t4 ; t1 )t2

qy (t) = b(t4 ; t1 )t4 t5(1 ; t)2 ; d(1 ; t0 )(1 ; t1 )(t4 ; t1 )t2 qz (t) = e(1 ; t0 )(1 ; t1 )(t4 ; t1 )t2: The formulae for the second coordinate system, P1xy z , are similar. 0

0

(4.10) (4.11)

4.4. Acuteness and local convexity z

z’

49 z

z

y

11111111111111111111111111111 00000000000000000000000000000 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 xz-plane 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 xy’-plane 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 xy-plane 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 xz’-plane 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000000 11111111111111111111111111111 00000000000000000000000000 11111111111111111111111111

z’

z’

qz

qy

q

y’

x

q

q q

y

y

y

q z’

lx

x

ly

y’ y’

l y’ l z’

lz

y’

y’

Figure 4.4: Subspaces that the two coordinate systems can create, and possible positions of the components of Q_ (u) and L(u).

The cross-product L(t) Q_ (t) can be analyzed in the cross-products of the components of L(t) and Q_ (t), which lie on the axes of xyz and xy z (Fig. 4.4). Calculating the nine cross-products, we see that the components of L(t) Q_ (t) lie within a subspace formed by x-axis, the negative z -axis and the positive z-axis. Then the only case that an inection could happen would be if the vector lied in x-axis, which is incongruous, since the y-component of that vector, Lz (t)Q_ x(t) ; Lx (t)Q_ y (t) is always non-zero. That is true, because Lz (t)Q_ x(t) is always negative, while Lx(t)Q_ y (t) is non-positive. So, the possible directions of normal vector in the segment lie within the convex spherical sector formed be the unit sphere at P1, the positive x;axis, the positive z;axis and the negative z ;axis. 0

0

0

0

Based on the results above, we can state that, if the control polygon of a B ;spline curve is locally acute, within the segment it inuences neither an inection nor even a latent inection occur. Furthermore, the Ff changes predictably therein.


50

4.5 Formulation of the problem In former sections the sucient conditions for the control polygon are set, so that the manipulation of the shape of the curve is guaranteed in any segment. It is also well known how one can create planar segments (zero torsion) or straight-line segments (zero curvature) by enforcing a special con guration on the control polygon (see, e.g. Farin '93], Hoschek & Lasser '93]). Having all these in mind, we set the following problem:

Problem (Pshape): Let Q(0) (u) u 2 u3 uN 3] be a C 2 cubic B ;spline with a knot vector U = fui i = 0(1)N g and a control polygon D(0) = fd(0) i i = 0(1)M g N = M + 4. Construct a cubic B ;spline Q(u) with the same knot vector U and a new control polygon D, which satis es the delity criterion: ;

M X i=0

2 kdi ; d(0) i k = ^min d IR

i

2

3

M X 2 kdî ; d(0) i k

(4.12)

i=0

and ful ls the shape requirements (i),(ii) and the tolerance constraints (iii): (i) Torsion constraints: Let It = fr 3 r N ; 4g be a set of indices, specifying the intervals ur ur+1], where the sign of torsion can be predetermined. For the corresponding segments the torsion can be positive, negative, or even zero (planar case), that is: jdr 3dr 2 dr 1j > 0 dl 2 dl 1 > 0 and (dl 3 dl 2 ) (dl 2 dl 1 ) > 0 (4.14) ;

;

;

;

;

;

;

;

4.6. Examples

51

or be co-linear: dl 3 dl 2 = 0 and dl 2 dl 1 = 0: ;

;

;

;

(4.15)

(iii) Tolerance constraints: each control point, di, should lie within a sphere with centre at d(0) i and user-speci ed radius, ri : 2 kdi ; d(0) i k ri :

(4.16)

(Pshape) is a Non-Linear Programming (NLP) problem, consisting of a quadratic objective function, bi- tri- and quarti-linear shape constraints and quadratic tolerance constraints. Because of the tolerance constrains, (Pshape) is not always solvable. The acuteness of the control polygon is materialized by replacing zero in the strict shape constraints with suciently small numbers : All these and ri are parameters for the problem. Any feasible data for the optimization problem guarantees the non-existence of latent points. If no feasible point is found, one should permit larger deviations from the initial control points and/or decrease of the -values. (Pshape) is solved with the Sequential Quadratic Programming (SQL) technique , where the whole problem is analysed in a number of Quadrartic Programming (QP) problems (see, e.g., Gill et. al '81]). The software implementation that has been used for solving (Pshape) is provided by Spellucci '95].

4.6 Examples In order that the performance of the algorithm can be tested, the employed data sets are the same to those of Chapter 3. Again, the initial curve is depicted with dashed line, while the nal one with solid line. The same convention stands for curvature and torsion plots. In both examples, the rst and the last control points remain unchanged. The gures are in exactly the same order as in the previous chapters, namely rst the orthogonal

52


projections, second the curvature and torsion plots and, nally, the distributions of the binormal vector and the curvature vector (scaled).

Example 1: For the chine data, the permissible deviation for the control points is set

to ri = 0:050m i = 2(1)14 (reference length: 10:000m), and the sign of torsion has to be negative. The maximum deviation between the initial and the nal curve is found to be 0:029m, the maximum nodal deviation 0:020m and the sum of nodal deviations is = 0:123m. The resulting curve is almost planar, with no inections or latent inections (Figs. 4.7, 4.8), though the changes in curvature plot are not impressive (Fig. 4.6). In Fig. 4.5, one can hardly distinguish the initial curve from the nal one.

Example 2: The permissible deviation for the control points of the GM data is kept to

the industry standards, namely ri = 1:000mm i = 2(1)21 (reference length: 700mm), and the sign of torsion is set negative. The resulting curve exhibits maximum nodal deviation 0:660mm and sum of nodal deviations = 4:729mm. The quality of the resulting curve can be checked through the torsion plot in Fig. 4.10, as well as with the distributions in Figs. 4.11, 4.12.

4.7 Conclusions - Future work Observing the results of existing fairing methods, the one of Chapter 3 included, one can realise that ill-conditioned areas on a curve, like latent inections, are very persistent and change with great diculty. The proposed method is { in some sense { constructive, since it forms a locally acute control polygon, and guarantees the elimination of the problem. Though the non-linearity of the inequalities sometimes make the solving process uncertain and time-consuming, the solution is very satisfactory and appropriate for re ned data close to planar, which, in most cases, exhibit acceptable curvature and local aberrations of torsion.


53

Linearization of the inequalities would be an obvious improvment for the method, since the problem would become a QP one, which guarantees the existence of solution. Also, this will enable the acceleration of the process and the determination of a good starting point for the existing algorithm.

0

0

Figure 4.5: Orthogonal projections of the initial (dashed) and nal (solid) chine curve.

0

1.6

0

0

1.6

1.6

0

1.6

10

10

54


Figures of examples

55

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 u3

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

20

0

-20

-40

-60

-80

-100

-120

u3

5

0

-5

-10

-15

u3

Figure 4.6: Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) chine curve.


56

2 S 0

-2 6 0

10

0

2 S 0

-2 6 0

10

0

Figure 4.7: Binormal vector distribution of the initial (upper) and nal (lower) chine curve.

Figures of examples

57

S 2 0 -2 0

10

6

S 2 0 -2 0

10

6

Figure 4.8: Curvature vector distribution of the initial (upper) and nal (lower) chine curve.

0

0

Figure 4.9: Orthogonal projections of the initial (dashed) and nal (solid) GM curve.

4650

4650 950 1050

5050

5050

950

1050

700

700

58


Figures of examples

59

0.012

0.01

0.008

0.006

0.004

0.002

0 u3

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

u4

u5

u6

u7

u8

u9

u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

2

1.5

1

0.5

0

-0.5

-1

u3

0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025

u3

Figure 4.10: Curvature (upper) and torsion (lower) plots of the initial (dashed) and nal (solid) GM curve.


60

1050 S

650

5100

700

4600

0

1050 S

650

5100 700 4600

0

Figure 4.11: Binormal vector distribution of the initial (upper) and nal (lower) GM curve.

Figures of examples

61

S 1050

5100

650

700

0

4600

S

1050

5100

650

700

4600

Figure 4.12: Curvature vector distribution of the initial (upper) and nal (lower) GM curve.

Chapter 5 Fairing of 2D curves under design constraints 5.1 Introduction The subject of this chapter is how one can produce fair planar curves, which possess a number of properties of practical interest. These properties may be of geometric nature { i.e., the convexity of the curve locally, the deviation of the curve from a given point set, the end conditions of the curve, but also general design constraints { i.e., the area under a curve and the area centroid as well as higher moments of it. This problem has been partially faced by Nowacki et al. '90], Nowacki & Lu '94], whereas here is set and solved in a more general frame. The resulting algorithm can be eciently used in the context of CAShipD: it o ers the ability of determining ship lines when only a rough description of geometry of the hull is given and the integrated characteristics (hydrostatics) of the ship are available. The proposed method is presented through a detailed practical example.

5.2 Prerequisites Some aspects connected with properties of B ;splines are examined, so that the constraints of the fairing problem can be set in terms of the control polygon of the curve, not 62

5.2. Prerequisites

63

a ecting the initial parametrization.

5.2.1 End conditions Based on eqs. (2.21), (2.22) and (2.23), it can be shown (see Kapniaris '95]) that the rst (second) derivative at both ends is controlled by only two (three) control points, provided that the curve is open and the end knots are of multiplicity n + 1. More speci cally, the rst-order boundary derivatives are given by:

Q_ (un) = u n; u d0 n+1 1 and

Q_ (uM +1) = u

n

M +n ; uM

(5.1)

dM 1:

(5.2)

;

Now, for the second-order bounaryderivatives, we have:

"

Q (un) = un(n ;; 1)u u 1; u d1 ; u 1; u d0 n+1 2 n+2 2 n+1 1 and

Q (uM +1) = u n(n ;; 1)u M +n 2 M ;

" 1

;

1

#

(5.3)

1

uM +n 1 ; uM 1 dM 1 ; uM +n 2 ; uM 2 dM ;

;

;

;

;

# ;

2

: (5.4)

Finally, for closed curves, periodic conditions (d`Q=du`ju=u0 = d`Q=du`ju=uM +n+1 ` = 0(1)n ; 1) can be achieved when:

d = dM

;

n+i+1

u = uM +n

i

; ;

1

i = 0(1)n ; 1

(5.5)

with ui = ui+1 ; ui:

5.2.2 Calculation of integrated measurs In case that the curve, Q(u) = (Qx(u) Qy (u))T , is a Jordan curve that bounds a planar compact domain D with boundary @D, the area of that domain and its rst- and


64

second-order moments, with respect to the axes of a Cartesian co-ordinate system, can be eventually expressed through the control polygon of the curve with the aid of the two-dimensional Stokes formula: ZZ @Q(x y) @P (x y) I ; dx dy = ; P ( x y ) dx ; Q ( x y ) dy (5.6) @x @y D @D H where P (x y)i + Q(x y)j is a di erentiable vector eld and ` ' denotes clockwise contour integration. The area A of D can be obtained from the above formula by setting Q(x y) = 0 and @P=@y = ;1, which gives: A = =

ZZ

D MX +n

dx dy =

Xi

i=0 j k=i n

I

@D

dyj dxk

;

y dx =

Z ui+1 ui

Z

U

MX +n Z ui+1 Qy (u) Q_ x(u) du Qy (u) Q_ x(u) du = i=0 ui

Njn(u) N_ kn(u) du:

(5.7)

The rst-order moments of D with respect to the Ox;/Oy;axis are obtained by setting Q(x y) = 0 and @P=@y = ;y /;x, respectively: ZZ I Z Mx = ydxdy = 12 y2 dx = 12 (Qy (u))2 Q_ x(u) du (5.8) D @D MX +n Z ui+1 1 = ; (Qy (u))2Q_ x(u) du 2 i=0 ui U

MX +n Xi y y x Z ui+1 n 1 dj dk dl Nj (u) Nkn(u) N_ ln (u) du = 2 u i i=0 j k l=i n

(5.9)

;

and My = = =

ZZ D

xdxdy =

I @D

x y dx =

Z U

Qx(u) Qy (u) Q_ x(u) du

MX +n Z ui+1 Qx(u) Qy (u) Q_ x (u) du i=0 ui MX +n i=0

Z ui+1 y x x dj dk dl Njn (u) Nkn (u) N_ ln (u) du: ui j k l =i n Xi

;

(5.10)

5.2. Prerequisites

65

The second-order moments, also called inertia, of D with respect to the Ox; /Oy;axis are (Q(x y) = 0 and @P=@y = ;y2 /;x2 respectively): Ix =

ZZ

I Z 1 1 3 y dx dy = 3 y dx = 3 (Qy (u))3 Q_ x(u) du D @D 2

U

MX +n Z ui+1 = 13 (Qy (u))3 Q_ x(u) du u i=0 i

MX +n Xi y y y x Z ui+1 n = 13 dj dk dl dr Nj (u)Nkn(u)Nln(u)N_ rn(u) du ui i=0 j k l r=i n

(5.11)

;

and Iy = = =

ZZ D

x dx dy = 2

I @D

Z

x y dx = (Qx(u))2 Qy (u) Q_ x (u) du 2

U

MX +n Z ui+1 (Qx(u))2 Qy (u) Q_ x(u) du u i i=0 MX +n

Xi

dxj dxk dyl dxr

i=0 j k l r=i n ;

Z ui+1 Njn (u)Nkn(u)Nln(u)N_ rn(u) du: u i

(5.12)

Now, if the curve, Q(u), is open and represents a function with respect to the Ox;axis, then formulae (5.7)-(5.12) have to be modi ed by simply restricting the range of index i from i = 0(1)(M + n) to i = 1(1)M: Furthermore, if the limits of integration do not coincide with knots of the given parametrization, U , then knot insertion can be employed (see, e.g., Hoschek & Lasser '93], pp. 190-195), so that the curve remains unchanged and the integration limits become knots. The integrals of the basis functions can be calculated exactly with Gauss quadrature (see, e.g., Hildebrand '56], ch. 8), which exhibits no loss of signi cance for any order or knot sequence, and the evaluation cost per segment is quite satisfactory Vermeulen et al. '92]. More about calculations of area and moments can be found in Kapniaris '95].


66

5.2.3 Local convexity of planar cubic B ;splines Referring to local convexity we mean the convexity of a segment of Q(u), namely the constancy of sign for the signed curvature in that segment. From eq. (2.10) one can see that the sign depends on the numerator of curvature, namely on L(u), which, for a B ;spline of arbitrary degree, n, can be expressed as:

L(u) = n2 (n ; 1)

i 1 X

Xi

;

j =i+1 n k=j +1

dj 1 dk 1 Anj k(u) ;

(5.13)

;

;

1 j +n ; uj )(uk+n ; uk )

where Anj k (u) = (u

h (uk ; uj )Nkn 2 (u)Njn 2(u) (u ; u )N n 2 (u)Njn+12(u) + k+n j+n k+1 + (uk+n 1 ; uk )(uj+n 1 ; uj ) (uk+n ; uk+1)(uj+n ; uj+1) (uk+n ; uj )Nkn+12 (u)Njn 2(u) i (uj+n ; uk )Nkn 2 (u)Njn+12(u) (uk+n 1 ; uk )(uj+n ; uj+1) ; (uk+n ; uk+1)(uj+n 1 ; uj ) (5.14) and j = i + 1 ; n(1)i ; 1 k = j + 1(1)i: ;

;

;

;

;

;

;

;

;

;

;

;

The function Anj k(u) is positive if the parametrization is uniform, because in that case the multiplier of the negative term in the right-hand side of (5.14)equals to the sum of the multipliers of the positive terms, while the products of the basis functions for the positives terms are greater or equal to the product of the negative term.

Proposition 5.1 The signed curvature of a planar B ;spline curve, Q(u), of degree n and uniform parametrization, is of constant sign in ui ui+1] if all cross products dj 1 dk 1 j = i + 1 ; n(1)i ; 1 k = j + 1(1)i are of the same direction. ;

;

In the particular case of n = 3, L(u) is analysed according to (4.2) and (4.3). Baring in mind that j (u) 0 j = 1 2 3 for an arbitrary parametrization, it can be stated that

Proposition 5.2 The signed curvature of a cubic B ;spline curve, Q(u), is of constant

5.3. Fairing of B ;spline curves under constraints sign in ui ui+1 ] if di 3 di direction. ;

2

;

di 2 di ;

67

and di 3 di 1 are of the same

1

;

;

;

So, if the sign is predetermined, then the unit vector i = (1 0 0)T can be introduced for de ning the sense that the cross products should share in order that the signed curvature is positive/negative in ui ui+1]:

i (di 3 di 2 ) 0 and i (di 2 di 1 ) 0 and i(di 3 di 1 ) 0: (5.15) ;

;

;

;

;

;

Also, it can be shown that (see Theorem 2 in Pigounakis & Kaklis '94]):

Proposition 5.3 If jjdi 3jj = jjdi 2jj = jjdi 1 jj, the knots uj j = i ; 2(1)i + 3 are equally spaced and the cross products di 3 di 2 di 2 di 1 are of the same direction, then the signed curvature of the cubic B ;spline curve, Q(u) is of constant sign ;

;

;

;

;

;

;

in ui ui+1]:

5.3 Fairing of B ;spline curves under constraints 5.3.1 The problem (Pdesign) The problem of fairing with constraints is approached as an optimization problem and set as follows:

Problem (Pdesign) : Let Q0 (u) be a C 2 cubic B ;spline curve, and D0, U its control polygon and knot vector, respectively. Retaining U , nd Q (u), which minimizes the

functional:

J (Q) = w0 J0(Q) + wr1 Jr1 (Q) + wr2 Jr2 (Q) w0 + wr1 + wr2 = 1 w0 wr1 wr2 0 (5.16) with

J0(Q) =

M X i=0

kdi ; d0i k2

(5.17)


68 and

Jr (Q) =

Z uM +1 un

kQ (u)k du = (r)

2

Z uM +1 un

Q(xr) 2(u) + Q(yr) 2(u)du

(5.18)

under the following constraints: (i) Boundary constraints: For closed curves, conditions (5.5) should hold. For open curves, the rst- and/or second-order derivative at each end, as presented in (5.1)(5.4), should be equal to speci ed vectors. (ii) Area constraints: For closed curves, the area of the included domain is given by (5.7), whereas for open curves the range of index i in (5.7) should be restricted from i = 0(1)(M + n) to i = 1(1)M: The area should not deviate from a speci ed value more than a certain tolerance. (iii) Moments-of-area constraints: For closed curves, rst- and second-order moments are calculated using the formulae (5.9)-(5.12), whereas for open curves the range of i in (5.9)-(5.12) should be restricted from i = 0(1)(M + n) to i = 1(1)M: The moments should not deviate from speci ed values more than a certain tolerance. (iv) Local-convexity constraints: For cubic splines, inequalities (5.15) can be set for every segment that its curvature sign i is predetermined. Also, for cubic splines with almost equally spaced vertices and uniform parametrization, it is safe to take advantage of Corollary 5.3. For arbitrary degree, the parametrization should be uniform in order to bene t of Proposition 5.1. (v) Tolerance constraints: The deviation of nodal points, Pij=i n dj Njn (ui), from the nodal points of the initial curve should be bounded: ;

j

Xi

(dxj ; dxj0)Njn (u)j xi

j =i n ;

j

Xi

(dyj ; dyj 0)Njn (u)j yi

j =i n ;

where i = n(1)M + 1, and xi yi are user-speci ed tolerances.

(5.19)

5.3. Fairing of B ;spline curves under constraints

69

The functional J0(Q) in (5.16) measures the delity of the curve Q(u) to the initial one, Q0 (u), while the other two functionals, Jr (Q) r 3, measure the smoothness of Q(u): J2 and J3 express the (simpli ed) bending energy and the (simpli ed)jerk of the curve, respectively, and are commonly for global fairing (see, e.g., Eck & Hadenfeld '95a], Nowacki & Lu '94], Reinsch '67], Reinsch '71]). Note that, in terms of the control polygon, D, the smoothness functionals are expressed as below:

Jr (Q) =

M Z ui+1 X i X

i=n ui

j =i n

dxjNjn (r)(u)

;

Xi k=i n ;

dxkNkn(r) (u)+

Xi j =i n ;

dyj Njn (r) (u)

Xi k=i n

dyk Nkn(r) (u) du

;

(5.20) and nally:

Jr (Q) =

Z ui+1 T dj dk u Njn (r) (u)Nkn(r)(u)du i i=n j k=i n M X

Xi

dTj dk = dxjdxk + dyj dyk :

(5.21)

;

5.3.2 Solution of the problem (Pdesign) - An example (Pdesign) is an NLP problem with respect to the vertices of the control polygon, D, of the sought-for curve Q(u). More speci cally, the objective function, J (Q), is quadratic with respect to the control vertices, di i = 0(1)M , while the family of constraints includes linear (boundary, tolerance), bilinear (local convexity), quadratic (area), cubic ( rst-order moments of area) and quartic (second-order moments of area) inequalities. If the feasible space, de ned by the constraints of the problem, is non-void, (Pdesign) is numerically solved by employing the SQP technique (see, e.g., Gill et. al '81]) in its NAg '90] implementation. In this implementation, the user should provide an initial point that is feasible with respect to the linear constraints of the problem. In our case, the starting point is chosen to be the C 2 cubic B ;spline, that interpolates the initial nodal points under the prescribed boundary conditions, thus satisfying the linear (boundary, tolerance) constraints of (Pdesign).

70


The various integrals, appearing in the analytic expression of the objective function (see eqs. (5.18), (5.21)), and the area and moments-of-area constraints (see eqs. (5.7)-(5.12)), retain constant values throughout the optimization procedure since the parametrization U remains xed and they are calculated accurately by means of the Gaussian quadrature. Finally, it is worth noticing that, when U is close to be uniform and the lengths of the legs of the initial control polygon, D0, are almost equal, then it is rather safe to appeal to Proposition 5.3 for simplifying (Pdesign) with respect to the local-convexity constraints. A number of examples are presented in Kapniaris '95], but no inertia constraints are em00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111111111111111111

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x

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Figure 5.1: Planar symmetrical section of a symmetric body. Axis of symmetry: Ox. ployed there. Here, the algorithm is tested for a curve which represents half the boundary of a section of a symmetric body! see Fig. 5.1. The curve is a C 2 cubic B ;spline interpolating eighteen (18) data points points obtained with digitizing, so the control polygon consists of twenty (20) vertices. The parametrization is with respect to chord length of the data points, and the end conditions are Q_ (u3) = (1 0)T and Q_ (u20) = (0 ;1)T . The task for this curve is to retain its shape and the sense of the end conditions and to extend it to the x-direction from 82m to 92m, while the y-direction should not extend more than 6m (half breadth). Furthermore, the area centroid should be located at about 52% of the x-length of the curve, namely at 47:84m from the starting point, and the area concluded by the new section should be A = 840m2. So, the moment of sectional area should equal to Myy = 40200m m2. The inertia of the nal section cannot be estimated so far. In Figs. 5.2, 5.3 at the top and with long dashes the control polygon and the curve is pre-

5.3. Fairing of B ;spline curves under constraints

71

sented, respectively. The curvature plot can be found in Fig. 5.4 also with long dashed line in two di erent scalings. The characteristics of that initial curve is shown in Table 5.1. Since the curve is a B ;spline, an obvious solution for extending the x-direction would be to transform the control polygon of the curve in the x-direction. Due to B ;spline properties, such a transformation results in a new curve, which retains the linear part around the middle and the sense of the end conditions. The characteristics of the transformed curve are given in Table 5.1 and the control polygon, the curve and its curvature plot are given in Figs. 5.2, 5.3 and 5.4, respectively, denoted with short dashed line. The transformed curve is the initial `vector' for the optimization algorithm. First, the deviation from the transformed curve is not bounded, as we are interested only in the area and the centre of area of the curve. The constraints for those are considered satis ed if the values exhibit a deviation of 1%. Then the algorithm gives Solution 1 (see Table 5.1) shown in Figs. 5.2 (control polygon) and 5.3 (curve), third from top with thin solid line. The curvature plot is given also with thin solid line in Fig. 5.4. As one can observe, Solution 1 is fair enough and ful ls the area and centroid constraints but its shape does not resemble to that of the transformed curve. The reason is that neither tolerance constraints nor inertia constraints are active. In order that the shape of the transformed curve is kept, tolerance constraints are set for the parallel part as well as for the area near the second end. The tolerances of the nodal points for the parallel part are 0:050m (y-direction) and 5:000m (x-direction), and for the parts of the curve near the ends are 0:100m (y-direction) and 2:500m (x-direction). Furthermore, the inertia constraints are activated and set to Iyy = 1:7 104 m m m2 and Ixx = 7:5 105 m m m2 for the whole sectional area, and they are considered satis ed when the values deviate up to 2%. The values for the inertia constraints are set so, because the distribution of area should be changed, i.e., area should be moved away of the axis of symmetry and gathered near the middle of the section along its length. The result of this run is Solution 2 (see Table


72

5.1 and Figs. 5.2, 5.3, 5.4). Both solutions are achieved within a limited number of iterations (between 10 and 20) and the real time neaded for optimization is about 30secs. The weight of the three terms in (5.16) are equal. Though (Pdesign) is not always solvable, an experienced user can set the constraints in such a way that the desirable solution is found in relatively short time. Curve Area A Moment Myy Centre Myy =A Inertia-x Ixx Inertia-y Iyy P

Initial Transformed Solution 1 Solution 2 Deviation 711.9 800.8 844.2 838.2 0.2% 28567. 36175. 20009. 40128.2 >0.1% 40.13 45.18 47.40 47.88 0.2% 13830. 15558. 16801. 17088. 0.5% 479630. 682672. 774024. 743586. 0.9% 0 Q jju=ui | | 6.499 15.382 R jjjjQQjj2; 2.068 2.356 1.873 2.169 ... 2du R jj Q jj du 3.118 3.567 1.854 3.436

Table 5.1: Characteristics of the boundary curves of the example section.

5.4 Conclusions - Future work In this chapter we presented a method of fairing planar curves under constraints which are sound for design and cover end, tolerance, area, moments of area and shape requirements. The innovation of the method lies mainly on the number of kinds of constraints and their nature (linear / non-linear). The performance of the developed algortihm is quite satisfactory, so it can be used in a CAD system, or even become a kernel of a more complex method for handling curve meshes (e.g. ship lines). This makes the particular tool very useful, as one can also see in the next chapter. It would be interesting if the geometric conditions of the method would enriched with conditions of uid mechanics, so that the solution would be optimalfor the ow round the


73

sought-for section. Also, the linearization of the problem would result to faster performance.

0

7

0

7

0

7

0

7

-5

-5

-5

-5

Solution 1 (thin solid), Solution 2 (thick solid).

Figure 5.2: Control polygons: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed),

95

95

95

95

74


(thin solid), Solution 2 (thick solid).

Figure 5.3: Curves: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed), Solution 1

95

0 -5 7

95

95

0 -5 7

0 -5

95

0 -5 7

7

Figures of example 75


76

1

0

-1

-2

-3

-4

-5 u3 u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u16

u17 u18u19u20

u3 u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

u14

u15

u16

u17 u18u19u20

0.15

0.1

0.05

0

-0.05

-0.1

Figure 5.4: Curvature Plot: Interpolant of the initial data (long dashed), Anely transformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid).

Chapter 6 Applications in Computer Aided Ship Design 6.1 Introduction

The majority of designers still prefer the traditional way of hull description, i.e. through a mesh of planar curves, to surface de nitions, which are currently available in commercial systems Wake '93]. There are at least two reasons for that: (i) the inherent diculty in handling a large and complex surface, like the one representing the ship hull, and (ii) the methodologies of naval architecture, which make extensive use of hull sections Comstock-PNA '67], Rawson & Tupper '68]. Nevertheless, Computational Geometry Nowacki et al. '95], Su & Liu '89], has started showing o methods that naval architects can bene t of in Ship Design. Taking advantage of the previously presented algorithms, this chapter proposes a process, which is a compromise of the old and the new trend in hull design. In order to describe the hull of a ship, sections of the hull along di erent directions are developed, forming a dense wireframe of fair planar lines { stations, waterlines, buttocks, which satisfy requirements for shape and hydrostatics. 77

78


6.2 Ship design and hull geometry Ships are quite expensive and complex constructions, with special needs and tasks, which service mainly in an inhospitable environment. Thus, a new building is decided only if an actual or forecast situation shows the ship's necessity and makes the investment sound. For merchant ships, the basic need is for transporting a payload { whether it is passengers or cargo { under certain speci ed conditions of speed, distance and environment. This information is coded in the term owner's requirements. The variety of payloads and service conditions leads to di erent types of ships, e.g. passenger ships, ferry boats, oil tankers, bulk carriers, etc., resulting to di erences in hull geometry. The naval architect applies special knowledge to meet the owner's requirements and ensure that the ship is optimum, from the engineering point of view. Hydrodynamics, structural analysis, as well as economics form the framework of the design, generating several interdepended studies and projects to be carried out by the design team. The endeavour is conveyed by the design spiral, each coil of which is of smaller uncertainty than the previous one, and nal design lies in the centre Papanikolaou '89], Rawson & Tupper '68]. Starting from a feasibility study, called concept design, the spiral goes on to preliminary design, where all main characteristics are determined, employing standard methodologies and techniques of naval architecture. The next stage is known as contract design, when the design team of the shipyard are bound for the successful materialization of their choices, and the spiral ends with the detailed design. As one can realize, preliminary design is vital for the success of the whole undertaking. According to Rawson & Tupper '68], an accuracy of about 5 per cent is desirable for this stage, where the naval architect concentrates in space allocation, main dimensions, dislacement and deadweight, form parameters, weather deck layout, and machinery type and layout. After having estimated the principle dimensions and the displacement, the classical approach demands the determination of the curve of areas for the below water

6.3. Development of ship lines

79

portion of the hull. This may be based on parent ships, or on systematic series data, which also provide the required powering characteristics. The curve of areas, or sectional area curve (SAC), must then translated into a hull geometry, that is, into a `body plan' (sections of the hull along the ship's length), which is also based on parent ships or systematic series (e.g. Guldhammer-FORMDATA]). Finally, other sets of sections along the rest principle dimensions are calculated and drawn. That phase is known as lines development.

6.3 Development of ship lines As stated above, the hull of a ship is traditionally described through a set of curves, called ship lines. These curves are mainly planar and represent sections of the ship hull along her length, her draught and her breadth and called stations, waterlines and buttocks, respectively. Sectional curves in the same direction are drawn together, namely stations compose the body plan of the hull, waterlines compose the half-breadth plan, and buttocks compose the sheer plan. Along with these lines, curves that represent intersections of the hull with inclined planes are also typical for the lines plan. Such curves are the diagonal lines, or the boundary curves of the at of side (FOS) and the at of bottom (FOB). Furthermore, a few spatial curves are also present, like deck or `knuckle' lines, which delimit regions of the ship. Details for the lines plan can be found in Comstock-PNA '67]. Baring in mind the capabilities of Computational Geometry, the problem of designing a hull is set as follows Nowacki et al. '95]:

Problem (Pmesh) : Develop a set of curves, most of them planar curves, such that the mesh of ship lines is formed by which a ship surface geometry of some desirable shape is de ned. The characteristics of the 'desired shape' are related not only to the geometric shape,

80


but mainly to form-parameters of the individual curves and some global properties of the hull. These are the hydrostatic characteristics of the hull, also called hydrostatics, which are to be determined before the development of lines, since they are closely related to the performance of the ship and derive almost explicitly from the owner's requirements. Among hydrostatics, the Bonjean curves are of vast importance, because they express the volume distribution of the hull. In the rst step of lines development, the resulting stations should satisfy at least the SAC, which expresses the intersection of the Bonjean curves with the still water plane. Starting from the body plan, one can obtain information and form the half-breadth plan and the sheer plan. In order to obtain the desirable hull description, the designer can either start an ab initio design, or distort an existing successful one. The rst choice used to be not so favourite, since testing a totally new hull is a rather dicult task. Nonetheless, modern CAShipD systems o er such an opportunity for quick and relatively accurate calculations of the performance of a new design, though hydrostatics are checked a posteriori and cannot be evaluated before or during the design process. The second choice seems more secure and is still followed by many designers, though experience has shown that small and medium sized CAShipD systems handle with diculty distortion problems, as they are oriented to the previous approach, and rely on the capabilities of the user. The use of systematic series can be also classi ed in the latter case, since non-dimensional descriptions of a model are adapted to the needs of the current designer. Systematic series provide good estimates for the shape, the hydrostatics and the resistance of the ship hull. In the proposed process, the designer can start either ab initio, or based on existing data. First, the main dimensions and hydrostatics are decided, and then any of the three set of lines of the hull is faired, subjected to the afore mentioned characteristics, which are de ned in the following subsections. A more complete approach would employ a continuous description of the Bonjean curves, namely a Bonjean surface. Such a surface includes all information needed for the development of a hull, so, along with shape, it

6.3. Development of ship lines

81

would consist the constraint for the hull surface, and would o er the ability of solving the problem (Pmesh) in a one-step process.

6.3.1 Main characteristics The basic geometric characterists of a ship are her main dimensions, namely

length LOA=LW : overall or measured on the design waterline (DW), draught T : vertical distance between the DW and the basic reference plane that the ship is laid on, measured at the middle of her length (midship section),

breadth B=BW : maximum breadth of the midship section / maximum breadth of the DW, respectively,

depth D: vertical distance between the basic reference plane and the main deck, measured at the midship section. From now on, reference to the main dimensions implies dimensions measured on or from the design waterline. The hydrostatics that are taken into consideration for the proposed process are:

Displacement, r, block coecient, CB : The displacement is the volume of the under water portion of the hull and CB = r=(LW BW T ). Maximum sectional area, AX , maximum section coecient, CX : The maximum area of station up to DW, and CX = AX =(BW T ). Sectional area curve (SAC) : The area of station up to DW vs. the length of the ship.


82

Sectional leverarm curve (SLC) : The vertical moment leverarm of transverse section up to DW vs. the length.

Longitudinal centre of buoyancy (LCB) : The longitudinal position of the centroid of the displacement.

Area of the design waterplane, AW , waterplane coecient, CW = AW =(B LW ). Longitudinal centre of otation (LCF) : The longitudinal position of the centroid of AW .

Inertia moments of AW , Ixx Iyy .

6.3.2 Process for solving the problem (Pmesh) The procedure proposed here for solving (Pmesh) is mainly based on the algorithm presented in Chapter 5, henceforth called the core algorithm, and follows a number of steps: 1. Determine or estimate the following characteristics: LW , BW , T , D, r, SAC, LCB, SLC, AX , AW , LCF. 2. Determine the necessary control lines, i.e. FOS, FOB, knuckle lines, deck line, pro le line etc., and fair them. 3. Choose positions for developing a set of sections. Estimate end conditions and shape and provide/obtain an initial point set for each section with the aid of the control curves of Step 2. Interpolate all data sets and fair the resulting curves with the core algorithm, so that all constraints, derived by the hydrostatics or the shape and the ends of each curve, are satis ed within a tolerance. 4. Choose positions for developing a set of sections, in a vertical direction to the one of the already developed one(s). For each position, obtain an initial point set from the

6.4. A CAShipD example

83

already developed set(s) and interpolate appropriately. Using the core algorithm, develop a fair set of sections while retaining within a tolerance all the requested characteristics. 5. Check the nodal points of the so far derived mesh. If the deviation between the sets of curves is less than a tolerance, stop the process, else return to Step 4. As it is mentioned, from the very start of a design the characteristics of Step 1 can be evaluated and remain more or less unchanged throughout the whole process. The estimates and choices of Step 2 remain unchanged, too. In Step 3 an initial geometry, compatible to the curves of Step 2, should be provided by the user or by a data base and imported into the system. Along with geometry, the available hydrostatics should be set as constraints. If no hydrostatics or distribution curves are available, the ones of the input geometry can be used. Step 4 is repeated for any set of sections until all sets coincide at the nodal points of the mesh (within a tolerance). This checking is done in Step 5. Though there is no evidence for convergence, this process has been used for the development of lines of a few hulls, which have presented good quality of the resulting meshes and satisfactory hydrostatics (deviation from the required ones less that 2 per cent).

6.4 A CAShipD example The development of a lines plan with the aid of a procedure, similar to the one proposed here, has been successfully applied in Kapniaris '95]. The derived lines plan is of high quality and has been developed in two loops of the process. In forth, another example is presented, where the hull of an existing ship is available in electronic form (digitized), as well as her main characteristics! see Table 6.1. The task is to develop a new hull, with greater displacement of about 20 ; 25%, say 530m3 and retain the breadth of the ship. For this purpose, Chapter 66{'Steps in the Preliminary Design' of Saunders '57] is


84

followed. From Fig. 66E, p.470 (ibid.), the length of the hull is estimated to be about 92m, which means an enlongation of 10m. According to Fig. 66A (ibid), the displacementlength quotient implies that the Froude number should be about Fn = 0:301 and the prismatic coecient should be near to CPX = 0:60. The midship section coecient is determined from the existing initial hull and set to CX = 0:894. Then, the block coecient is about CB = 0:54 and the draught (DW) is calculated at T = 4m3. Also, according to Saunders '57], LCB should be located at 48:5 ; 50:5% of LW , measured from stern end (Fig. 66.N), while the midship section should be located at 46 ; 50% of LW (Fig. 66L). The centre of otation is normally between the midship section position and the position of LCB. The higher the Froude number, the more abaft all three positions should lie. Additionally, an estimation can be made for the waterline coecient and the inertia moments, but these magnitudes can vary more than others. The depth of the new hull remains unchanged, since the change of draught is rather small, and the remaining freeboard is judged enough, according to holding regulations. The desirable characteristics of the nal hull are also given in Table 6.1, and mean values are chosen in the cases of magnitudes within a range.

Analysis of the design process In this example only the body plan and the half-breadth plan are involved in the design process. This choice is based on the observation that, usually, no special requirements for areas or moment of areas are set for buttocks. Additionally, the shape cannot be predetermined, at least not so much as the shape of a waterline or a station. Therefore, buttocks are not adequate for evaluating the process. On the other hand, an example that makes use of the sheer plan can be found in Kapniaris '95]. First, the body plan of the initial ship is digitized and the derived points are interpolated Due to resistance reasons, the area near this specic Froude number is generally avoided in designs. Nonetheless, the derived hull is geometrically acceptable, and the ship can nally service with a speed of 20 knots, i.e., Fn = 0:34, out of the undesirable area. 1

6.4. A CAShipD example

85

Characteristics Initial ship hull New ship hull LW 82:00m 92:00m B 14:65m 14:65m T 3:85m 4:00m D 5:60m 5:60m 3 r 2368m 2900m3 CB 0:516 0:54 LCB 42:43m 45:0m 2 AX 50:40m 52:75m2 CX 0:894 0:894 CPX 0:578 0:60 2 AW 881:2m 1025m2 CW 0:734 0:76 LCF 40:14m 43:5m 4 Ixx 11485m 14000m4 4 Iyy 328900m 500000m4

Table 6.1: Initial and required characteristics of the hull. Characteristics Hull0.1 Hull0.2 LW m] 81:71 81:71 B m] 14:57 14:54 T m] 3:85 3:85 D m] 5:60 5:59 r m3 ] 2368 2378 CB 0:517 0:520 LCB m] 42:14 41:04 AX m2 ] 50:60 50:04 CX 0:902 0:894 CPX 0:573 0:582 AW m2 ] 863:2 864:1 CW 0:725 0:727 LCF m] 40:24 40:01 Ixx m4 ] 11139 11129 Iyy m4 ] 318073 319333

Hull1.1 92:07 14:62 4:03 5:58 2904 0:535 45:01 52:70 0:894 0:598 1030:8 0:766 43:53 13932 515177

Hull1.2 92:07 14:60 4:03 5:58 2910 0:537 45:11 52:71 0:896 0:598 1029:7 0:766 43:49 13750 520491

Hull1.3 Deviation 92:07 > 0.1% 14:64 > 0.1% 4:03 0.8% 5:58 0.4% 2909 0.3% 0:536 0.7% 45:08 0.2% 52:78 > 0.1% 0:895 0.1% 0:599 0.2% 1030:4 0.5% 0:764 0.5% 43:58 0.2% 13892 0.8% 516572 3.3%

Table 6.2: The characteristics of hulls calculated through the development process. (St0.0) and faired (St0.1) with the core algorithm. The deck line has been also faired by employing the algorithm of Chapter 2 (Fig. 6.1). For St0.0, the curves SAC (SAC-

86


I) and SLC (SLC-I) are calculated and retained throughout the fairing process, which results St0.1. From St0.1, the derived waterline points are interpolated with appropriate end conditions (Wl0.1). St0.1 and Wl0.1 consist the mesh Hull0.1, the characteristics of which are given in Table 6.2. Wl0.1 is faired and results Wl0.2. In Figures 6.3, 6.4 the comparison of Wl0.1 and Wl0.2 is given, where one can notice how the core algorithm changed the undesirable areas aftwords and forewards. Next, Wl0.2 is used for deriving sections again (St0.3), which possess slightly di erent distributions, SAC-II, SLC-II (see Fig. 6.5). From St0.3, SAC-II and SLC-II, a denser set of faired stations is derived and, since the tolerance constraints are rather strict (2:5cm globally), one can consider that the waterline characteristics are retained, and the sets St0.4 (Fig. 6.2), and Wl0.2 form Hull0.2, the characteristics of which are also given in Table 6.2. Hull0.2 is considered the parent hull mesh, the stations of which are used for the generation of the rst body plan, St1.1, of the new hull (Fig. 6.7). St1.1 enables the calculation of a set of waterlines, Wl1.1 (Fig. 6.10), and the two sets form Hull1.1, which exhibits the distribution curves, SAC-1, SLC-1. The characteristics of Hull1.1, are shown in Table 6.2. Fairing Wl1.1, we get Wl1.2 (see Fig. 6.10), and from that new set the distribution of the area of waterlines, as well as the longitudinal centroid and the inertia moments around the two basic axes, Ox and Oy, are calculated and faired. The resulting distributions, G, are given in Figure 6.9. These magnitudes are going to be active constraints for waterlines. From Wl1.2 a new set of stations is derived, St1.2, which, nally results to St1.3, satisfying SAC-1, SLC-1 (Hull1.2- Table 6.2). Next, Wl1.3 results from St1.3, and it is faired to Wl1.4, satisfying G and tolerance of 1cm between the corresponding nodal points. Wl1.4 is used for St1.4, which, after fairing with tolerance 1cm, results St1.5. The deviation between St1.4 and St1.5 is less than 1cm for all nodal points, so St1.5 and Wl1.4 form the nal mesh Hull1.3, (see Figs. 6.8, 6.11),


87

which satis es G, and its contributions, SAC-F and SLC-F, are almost identical to SAC-1 and SLC-1, respectively (see Fig. 6.6). The characteristics of the nal hull are given also in Table 6.2.

6.5 Conclusions - Future work The proposed process enables the generation of curves from any existing data, though in few cases the core algorithm cannot reach an optimum, due to stringent conditions. Another drawback is the interpolation phase, which depends on the end conditions, now given as an input. For these two reasons, the user must be fully aware of the CAGD principles and techniques. Nevertheless, comparing the results of this process to those of other proposed methods { e.g., Standerski '88], one ascertains that the quality of the developed lines is higher and the requirements for integrated quantities are met very satisfactorily. The process can be automated and employed either for modi cation of an existing hull (the example above), or for ab initio design, based, e.g., on some systematic series (example in Kapniaris '95]). An open data base, where meshes of existing ships and/or systematic series forms would be stored, could enable the fast and accurate lines development, at least for the needs of the preliminary design.


88

7

7

6

80

5

6 78

-2

4

5

76

3 2 28 32

1 BL

24

20

8

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70

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3

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80

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52

2 44

38

38

8

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6

5

4

3

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1

CL

1

2

3

5

4

6

7

1 BL 8

Figure 6.1: Initial Hull: Body plan St0.1 of Hull0.1.

7

7

6

80 79

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6

78

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76 -2

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73 0 2 5

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80 81

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1 BL

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1 BL

40

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CL

8

CL

8

-2

0

5.567

5.000

3.8 50 4.2 50

50 4.2 50 3.8

5.567 5.000

44

32

-2 0 2 5 8

2

64

58

52

38

28

24

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0.500 0.750 1.000 1.500 2.000 2.500

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0.500 0.750 1.000 1.500 2.000 2.500 3.000 3.850 5.000 5.567

CL

2.500 2.000 1.500 1.000 0.750 0.500

64

3.000

0 4.25

4.2 50

70

73

76

Figure 6.4: Aft (left) and fore (right) of Wl0.1 (upper) and Wl0.2 (lower).

5

00

3.0

00

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2.500 2.000 1.500 1.000 0.750 0.500

8

5.567 5.000 3.850

78

80

70 73 76 78 80

Figure 6.3: Sets of waterlines of the initial hull: Wl0.1 (upper) and Wl0.2 (lower).


-10

0

10

20

30

40

Leverarm 50

Sectional Area

60

70

80

90

Leverarm

Sectional Area

Figure 6.6: Comparison between SAC-1/SLC-1 (dashed) and SAC-F/SLC-F (solid).

0 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

10

20

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50

60

Figure 6.5: Initial (dashed-I) and modi ed (solid-II) distributions, SAC and SLC, of the initial hull.

0

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90


Figures of example

91

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7 6

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85 -2.

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.2 5 87 .5 86 .7 5

6 -0 -1 -2.2 . .5 5 0 75

DWL

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22 20 24 26 28 30 32 35 42.5

14 18 16

75 2 4 6 8

.8

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10

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0.

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.5 89

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86 85 8483 82 .5 80 79 5 . 77 6 7

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CL

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91

70 68 66 64 62 60 58 56

2

5

1

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2 42.5

1 BL

45 53

5

Figure 6.8: Final Hull: Body plan St1.5 of Hull1.3.

50

o

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8


92

T [m] 6 A wl/2 LCF

5

Ixx Iyy

4

3

2

1 BL 250

300

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40.5

41

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42

42.5

43

43.5

]

0

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]

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LCF [m]

I

I

[x10

2

xx

[x10

yy

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m

m

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350

400

450

600

650

44

44.5

45

17.5

20

22.5

25

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64

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550

Figure 6.9: Distributions of Aw , LCF , Ixx and Iyy with respect to the draught of the ship (G-distributions).

8

8

1.5

1.0

0.5

0.3

4 5. 5.5 3 3.45.03.5 0 1.52.20.5 .0 1. 0.50

2.0 1.5

0.5

5 5.5 3 4.04.5 .0 2.2.53.0 .5 3 1.10.5 0 0.5

Figure 6.11: Waterlines: Wl1.4 (upper and lower).

1.0

Flat bottom line

Figure 6.10: Sets of waterlines: Wl1.1 (upper) and Wl1.2 (lower).

2.0

5 5.5.40.5 03 4. 3.5 0 3. 2.5

5 5.5.40.5 03 4. 3.5 0 3. 2.5

-5

-5

8

CL

CL

8

Deck line

Deck line

95

95


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