Geometric Methods and Applications for Computer Science - CiteSeerX

Jean Gallier

Geometric Methods and Applications for Computer Science and Engineering, Second Edition March 28, 2011

Springer

To my wife, Anne, my children, Mia, Philippe, and Sylvie, and my grandchildren, Bahari and Demetrius

Preface

This book is an introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature with a computer. Our main goal is to present a collection of tools that can be used to solve problems in computer vision, robotics, machine learning, computer graphics, and geometric modeling. During the ten years following the publication of the first edition of this book, optimization techniques have made a huge comeback, especially in the fields of computer vision and machine learning. In particular, convex optimization and its special incarnation, semidefinite programming (SDP), are now widely used techniques in computer vision and machine learning, as one may verify by looking at the proceedings of any conference in these fields. Therefore, we felt that it would be useful to include some material (especially on convex geometry) to prepare the reader for more comprehensive expositions of convex optimization, such as Boyd and Vandenberghe [2], a masterly and encyclopedic account of the subject. In particular, we added Chapter 7, which covers separating and supporting hyperplanes. We also realized that the importance of the SVD (singular value decomposition) and of the pseudo-inverse had not been sufficiently stressed in the first edition of this book, and we rectified this situation in the second edition. In particular, we added sections on PCA (principal component analysis) and on best affine approximations and showed how they are efficienlty computed using SVD. We also added a section on quadratic optimization and a section on the Schur complement, showing the usefulness of the pseudo-inverse. In this second edition, many typos and small mistakes have been corrected, some proofs have been shortened, some problems have been added, and some references have been added. Here is a list containing brief descriptions of the chapters that have been modified or added. • Chapter 3, on the basic properties of convex sets, has been expanded. In particular, we state a version of Carathéodory’s theorem for convex cones (Theorem 3.2), a version of Radon’s theorem for pointed cones (Theorem 3.6), and Tverberg’s theorem (Theorem 3.7), and we define centerpoints and prove their existence (Theorem 3.9).

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• Chapter 7 is new. This chapter deals with separating hyperplanes, versions of Farkas’s lemma, and supporting hyperplanes. Following Berger [1], various versions of the separation of open or closed convex subsets by hyperplanes are proved as consequences of a geometric version of the Hahn–Banach theorem (Theorem 7.1). We also show how various versions of Farkas’s lemma (Lemmas 7.3, 7.4, and 7.5) can be easily deduced from separation results (Corollary 7.4 and Proposition 7.3). Farkas’s lemma plays an important result in linear programming. Indeed, it can be used to give a quick proof of so-called strong duality in linear programming. We also prove the existence of supporting hyperplanes for boundary points of closed convex sets (Minkowski’s lemma, Proposition 7.4). Unfortunately, lack of space prevents us from discussing polytopes and polyhedra. The reader will find a masterly exposition of these topics in Ziegler [3]. • Chapter 14 is a major revision of Chapter 13 (Applications of Euclidean Geometry to Various Optimization Problems) from the first edition of this book and has been renamed “Applications of SVD and Pseudo-Inverses.” Section 14.1, about least squares problems, and the pseudo-inverse has not changed much, but we have added the fact that AA+ is the orthogonal projection onto the range of A and that A+ A is the orthogonal projection onto Ker(A)⊥ , the orthogonal complement of Ker(A). We have also added Proposition 14.1, which shows how the pseudoinverse of a normal matrix A can be obtained from a block diagonalization of A (see Theorem 12.7). Sections 14.2, 14.3, and 14.4 are new. In Section 14.2, we define various matrix norms, including operator norms, and we prove Proposition 14.4, showing how a matrix can be best approximated by a rank-k matrix (in the " "2 norm). Section 14.3 is devoted to principal component analysis (PCA). PCA is a very important statistical tool, yet in our experience, most presentations of this concept lack a crisp definition. Most presentations identify the notion of principal components with the result of applying SVD and do not prove why SVD does in fact yield the principal components and directions. To rectify this situation, we give a precise definition of PCAs (Definition 14.3), and we prove rigorously how SVD yields PCA (Theorem 14.3), using the Rayleigh–Ritz ratio (Lemma 14.2). In Section 14.4, it is shown how to best approximate a set of data with an affine subspace in the least squares sense. Again, SVD can used to find solutions. • Chapter 15 is new, except for Section 15.1, which reproduces Section 13.2 from the first edition of this book. We added the definition of the positive semidefinite cone ordering, #, on symmetric matrices, since it is extensively used in convex optimization. In Section 15.2, we find a necessary and sufficient condition (Proposition 15.2) for the quadratic function f (x) = 12 x$ Ax + x$ b to have a minimum in terms of the pseudo-inverse of A (where A is a symmetric matrix). We also show how to accommodate linear constraints of the form C$ x = 0 or affine constraints of the form C$ x = t (where t %= 0). In Section 15.3, we consider the problem of maximizing f (x) = x$ Ax on the unit sphere x$ x = 1 or, more generally, on the ellipsoid x$ Bx = 1, where A is a symmetric matrix and B is symmetric, positive definite. We show that these

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problems are completely solved by diagonalizing A with respect to an orthogonal matrix. We also briefly consider the effect of adding linear constraints of the form C$ x = 0 or affine constraints of the form C$ x = t (where t %= 0). Chapter 16 is new. In this chapter, we define the notion of Schur complement, and we use it to characterize when a symmetric 2 × 2 block matrix is either positive semidefinite or positive definite (Proposition 16.1, Proposition 16.2, and Theorem 16.1). Chapter 17 is also brand new. In this chapter, we show how a computer vision problem, contour grouping, can be formulated as a quadratic optimization problem involving a Hermitian matrix. Because of the extra dependency on an angle, this optimization problem leads to finding the derivative of eigenvalues and eigenvectors of a normal matrix X. We derive explicit formulas for these derivatives (in the case of eigenvectors, the formula involves the pseudo-inverse of X) and we prove their correctness. It appears to be difficult to find these formulas together with a clean and correct proof in the literature. Our optimization problem leads naturally to the consideration of the field of values (or numerical range) F(A) of a complex matrix A. A remarkable property of the field of values is that it is a convex subset of the plane, a theorem due to Toeplitz and Hausdorff, for which we give a short proof using a deformation argument (Theorem 17.1). Properties of the fields of values can be exploited to solve our optimization problem. This chapter describes current and exciting research in computer vision. Chapter 18 (which used to be Chapter 14 in the first edition) has been slightly expanded and improved. Our experience in teaching the material of this chapter, an introduction to manifolds and Lie groups, is that it is helpful to review carefully the notion of the derivative of a function f : E → F where E and F are normed vector spaces. Thus we added Section 18.7, which provides such a review. We also state the inverse function theorem and define immersions and submersions. Section 18.8 has also been slightly expanded. We added Proposition 18.6 and Theorem 18.7, which are often useful in proving that various spaces are manifolds; we defined critical and regular values and defined Morse functions; and we made a few cosmetic improvements in the paragraphs following Definition 18.20. A number of new problems on manifolds have been added. The only change to Chapter 19 (Chapter 15 in the first edition) is the inclusion of a more complete treatment of the Frenet frame for nD curves in Section 19.10. Similarly, the only change to Chapter 20 (Chapter 16 in the first edition) is the addition of Section 20.12, on covariant derivatives and the parallel transport.

Besides adding problems to all the chapters listed above we added one more problem to Chapter 2. As in the first edition, there is some additional material on the web site http: //www.cis.upenn.edu/˜jean/gbooks/geom2.html This material has not changed, and the chapter and section numbers are those of the first edition. A graph showing the dependencies of chapters is shown in Figure 0.1.

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Fig. 0.1 Dependency of chapters.

Acknowledgments Since the publication of the first edition of this book I have received valuable comments from Kostas Daniilidis, Marcelo Siqueira, Jianbo Shi, Ben Taskar, CJ Taylor, Mickey Brautbar, Katerina Fragiadaki, Ryan Kennedy, Oleg Naroditsky, and Weiyu Zhang. I also want to extend special thanks to David Kramer, who copyedited the first edition of this book over ten years ago, and did a superb job on this second edition.

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References 1. Marcel Berger. Géométrie 2. Nathan, 1990. English edition: Geometry 2, Universitext, Springer-Verlag. 2. Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, first edition, 2004. 3. Gunter Ziegler. Lectures on Polytopes. GTM No. 152. Springer Verlag, first edition, 1997.

Philadelphia, March 2011

Jean Gallier

Preface to the First Edition

Many problems arising in engineering, and notably in computer science and mechanical engineering, require geometric tools and concepts. This is especially true of problems arising in computer graphics, geometric modeling, computer vision, and motion planning, just to mention some key areas. This book is an introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature with a computer. In a previous text, Gallier [24], we focused mostly on affine geometry and on its applications to the design and representation of polynomial curves and surfaces (and B-splines). The main goal of this book is to provide an introduction to more sophisticated geometric concepts needed in tackling engineering problems of a geometric nature. Many problems in the above areas require some nontrivial geometric knowledge, but in our opinion, books dealing with the relevant geometric material are either too theoretical, or else rather specialized. For example, there are beautiful texts entirely devoted to projective geometry, Euclidean geometry, and differential geometry, but reading each one represents a considerable effort (certainly from a nonmathematician!). Furthermore, these topics are usually treated for their own sake (and glory), with little attention paid to applications. This book is an attempt to fill this gap. We present a coherent view of geometric methods applicable to many engineering problems at a level that can be understood by a senior undergraduate with a good math background. Thus, this book should be of interest to a wide audience including computer scientists (both students and professionals), mathematicians, and engineers interested in geometric methods (for example, mechanical engineers). In particular, we provide an introduction to affine geometry, projective geometry, Euclidean geometry, basics of differential geometry and Lie groups, and a glimpse of computational geometry (convex sets, Voronoi diagrams, and Delaunay triangulations). This material provides the foundations for the algorithmic treatment of curves and surfaces, some basic tools of geometric modeling. The right dose of projective geometry also leads to a rigorous and yet smooth presentation of rational curves and surfaces. However, to keep the size of this book reasonable, a number of topics could not be included. Nevertheless, they can be found in the additional material on the web site: see http://www.cis.

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upenn.edu/˜jean/gbooks/geom2.html. This is the case of the material on rational curves and surfaces. This book consists of sixteen chapters and an appendix. The additional material on the web site consists of eight chapters and an appendix: see http://www. cis.upenn.edu/˜jean/gbooks/geom2.html. • The book starts with a brief introduction (Chapter 1). • Chapter 2 provides an introduction to affine geometry. This ensures that readers are on firm ground to proceed with the rest of the book, in particular, projective geometry. This is also useful to establish the notation and terminology. Readers proficient in geometry may omit this section, or use it as needed. On the other hand, readers totally unfamiliar with this material will probably have a hard time with the rest of the book. These readers are advised do some extra reading in order to assimilate some basic knowledge of geometry. For example, we highly recommend Pedoe [42], Coxeter [9], Snapper and Troyer [52], Berger [2, 3], Fresnel [22], Samuel [51], Hilbert and Cohn–Vossen [31], Boehm and Prautzsch [5], and Tisseron [54]. • Basic properties of convex sets and convex hulls are discussed in Chapter 3. Three major theorems are proved: Carthéodory’s theorem, Radon’s theorem, and Helly’s theorem. • Chapter 4 presents a construction (the “hat construction”) for embedding an affine space into a vector space. An important application of this construction is the projective completion of an affine space, presented in the next chapter. Other applications are treated in Chapter 20 on the web site, see http: //www.cis.upenn.edu/˜jean/gbooks/geom2.html. • Chapter 5 provides an introduction to projective geometry. Since we are not writing a treatise on projective geometry, we cover only the most fundamental concepts, including projective spaces and subspaces, frames, projective maps, multiprojective maps, the projective completion of an affine space, cross-ratios, duality, and the complexification of a real projective space. This material also provides the foundations for our algorithmic treatment of rational curves and surfaces, to be found on the web site (Chapters 18, 19, 21, 22, 23, 24); see http://www.cis.upenn.edu/˜jean/gbooks/geom2.html. • Chapters 6, 8, and 9, provide an introduction to Euclidean geometry, to the groups of isometries O(n) and SO(n), the groups of affine rigid motions Is(n) and SE(n), and to the quaternions. Several versions of the Cartan–Dieudonné theorem are proved in Chapter 8. The QR-decomposition of matrices is explained geometrically, both in terms of the Gram–Schmidt procedure and in terms of Householder transformations. These chapters are crucial to a firm understanding of the differential geometry of curves and surfaces, and computational geometry. • Chapter 10 gives a short introduction to some fundamental topics in computational geometry: Voronoi diagrams and Delaunay triangulations. • Chapter 11 provides an introduction to Hermitian geometry, to the groups of isometries U(n) and SU(n), and the groups of affine rigid motions Is(n, C) and SE(n, C). The generalization of the Cartan–Dieudonné theorem to Hermitian spaces can be found on the web site, Chapter 25; see http://www.


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cis.upenn.edu/˜jean/gbooks/geom2.html. A short introduction to Hilbert spaces, including the projection theorem, and the isomorphism of every Hilbert space with some space l 2 (K), can also be found on the web site in Chapter 26, see http://www.cis.upenn.edu/˜jean/gbooks/geom2.html. Chapter 12 provides a presentation of the spectral theorems in Euclidean and Hermitian spaces, including normal, self-adjoint, skew self-adjoint, and orthogonal linear maps. Normal form (in terms of block diagonal matrices) for various types of linear maps are presented. The singular value decomposition (SVD) and the polar form of linear maps are discussed quite extensively in Chapter 13. The pseudo-inverse of a matrix and its characterization using the Penrose properties are presented. Chapter 14 presents some applications of Euclidean geometry to various optimization problems. The method of least squares is presented, as well as the applications of the SVD and QR-decomposition to solve least squares problems. We also describe a method for minimizing positive definite quadratic forms, using Lagrange multipliers. Chapter 18 provides an introduction to the linear Lie groups, via a presentation of some of the classical groups and their Lie algebras, using the exponential map. The surjectivity of the exponential map is proved for SO(n) and SE(n). An introduction to the local differential geometry of curves is given in Chapter 19 (curvature, torsion, the Frenet frame, etc). An introduction to the local differential geometry of surfaces based on some lectures by Eugenio Calabi is given in Chapter 20. This chapter is rather unique, as it reflects decades of experience from a very distinguished geometer. Chapter 21 is an appendix consisting of short sections consisting of basics of linear algebra and analysis. This chapter has been included to make the material self-contained. Our advice is to use it as needed!

A very elegant presentation of rational curves and surfaces can be given using some notions of affine and projective geometry. We push this approach quite far in the material on the web; see http://www.cis.upenn.edu/˜jean/ gbooks/geom2.html. However, we provide only a cursory coverage of CAGD methods. Luckily, there are excellent texts on CAGD, including Bartels, Beatty, and Barsky [1], Farin [17, 18], Fiorot and Jeannin [20, 21], Riesler [50], Hoschek and Lasser [33], and Piegl and Tiller [43]. Although we cover affine, projective, and Euclidean geometry in some detail, we are far from giving a comprehensive treatment of these topics. For such a treatment, we highly recommend Berger [2, 3], Samuel [51], Pedoe [42], Coxeter [11, 10, 8, 9], Snapper and Troyer [52], Fresnel [22], Tisseron [54], Sidler [45], Dieudonné [13], and Veblen and Young [57, 58], a great classic. Similarly, although we present some basics of differential geometry and Lie groups, we only scratch the surface. For instance, we refrain from discussing manifolds in full generality. We hope that our presentation is a good preparation for more advanced texts, such as Gray [27], do Carmo [14], Berger and Gostiaux [4], and Lafontaine [36]. The above are still fairly elementary. More advanced texts on differential geometry include do Carmo [15, 16], Guillemin and Pollack [29], Warner

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[59], Lang [37], Boothby [6], Lehmann and Sacré [38], Stoker [53], Gallot, Hulin, and Lafontaine [25], Milnor [41], Sharpe [44], Malliavin [39], and Godbillon [26]. It is often possible to reduce interpolation problems involving polynomial curves or surfaces to solving systems of linear equations. Thus, it is very helpful to be aware of efficient methods for numerical matrix analysis. For instance, we present the QR-decomposition of matrices, both in terms of the (modified) Gram–Schmidt method and in terms of Householder transformations, in a novel geometric fashion. For further information on these topics, readers are referred to the excellent texts by Strang [48], Golub and Van Loan [28], Trefethen and Bau [55], Ciarlet [7], and Kincaid and Cheney [34]. Strang’s beautiful book on applied mathematics is also highly recommended as a general reference [46]. There are other interesting applications of geometry to computer vision, computer graphics, and solid modeling. Some good references are Trucco and Verri [56], Koenderink [35], and Faugeras [19] for computer vision; Hoffman [32] for solid modeling; and Metaxas [40] for physics-based deformable models.

Novelties As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry, projective geometry, Euclidean geometry, Hermitian geometry, basics of Hilbert spaces with a touch of Fourier series, basics of Lie groups and Lie algebras, as well as a presentation of curves and surfaces both from the standard differential point of view and from the algorithmic point of view in terms of control points (in the polynomial and rational case).

New Treatment, New Results This books provides an introduction to affine geometry, projective geometry, Euclidean geometry, Hermitian geometry, Hilbert spaces, a glimpse at Lie groups and Lie algebras, and the basics of local differential geometry of curves and surfaces. We also cover some classics of convex geometry, such as Carathéodory’s theorem, Radon’s theorem, and Helly’s theorem. However, in order to help the reader assimilate all these concepts with the least amount of pain, we begin with some basic notions of affine geometry in Chapter 2. Basic notions of Euclidean geometry come later only in Chapters 6, 8, 9. Generally, noncore material is relegated to appendices or to the web site: see http://www.cis.upenn.edu/˜jean/ gbooks/geom2.html. We cover the standard local differential properties of curves and surfaces at an elementary level, but also provide an in-depth presentation of polynomial and rational curves and surfaces from an algorithmic point of view. The approach (sometimes called blossoming) consists in multilinearizing everything in sight (getting polar forms), which leads very naturally to a presentation of polynomial and rational curves and surfaces in terms of control points (Bézier curves and surfaces). We present many algorithms for subdividing and drawing curves and surfaces, all implemented in Mathematica. A clean and elegant presentation of control points with weights (and control vectors) is obtained by using a construction for embedding


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an affine space into a vector space (the so-called “hat construction,” originating in Berger [2]). We also give several new methods for drawing efficiently closed rational curves and surfaces, and a method for resolving base points of triangular rational surfaces. We give a quick introduction to the concepts of Voronoi diagrams and Delaunay triangulations, two of the most fundamental concepts in computational geometry. As a general rule, we try to be rigorous, but we always keep the algorithmic nature of the mathematical objects under consideration in the forefront. Many problems and programming projects are proposed (over 230). Some are routine, some are (very) difficult.

Applications Although it is core mathematics, geometry has many practical applications. Whenever possible, we point out some of these applications, For example, we mention some (perhaps unexpected) applications of projective geometry to computer vision (camera calibration), efficient communication, error correcting codes, and cryptography (see Section 5.13). As applications of Euclidean geometry, we mention motion interpolation, various normal forms of matrices including QR-decomposition in terms of Householder transformations and SVD, least squares problems (see Section 14.1), and the minimization of quadratic functions using Lagrange multipliers (see Section 15.1). Lie groups and Lie algebras have applications in robot kinematics, motion interpolation, and optimal control. They also have applications in physics. As applications of the differential geometry of curves and surfaces, we mention geometric continuity for splines, and variational curve and surface design (see Section 19.11 and Section 20.13). Finally, as applications of Voronoi diagrams and Delaunay triangulations, we mention the nearest neighbors problem, the largest empty circle problem, the minimum spanning tree problem, and motion planning (see Section 10.5). Of course, rational curves and surfaces have many applications to computer-aided geometric design (CAGD), manufacturing, computer graphics, and robotics.

Many Algorithms and Their Implementation Although one of our main concerns is to be mathematically rigorous, which implies that we give precise definitions and prove almost all of the results in this book, we are primarily interested in the representation and the implementation of concepts and tools used to solve geometric problems. Thus, we devote a great deal of efforts to the development and implemention of algorithms to manipulate curves, surfaces, triangulations, etc. As a matter of fact, we provide Mathematica code for most of the geometric algorithms presented in this book. We also urge the reader to write his own algorithms, and we propose many challenging programming projects.

Open Problems Not only do we present standard material (although sometimes from a fresh point of view), but whenever possible, we state some open problems, thus taking the reader to the cutting edge of the field. For example, we describe very clearly the problem of resolving base points of rectangular rational surfaces (this material is on the web site, see http://www.cis.upenn.edu/˜jean/gbooks/geom2.html).

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What’s Not Covered in This Book Since this book is already quite long, we have omitted solid modeling techniques, methods for rendering implicit curves and surfaces, the finite elements method, and wavelets. The first two topics are nicely covered in Hoffman [32], and the finite element method is the subject of so many books that we will not attempt to mention any references besides Strang and Fix [47]. As to wavelets, we highly recommend the classics by Daubechies [12], and Strang and Truong [49], among the many texts on this subject. It would also have been nice to include chapters on the algebraic geometry of curves and surfaces. However, this is a very difficult subject that requires a lot of algebraic machinery. Interested readers may consult Fulton [23] or Harris [30].

How to Use This Book for a Course This books covers three complementary but fairly disjoint topics: (1) Projective geometry and its applications to rational curves and surfaces (Chapter 5, and on the web page, Chapters 18, 19, 21, 22, 23, 24); (2) Euclidean geometry, Voronoi diagrams, and Delaunay triangulations, Hermitian geometry, basics of Hilbert spaces, spectral theorems for special kinds of linear maps, SVD, polar form, and basics of Lie groups and Lie algebras (Chapters 6, 8, 9, 10, 11, 12, 13, 14, 18); (3) Basics of the differential geometry of curves and surfaces (Chapters 19 and 20). Chapter 21 is an appendix consisting of background material and should be used only as needed. Our experience is that there is too much material to cover in a one–semester course. The ideal situation is to teach the material in the entire book in two semesters. Otherwise, a more algebraically inclined teacher should teach the first or second topic, whereas a more differential-geometrically inclined teacher should teach the third topic. In either case, Chapter 2 on affine geometry should be covered. Chapter 4 is required for the first topic, but not for the second. Problems are found at the end of each chapter. They range from routine to very difficult. Some programming assignments have been included. They are often quite open-ended, and may require a considerable amount of work. The end of a proof is indicated by a square box ( ). The word iff is an abbreviation for if and only if . References to the web page http://www.cis.upenn.edu/˜jean/gbooks/ geom2.html will be abbreviated as web page. Hermann Weyl made the following comment in the preface (1938) of his beautiful book [60]: The gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle . . . . Nobody is more aware than myself of the attendant loss in vigor, ease and lucidity of expression.

Being in a similar position, I hope that I was at least successful in conveying my enthusiasm and passion for geometry, and that I have inspired my readers to study some of the books that I respect and admire.


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Acknowledgments This book grew out of lectures notes that I have written as I have been teaching CIS610, Advanced Geometric Methods in Computer Science, for the past two years. Many thanks to the copyeditor, David Kramer, who did a superb job. I also wish to thank some students and colleagues for their comments, including Koji Ashida, Doug DeCarlo, Jaydev Desai, Will Dickinson, Charles Erignac, Steve Frye, Edith Haber, Andy Hicks, Paul Hughett, David Jelinek, Marcus Khuri, Hartmut Liefke, Shih-Schon Lin, Ying Liu, Nilesh Mankame, Dimitris Metaxas, Viorel Mihalef, Albert Montillo, Youg-jin Park, Harold Sun, Deepak Tolani, Dianna Xu, and Hui Zhang. Also thanks to Norm Badler for triggering my interest in geometric modeling, and to Marcel Berger, Chris Croke, Ron Donagi, Herman Gluck, David Harbater, Alexandre Kirillov, and Steve Shatz for sharing some of their geometric secrets with me. Finally, many thanks to Eugenio Calabi for teaching me what I know about differential geometry (and much more!). I am very grateful to Professor Calabi for allowing me to write up his lectures on the differential geometry of curves and surfaces given in an undergraduate course in Fall 1994 (as Chapter 20).

References 1. Richard H. Bartels, John C. Beatty, and Brian A. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. Morgan Kaufmann, first edition, 1987. 2. Marcel Berger. Géométrie 1. Nathan, 1990. English edition: Geometry 1, Universitext, Springer-Verlag. 3. Marcel Berger. Géométrie 2. Nathan, 1990. English edition: Geometry 2, Universitext, Springer-Verlag. 4. Marcel Berger and Bernard Gostiaux. Géométrie différentielle: variétés, courbes et surfaces. Collection Mathématiques. Puf, second edition, 1992. English edition: Differential geometry, manifolds, curves, and surfaces, GTM No. 115, Springer-Verlag. 5. W. Boehm and H. Prautzsch. Geometric Concepts for Geometric Design. AK Peters, first edition, 1994. 6. William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, second edition, 1986. 7. P.G. Ciarlet. Introduction to Numerical Matrix Analysis and Optimization. Cambridge University Press, first edition, 1989. French edition: Masson, 1994. 8. H.S.M. Coxeter. Non-Euclidean Geometry. The University of Toronto Press, first edition, 1942. 9. H.S.M. Coxeter. Introduction to Geometry. Wiley, second edition, 1989. 10. H.S.M. Coxeter. The Real Projective Plane. Springer-Verlag, third edition, 1993. 11. H.S.M. Coxeter. Projective Geometry. Springer-Verlag, second edition, 1994. 12. Ingrid Daubechies. Ten Lectures on Wavelets. SIAM Publications, first edition, 1992. 13. Jean Dieudonné. Algèbre Linéaire et Géométrie Elémentaire. Hermann, second edition, 1965. 14. Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. 15. Manfredo P. do Carmo. Riemannian Geometry. Birkhäuser, second edition, 1992. 16. Manfredo P. do Carmo. Differential Forms and Applications. Universitext. Springer-Verlag, first edition, 1994. 17. Gerald Farin. Curves and Surfaces for CAGD. Academic Press, fourth edition, 1998. 18. Gerald Farin. NURB Curves and Surfaces, from Projective Geometry to Practical Use. AK Peters, first edition, 1995.

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19. Olivier Faugeras. Three-Dimensional Computer Vision, A Geometric Viewpoint. MIT Press, first edition, 1996. 20. J.-C. Fiorot and P. Jeannin. Courbes et Surfaces Rationelles. RMA 12. Masson, first edition, 1989. 21. J.-C. Fiorot and P. Jeannin. Courbes Splines Rationelles. RMA 24. Masson, first edition, 1992. 22. Jean Fresnel. Méthodes Modernes en Géométrie. Hermann, first edition, 1998. 23. William Fulton. Algebraic Curves. Advanced Book Classics. Addison-Wesley, first edition, 1989. 24. Jean H. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann, first edition, 1999. 25. S. Gallot, D. Hulin, and J. Lafontaine. Riemannian Geometry. Universitext. Springer-Verlag, second edition, 1993. 26. Claude Godbillon. Géométrie Différentielle et Mécanique Analytique. Collection Méthodes. Hermann, first edition, 1969. 27. A. Gray. Modern Differential Geometry of Curves and Surfaces. CRC Press, second edition, 1997. 28. Gene H. Golub and Charles F. Van Loan. Matrix Computations. The Johns Hopkins University Press, third edition, 1996. 29. Victor Guillemin and Alan Pollack. Differential Topology. Prentice-Hall, first edition, 1974. 30. Joe Harris. Algebraic Geometry, A First Course. GTM No. 133. Springer-Verlag, first edition, 1992. 31. D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chelsea Publishing Co., 1952. 32. Christoph M. Hoffman. Geometric and Solid Modeling. Morgan Kaufmann, first edition, 1989. 33. J. Hoschek and D. Lasser. Computer-Aided Geometric Design. AK Peters, first edition, 1993. 34. D. Kincaid and W. Cheney. Numerical Analysis. Brooks/Cole Publishing, second edition, 1996. 35. Jan J. Koenderink. Solid Shape. MIT Press, first edition, 1990. 36. Jacques Lafontaine. Introduction aux Variétés Différentielles. PUG, first edition, 1996. 37. Serge Lang. Differential and Riemannian Manifolds. GTM No. 160. Springer-Verlag, third edition, 1995. 38. Daniel Lehmann and Carlos Sacré. Géométrie et Topologie des Surfaces. Puf, first edition, 1982. 39. Paul Malliavin. Géométrie Différentielle Intrinsèque. Enseignement des Sciences, No. 14. Hermann, first edition, 1972. 40. Dimitris N. Metaxas. Physics-Based Deformable Models. Kluwer Academic Publishers, first edition, 1997. 41. John W. Milnor. Topology from the Differentiable Viewpoint. The University Press of Virginia, second edition, 1969. 42. Dan Pedoe. Geometry, A Comprehensive Course. Dover, first edition, 1988. 43. Les Piegl and Wayne Tiller. The NURBS Book. Monograph in Visual Communications. Springer-Verlag, first edition, 1995. 44. Richard W. Sharpe. Differential Geometry. Cartan’s Generalization of Klein’s Erlangen Program. GTM No. 166. Springer-Verlag, first edition, 1997. 45. J.-C. Sidler. Géométrie Projective. InterEditions, first edition, 1993. 46. Gilbert Strang. Introduction to Applied Mathematics. Wellesley–Cambridge Press, first edition, 1986. 47. Gilbert Strang and Fix George. An Analysis of the Finite Element Method. Wellesley– Cambridge Press, first edition, 1973. 48. Gilbert Strang. Linear Algebra and Its Applications. Saunders HBJ, third edition, 1988. 49. Gilbert Strang and Nguyen Truong. Wavelets and Filter Banks. Wellesley–Cambridge Press, second edition, 1997.


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50. J.-J. Risler. Mathematical Methods for CAD. Masson, first edition, 1992. 51. Pierre Samuel. Projective Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, first edition, 1988. 52. Ernst Snapper and Troyer Robert J. Metric Affine Geometry. Dover, first edition, 1989. 53. J.J. Stoker. Differential Geometry. Wiley Classics. Wiley-Interscience, first edition, 1989. 54. Claude Tisseron. Géométries Affines, Projectives, et Euclidiennes. Hermann, first edition, 1994. 55. L.N. Trefethen and D. Bau III. Numerical Linear Algebra. SIAM Publications, first edition, 1997. 56. Emanuele Trucco and Alessandro Verri. Introductory Techniques for 3D Computer Vision. Prentice-Hall, first edition, 1998. 57. O. Veblen and J. W. Young. Projective Geometry, Vol. 1. Ginn, second edition, 1938. 58. O. Veblen and J. W. Young. Projective Geometry, Vol. 2. Ginn, first edition, 1946. 59. Frank Warner. Foundations of Differentiable Manifolds and Lie Groups. GTM No. 94. Springer-Verlag, first edition, 1983. 60. Hermann Weyl. The Classical Groups. Their Invariants and Representations. Princeton Mathematical Series, No. 1. Princeton University Press, second edition, 1946.

Philadelphia

Jean Gallier

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Geometries: Their Origin, Their Uses . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Prerequisites and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 5

2

Basics of Affine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Chasles’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Affine Combinations, Barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Affine Independence and Affine Frames . . . . . . . . . . . . . . . . . . . . . . 2.7 Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Affine Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Affine Geometry: A Glimpse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Affine Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Intersection of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 15 16 17 21 26 32 38 41 45 46 48 62

3

Basic Properties of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Carathéodory’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Vertices, Extremal Points, and Krein and Milman’s Theorem . . . . . 3.4 Radon’s, Helly’s, Tverberg’s Theorems and Centerpoints . . . . . . . . 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 67 70 76 82 83

4

Embedding an Affine Space in a Vector Space . . . . . . . . . . . . . . . . . . . . . 85 4.1 The “Hat Construction,” or Homogenizing . . . . . . . . . . . . . . . . . . . . 85 4.2 Affine Frames of E and Bases of Eˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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4.3 Another Construction of Eˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Extending Affine Maps to Linear Maps . . . . . . . . . . . . . . . . . . . . . . . 97 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5

Basics of Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1 Why Projective Spaces? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3 Projective Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Projective Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5 Projective Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6 Projective Completion of an Affine Space, Affine Patches . . . . . . . . 126 5.7 Making Good Use of Hyperplanes at Infinity . . . . . . . . . . . . . . . . . . 133 5.8 The Cross-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.9 Duality in Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.10 Cross-Ratios of Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.11 Complexification of a Real Projective Space . . . . . . . . . . . . . . . . . . . 144 5.12 Similarity Structures on a Projective Space . . . . . . . . . . . . . . . . . . . . 146 5.13 Some Applications of Projective Geometry . . . . . . . . . . . . . . . . . . . . 151 5.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6

Basics of Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.1 Inner Products, Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . . 183 6.3 Linear Isometries (Orthogonal Transformations) . . . . . . . . . . . . . . . 195 6.4 The Orthogonal Group, Orthogonal Matrices . . . . . . . . . . . . . . . . . . 198 6.5 QR-Decomposition for Invertible Matrices . . . . . . . . . . . . . . . . . . . . 200 6.6 Some Applications of Euclidean Geometry . . . . . . . . . . . . . . . . . . . . 202 6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7

Separating and Supporting Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.1 Separation Theorems and Farkas’s Lemma . . . . . . . . . . . . . . . . . . . . 213 7.2 Supporting Hyperplanes and Minkowski’s Proposition . . . . . . . . . . 227 7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8

The Cartan–Dieudonné Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1 Orthogonal Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.2 The Cartan–Dieudonné Theorem for Linear Isometries . . . . . . . . . . 235 8.3 QR-Decomposition Using Householder Matrices . . . . . . . . . . . . . . . 246 8.4 Affine Isometries (Rigid Motions) . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.5 Fixed Points of Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.6 Affine Isometries and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 8.7 The Cartan–Dieudonné Theorem for Affine Isometries . . . . . . . . . . 260

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8.8 Orientations of a Euclidean Space, Angles . . . . . . . . . . . . . . . . . . . . . 264 8.9 Volume Forms, Cross Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9

The Quaternions and the Spaces S3 , SU(2), SO(3), and RP3 . . . . . . . . . 281 9.1 The Algebra H of Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.2 Quaternions and Rotations in SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . 285 9.3 Quaternions and Rotations in SO(4) . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.4 Applications to Motion Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 296 9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

10

Dirichlet–Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 10.1 Dirichlet–Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 10.2 Simplicial Complexes and Triangulations . . . . . . . . . . . . . . . . . . . . . 308 10.3 Delaunay Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10.4 Delaunay Triangulations and Convex Hulls . . . . . . . . . . . . . . . . . . . . 314 10.5 Applications of Voronoi Diagrams and Delaunay Triangulations . . 317 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

11

Basics of Hermitian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 11.1 Hermitian Spaces, Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 321 11.2 Orthogonality, Duality, Adjoint of a Linear Map . . . . . . . . . . . . . . . . 328 11.3 Linear Isometries (Also Called Unitary Transformations) . . . . . . . . 331 11.4 The Unitary Group, Unitary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

12

Spectral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 12.1 Introduction: What’s with Lie Groups and Lie Algebras? . . . . . . . . 343 12.2 Normal Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 12.3 Self-Adjoint and Other Special Linear Maps . . . . . . . . . . . . . . . . . . . 351 12.4 Normal and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 356 12.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

13

Singular Value Decomposition (SVD) and Polar Form . . . . . . . . . . . . . . 367 13.1 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 13.2 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . . . . 374 13.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

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Applications of SVD and Pseudo-inverses . . . . . . . . . . . . . . . . . . . . . . . . . 387 14.1 Least Squares Problems and the Pseudo-inverse . . . . . . . . . . . . . . . . 387 14.2 Data Compression and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 14.3 Principal Components Analysis (PCA) . . . . . . . . . . . . . . . . . . . . . . . 398 14.4 Best Affine Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

15

Quadratic Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 15.1 Quadratic Optimization: The Positive Definite Case . . . . . . . . . . . . . 411 15.2 Quadratic Optimization: The General Case . . . . . . . . . . . . . . . . . . . . 419 15.3 Maximizing a Quadratic Function on the Unit Sphere . . . . . . . . . . . 423 15.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

16

Schur Complements and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 16.1 Schur Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 16.2 SPD Matrices and Schur Complements . . . . . . . . . . . . . . . . . . . . . . . 434 16.3 Symmetric Positive Semidefinite Matrices and Schur Complements435 16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

17

Quadratic Optimization and Contour Grouping . . . . . . . . . . . . . . . . . . . 439 17.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 17.2 Derivatives of Eigenvalues and Eigenvectors for Normal Matrices . 443 17.3 Relationship between the Eigenvectors of P and H(δ ) . . . . . . . . . . . 446 17.4 Study of the Continuous Relaxation of the Problem . . . . . . . . . . . . . 449 17.5 The Field of Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 17.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

18

Basics of Manifolds and Classical Lie Groups . . . . . . . . . . . . . . . . . . . . . 459 18.1 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 18.2 Some Classical Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 18.3 Symmetric and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . 472 18.4 Exponential of Some Complex Matrices . . . . . . . . . . . . . . . . . . . . . . 475 18.5 Hermitian and Other Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . 478 18.6 The Lie Group SE(n) and the Lie Algebra se(n) . . . . . . . . . . . . . . . . 479 18.7 The Derivative of a Function Between Normed Spaces . . . . . . . . . . 483 18.8 Finale: Manifolds, Lie Groups, and Lie Algebras . . . . . . . . . . . . . . . 491 18.9 Applications of Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . 511 18.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

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19

Basics of the Differential Geometry of Curves . . . . . . . . . . . . . . . . . . . . . 529 19.1 Introduction: Parametrized Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 19.2 Tangent Lines and Osculating Planes . . . . . . . . . . . . . . . . . . . . . . . . . 534 19.3 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 19.4 Curvature and Osculating Circles (Plane Curves) . . . . . . . . . . . . . . . 540 19.5 Normal Planes and Curvature (3D Curves) . . . . . . . . . . . . . . . . . . . . 553 19.6 The Frenet Frame (3D Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 19.7 Torsion (3D Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 19.8 The Frenet Equations (3D Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 19.9 Osculating Spheres (3D Curves) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 19.10 The Frenet Frame for nD Curves (n ≥ 4) . . . . . . . . . . . . . . . . . . . . . . 564 19.11 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 19.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

20

Basics of the Differential Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . 585 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 20.2 Parametrized Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 20.3 The First Fundamental Form (Riemannian Metric) . . . . . . . . . . . . . . 592 20.4 Normal Curvature and the Second Fundamental Form . . . . . . . . . . . 597 20.5 Geodesic Curvature and the Christoffel Symbols . . . . . . . . . . . . . . . 602 20.6 Principal Curvatures, Gaussian Curvature, Mean Curvature . . . . . . 606 20.7 The Gauss Map and Its Derivative dN . . . . . . . . . . . . . . . . . . . . . . . . 613 20.8 The Dupin Indicatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 20.9 The Theorema Egregium of Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 20.10 Lines of Curvature, Geodesic Torsion, Asymptotic Lines . . . . . . . . 626 20.11 Geodesic Lines, Local Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . 631 20.12 Covariant Derivative, Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . 637 20.13 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 20.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 21.1 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 21.2 Metric Spaces and Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . 656 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665