Curves and Surfaces

In Geometric Modeling:

Theory And Algorithms

Jean GallierDepartment of Computer and Information Science

University of PennsylvaniaPhiladelphia, PA 19104, USA

e-mail: jean@cis.upenn.edu

c Jean GallierPlease, do not reproduce without permission of the author

November 7, 2011

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iii

To my new daughter Mia, my wife Anne,

my son Philippe, and my daughter Sylvie.

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Contents

1 Introduction 71.1 Geometric Methods in Engineering . . . . . . . . . . . . . . . . . . . . . . . 71.2 Examples of Problems Using Geometric Modeling . . . . . . . . . . . . . . . 8

Part I Basics of Affine Geometry 11

2 Basics of Affine Geometry 132.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Examples of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Chasles Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Affine Combinations, Barycenters . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Affine Independence and Affine Frames . . . . . . . . . . . . . . . . . . . . . 322.7 Affine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 Affine Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.9 Affine Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Part II Polynomial Curves and Spline Curves 61

3 Introduction to Polynomial Curves 633.1 Why Parameterized Polynomial Curves? . . . . . . . . . . . . . . . . . . . . 633.2 Polynomial Curves of degree 1 and 2 . . . . . . . . . . . . . . . . . . . . . . 743.3 First Encounter with Polar Forms (Blossoming) . . . . . . . . . . . . . . . . 773.4 First Encounter with the de Casteljau Algorithm . . . . . . . . . . . . . . . 813.5 Polynomial Curves of Degree 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 863.6 Classification of the Polynomial Cubics . . . . . . . . . . . . . . . . . . . . . 933.7 Second Encounter with Polar Forms (Blossoming) . . . . . . . . . . . . . . . 983.8 Second Encounter with the de Casteljau Algorithm . . . . . . . . . . . . . . 1013.9 Examples of Cubics Defined by Control Points . . . . . . . . . . . . . . . . . 1063.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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4 Multiaffine Maps and Polar Forms 1194.1 Multiaffine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.2 Affine Polynomials and Polar Forms . . . . . . . . . . . . . . . . . . . . . . . 1254.3 Polynomial Curves and Control Points . . . . . . . . . . . . . . . . . . . . . 1314.4 Uniqueness of the Polar Form of an Affine Polynomial Map . . . . . . . . . . 1334.5 Polarizing Polynomials in One or Several Variables . . . . . . . . . . . . . . 1344.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5 Polynomial Curves as Bezier Curves 1435.1 The de Casteljau Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2 Subdivision Algorithms for Polynomial Curves . . . . . . . . . . . . . . . . . 1565.3 The Progressive Version of the de Casteljau Algorithm . . . . . . . . . . . . 1695.4 Derivatives of Polynomial Curves . . . . . . . . . . . . . . . . . . . . . . . . 1745.5 Joining Affine Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . 1765.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6 B-Spline Curves 1876.1 Introduction: Knot Sequences, de Boor Control Points . . . . . . . . . . . . 1876.2 Infinite Knot Sequences, Open B-Spline Curves . . . . . . . . . . . . . . . . 1966.3 Finite Knot Sequences, Finite B-Spline Curves . . . . . . . . . . . . . . . . . 2096.4 Cyclic Knot Sequences, Closed (Cyclic) B-Spline Curves . . . . . . . . . . . 2146.5 The de Boor Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2306.6 The de Boor Algorithm and Knot Insertion . . . . . . . . . . . . . . . . . . . 2336.7 Polar forms of B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2376.8 Cubic Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Part III Polynomial Surfaces and Spline Surfaces 259

7 Polynomial Surfaces 2617.1 Polarizing Polynomial Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.2 Bipolynomial Surfaces in Polar Form . . . . . . . . . . . . . . . . . . . . . . 2707.3 The de Casteljau Algorithm; Rectangular Surface Patches . . . . . . . . . . 2757.4 Total Degree Surfaces in Polar Form . . . . . . . . . . . . . . . . . . . . . . 2797.5 The de Casteljau Algorithm for Triangular Surface Patches . . . . . . . . . . 2827.6 Directional Derivatives of Polynomial Surfaces . . . . . . . . . . . . . . . . . 2857.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

8 Subdivision Algorithms for Polynomial Surfaces 2958.1 Subdivision Algorithms for Triangular Patches . . . . . . . . . . . . . . . . . 2958.2 Subdivision Algorithms for Rectangular Patches . . . . . . . . . . . . . . . . 3238.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

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9 Polynomial Spline Surfaces and Subdivision Surfaces 3379.1 Joining Polynomial Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 3379.2 Spline Surfaces with Triangular Patches . . . . . . . . . . . . . . . . . . . . . 3439.3 Spline Surfaces with Rectangular Patches . . . . . . . . . . . . . . . . . . . . 3499.4 Subdivision Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3529.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

10 Embedding an Affine Space in a Vector Space 37110.1 The Hat Construction, or Homogenizing . . . . . . . . . . . . . . . . . . . 37110.2 Affine Frames of E and Bases of E . . . . . . . . . . . . . . . . . . . . . . . 37910.3 Extending Affine Maps to Linear Maps . . . . . . . . . . . . . . . . . . . . . 38110.4 From Multiaffine Maps to Multilinear Maps . . . . . . . . . . . . . . . . . . 38510.5 Differentiating Affine Polynomial Functions . . . . . . . . . . . . . . . . . . . 38810.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

11 Tensor Products and Symmetric Tensor Products 39711.1 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39711.2 Symmetric Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 40411.3 Affine Symmetric Tensor Products . . . . . . . . . . . . . . . . . . . . . . . 40711.4 Properties of Symmetric Tensor Products . . . . . . . . . . . . . . . . . . . . 40911.5 Polar Forms Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41211.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Part IV Appendices 423

A Linear Algebra 425A.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425A.2 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432A.3 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436A.4 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437A.5 Hyperplanes and Linear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 445

B Complements of Affine Geometry 447B.1 Affine and Multiaffine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 447B.2 Homogenizing Multiaffine Maps . . . . . . . . . . . . . . . . . . . . . . . . . 453B.3 Intersection and Direct Sums of Affine Spaces . . . . . . . . . . . . . . . . . 456B.4 Osculating Flats Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

C Topology 467C.1 Metric Spaces and Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . 467C.2 Continuous Functions, Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 471C.3 Normed Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

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D Differential Calculus 475D.1 Directional Derivatives, Total Derivatives . . . . . . . . . . . . . . . . . . . . 475D.2 Jacobian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

Preface

This book is primarily an introduction to geometric concepts and tools needed for solvingproblems of a geometric nature with a computer. Our main goal is to provide an introduc-tion to the mathematical concepts needed in tackling problems arising notably in computergraphics, geometric modeling, computer vision, and motion planning, just to mention somekey areas. Many problems in the above areas require some geometric knowledge, but in ouropinion, books dealing with the relevant geometric material are either too theoretical, or elserather specialized and application-oriented. This book is an attempt to fill this gap. Wepresent a coherent view of geometric methods applicable to many engineering problems ata 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 (bothstudents and professionals), mathematicians, and engineers interested in geometric methods(for example, mechanical engineers). In particular, we provide an introduction to affine ge-ometry. This material provides the foundations for the algorithmic treatment of polynomialcurves and surfaces, which is a main theme of this book. We present some of the main toolsused in computer aided geometric design (CAGD), but our goal is not to write another texton CAGD. In brief, we are writing about

Geometric Modeling Methods in Engineering

We refrained from using the expression computational geometry because it has a wellestablished meaning which does not correspond to what we have in mind. Although we willtouch some of the topics covered in computational geometry (for example, triangulations),we are more interested in dealing with curves and surfaces from an algorithmic point ofview . In this respect, we are flirting with the intuitionists ideal of doing mathematics froma constructive point of view. Such a point of view is of course very relevant to computerscience.

This book consists of four parts.

Part I provides an introduction to affine geometry. This ensures that readers are onfirm grounds to proceed with the rest of the book, in particular the study of curvesand surfaces. This is also useful to establish the notation and terminology. Readers

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2 CONTENTS

proficient in geometry may omit this section, or use it by need . On the other hand,readers totally unfamiliar with this material will probably have a hard time with therest of the book. These readers are advised do some extra reading in order to assimilatesome basic knowledge of geometry. For example, we highly recommend Berger [5, 6],Pedoe [59], Samuel [69], Hilbert and Cohn-Vossen [42], do Carmo [26], Berger andGostiaux [7], Boehm and Prautzsch [11], and Tisseron [83].

Part II deals with an algorithmic treatment of polynomial curves (Bezier curves) andspline curves.

Part III deals with an algorithmic treatment of polynomial surfaces (Bezier rectangularor triangular surfaces), and spline surfaces. We also include a section on subdivisionsurfaces, an exciting and active area of research in geometric modeling and animation,as attested by several papers in SIGGRAPH98, especially the paper by DeRose et al[24] on the animated character Geri, from the short movie Geris game.

Part IV consists of appendices consisting of basics of linear algebra, certain technicalproofs that were omitted earlier, complements of affine geometry, analysis, and dif-ferential calculus. This part has been included to make the material of parts IIIIself-contained. Our advice is to use it by need !

Our goal is not to write a text on the many specialized and practical CAGD methods.Our main goal is to provide an introduction to the concepts needed in tackling problemsarising in computer graphics, geometric modeling, computer vision, and motion planning,just to mention some key areas. As it turns out, one of the most spectacular applicationof these concepts is the treatment of curves and surfaces in terms of control points, a toolextensively used in CAGD. This is why many pages are devoted to an algorithmic treatmentof curves and surfaces. However, we only provide a cursory coverage of CAGD methods.Luckily, there are excellent texts on CAGD, including Bartels, Beatty, and Barsky [4], Farin[32, 31], Fiorot and Jeannin [35, 36], Riesler [68], Hoschek and Lasser [45], and Piegl andTiller [62]. Similarly, although we cover affine geometry in some detail, we are far from givinga comprehensive treatments of these topics. For such a treatment, we highly recommendBerger [5, 6], Pedoe [59], Tisseron [83], Samuel [69], Dieudonne [25], Sidler [76], and Veblenand Young [85, 86], a great classic. Several sections of this book are inspired by the treatmentin one of several of the above texts, and we are happy to thank the authors for providingsuch inspiration.

Lyle Ramshaws remarkably elegant and inspirational DEC-SRC Report, Blossoming: Aconnectthedots approach to splines [65], radically changed our perspective on polynomialcurves and surfaces. We have happily and irrevocably adopted the view that the mosttransparent manner for presenting much of the theory of polynomial curves and surfaces isto stress the multilinear nature (really multiaffine) of these curves and surfaces. This is incomplete agreement with de Casteljaus original spirit, but as Ramshaw, we are more explicit

CONTENTS 3

in our use of multilinear tools. As the reader will discover, much of the algorithmic theoryof polynomial curves and surfaces is captured by the three words:

Polarize, homogenize, tensorize!

We will be dealing primarily with the following kinds of problems:

Approximating a shape (curve or surface).We will see how this can be done using polynomial curves or surfaces (also called Beziercurves or surfaces), spline curves or surfaces.

Interpolating a set of points, by a curve or a surface.Again, we will see how this can be done using spline curves or spline surfaces.

Drawing a curve or a surface.The tools and techniques developed for solving the approximation problem will be veryuseful for solving the other two problems.

The material presented in this book is related to the classical differential geometry ofcurves and surfaces, and to numerical methods in matrix analysis. In fact, it is often pos-sible to reduce problems involving certain splines to solving systems of linear equations.Thus, it is very helpful to be aware of efficient methods for numerical matrix analysis. Forfurther information on these topics, readers are referred to the excellent texts by Gray [39],Strang [81], and Ciarlet [19]. Strangs beautiful book on applied mathematics is also highlyrecommended as a general reference [80]. There are other interesting applications of geom-etry to computer vision, computer graphics, and solid modeling. Some great references areKoenderink [46] and Faugeras [33] for computer vision, Hoffman [43] for solid modeling, andMetaxas [53] for physics-based deformable models.

Novelties

As far as we know, there is no fully developed modern exposition integrating the basicconcepts of affine geometry as well as a presentation of curves and surfaces from the algo-rithmic point of view in terms of control points (in the polynomial case). There is also noreasonably thorough textbook presentation of the main surface subdivision schemes (Doo-Sabin, Catmull-Clark, Loop), and a technical discussion of convergence and smoothness.

New Treatment, New Results

This books provides an introduction to affine geometry. Generally, background materialor rather technical proofs are relegated to appendices.

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We give an in-depth presentation of polynomial curves and surfaces from an algorith-mic point of view. The approach (sometimes called blossoming) consists in multilinearizingeverything in sight (getting polar forms), which leads very naturally to a presentation ofpolynomial curves and surfaces in terms of control points (Bezier curves and surfaces). Wepresent many algorithms for subdividing and drawing curves and surfaces, all implementedin Mathematica. A clean and elegant presentation of control points is obtained by usinga construction for embedding an affine space into a vector space (the so-called hat con-struction, originating in Berger [5]). We even include an optional chapter (chapter 11)covering tensors and symmetric tensors to provide an in-depth understanding of the foun-dations of blossoming and a more conceptual view of the computational material on curvesand surfaces. The continuity conditions for spline curves and spline surfaces are expressedin terms of polar forms, which yields both geometric and computational insights into thesubtle interaction of knots and de Boor control points.

Subdivision surfaces are the topic of Chapter 9 (section 9.4). Subdivision surfaces forman active and promising area of research. They provide an attractive alternative to splinesurfaces in modeling applications where the topology of surfaces is rather complex, andwhere the initial control polyhedron consists of various kinds of faces, not just trianglesor rectangles. As far as we know, this is the first textbook presentation of three popularmethods due to Doo and Sabin [27, 29, 28], Catmull and Clark [17], and Charles Loop [50].We discuss Loops convergence proof in some detail, and for this, we give a crash course ondiscrete Fourier transforms and (circular) discrete convolutions. A glimpse at subdivisionsurfaces is given in a new Section added to Farins Fourth edition [32]. Subdivision surfacesare also briefly covered in Stollnitz, DeRose, and Salesin [79], but in the context of waveletsand multiresolution representation.

A a general rule, we try to be rigorous, but we always keep the algorithmic nature of themathematical objects under consideration in the forefront.

Many problems and programming projects are proposed (over 200). Some are routine,some are (very) difficult.

Many algorithms and their implementation

Although one of our main concerns is to be mathematically rigorous, which impliesthat we give precise definitions and prove almost all of the results in this book, we areprimarily interested in the repesentation and the implementation of concepts and tools usedto solve geometric problems. Thus, we devote a great deal of efforts to the development andimplemention of algorithms to manipulate curves, surfaces, triangulations, etc. As a matterof fact, we provide Mathematica code for most of the geometric algorithms presented in thisbook. These algorithms were used to prepare most of the illustrations of this book. We alsourge the reader to write his own algorithms, and we propose many challenging programmingprojects.

Open Problems

CONTENTS 5

Not only do we present standard material (although sometimes from a fresh point ofview), but whenever possible, we state some open problems, thus taking the reader to thecutting edge of the field. For example, we describe very clearly the problem of finding anefficient way to compute control points for Ck-continuous triangular surface splines. We alsodiscuss some of the problems with the convergence and smoothness of subdivision surfacemethods.

Whats not covered in this book

Since this book is already quite long, we have omitted rational curves and rational sur-faces, and projective geometry. A good reference on these topics is [31]. We are also writinga text covering these topics rather extensively (and more). We also have omitted solidmodeling techniques, methods for rendering implicit curves and surfaces, the finite elementsmethod, and wavelets. The first two topics are nicely covered in Hoffman [43], a remarkablyclear presentation of wavelets is given in Stollnitz, DeRose, and Salesin [79], and a moremathematical presentation in Strang [82], and the finite element method is the subject of somany books that we will not even attempt to mention any references.

Acknowledgement

This book grew out of lectures notes that I wrote as I have been teaching CIS510, In-troduction to Geometric Methods in Computer Science, for the past four years. I wish tothank some students and colleagues for their comments, including Doug DeCarlo, JaydevDesai, Will Dickinson, Charles Erignac, Hany Farid, Steve Frye, Edith Haber, Andy Hicks,David Jelinek, Ioannis Kakadiaris, Hartmut Liefke, Dimitris Metaxas, Jeff Nimeroff, RichPito, Ken Shoemake, Bond-Jay Ting, Deepak Tolani, Dianna Xu, and most of all Raj Iyer,who screened half of the manuscript with a fine tooth comb. Also thanks to Norm Badler fortriggering my interest in geometric modeling, and to Marcel Berger, Chris Croke, Ron Don-agi, Gerald Farin, Herman Gluck, and David Harbater, for sharing some of their geometricsecrets with me. Finally, many thanks to Eugenio Calabi for teaching me what I know aboutdifferential geometry (and much more!).

6 CONTENTS

Chapter 1

Introduction

1.1 Geometric Methods in Engineering

Geometry, what a glorious subject! For centuries, geometry has played a crucial role inthe development of many scientific and engineering disciplines such as astronomy, geodesy,mechanics, balistics, civil and mechanical engineering, ship building, architecture, and inthis century, automobile and aircraft manufacturing, among others. What makes geometrya unique and particularly exciting branch of mathematics is that it is primarily visual . Onemight say that this is only true of geometry up to the end of the nineteenth century, buteven when the objects are higher-dimensional and very abstract, the intuitions behind thesefancy concepts almost always come from shapes that can somehow be visualized. On theother hand, it was discovered at the end of the nineteenth century that there was a danger inrelying too much on visual intuition, and that this could lead to wrong results or fallaciousarguments. What happened then is that mathematicians started using more algebra andanalysis in geometry, in order to put it on firmer grounds and to obtain more rigorousproofs. The consequence of the strive for more rigor and the injection of more algebra ingeometry is that mathematicians of the beginning of the twentieth century began suppressinggeometric intuitions from their proofs. Geometry lost some of its charm and became a ratherinpenetrable discipline, except for the initiated. It is interesting to observe that most Collegetextbooks of mathematics included a fair amount of geometry up to the fourties. Beginningwith the fifties, the amount of geometry decreases to basically disappear in the seventies.

Paradoxically, with the advent of faster computers, starting in the early sixties, automo-bile and plane manufacturers realized that it was possible to design cars and planes usingcomputer-aided methods. These methods pioneered by de Casteljau, Bezier, and Ferguson,used geometric methods. Although not very advanced, the type of geometry used is very el-egant. Basically, it is a branch of affine geometry, and it is very useful from the point of viewof applications. Thus, there seems to be an interesting turn of events. After being neglectedfor decades, stimulated by computer science, geometry seems to be making a come-back asa fundamental tool used in manufacturing, computer graphics, computer vision, and motionplanning, just to mention some key areas.

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8 CHAPTER 1. INTRODUCTION

We are convinced that geometry will play an important role in computer science andengineering in the years to come. The demand for technology using 3D graphics, virtualreality, animation techniques, etc, is increasing fast, and it is clear that storing and processingcomplex images and complex geometric models of shapes (face, limbs, organs, etc) will berequired. We will need to understand better how to discretize geometric objects such ascurves, surfaces, and volumes. This book represents an attempt at presenting a coherentview of geometric methods used to tackle problems of a geometric nature with a computer.We believe that this can be a great way of learning about curves and surfaces, while havingfun. Furthermore, there are plenty of opportunities for applying these methods to real-worldproblems.

Our main focus is on curves and surfaces, but our point of view is algorithmic. Weconcentrate on methods for discretizing curves and surfaces in order to store them anddisplay them efficiently. Thus, we focus on polynomial curves defined in terms of controlpoints, since they are the most efficient class of curves and surfaces from the point of viewof design and representation. However, in order to gain a deeper understanding of thistheory of curves and surfaces, we present the underlying geometric concepts in some detail,in particular, affine geometry. In turn, since this material relies on some algebra and analysis(linear algebra, directional derivatives, etc), in order to make the book entirely self-contained,we provide some appendices where this background material is presented.

In the next section, we list some problems arising in computer graphics and computervision that can be tackled using the geometric tools and concepts presented in this book.

1.2 Examples of Problems Using Geometric Modeling

The following is a nonexhaustive listing of several different areas in which geometric methods(using curves and surfaces) play a crucial role.

Manufacturing Medical imaging Molecular modeling Computational fluid dynamics Physical simulation in applied mechanics Oceanography, virtual oceans Shape reconstruction Weather analysis

1.2. EXAMPLES OF PROBLEMS USING GEOMETRIC MODELING 9

Computer graphics (rendering smooth curved shapes) Computer animation Data compression Architecture Art (sculpture, 3D images, ...)

A specific subproblem that often needs to be solved, for example in manufacturing prob-lems or in medical imaging, is to fit a curve or a surface through a set of data points. Forsimplicity, let us discuss briefly a curve fitting problem.

Problem: Given N +1 data points x0, . . . , xN and a sequence of N +1 reals u0, . . . , uN ,with ui < ui+1 for all i, 0 i N 1, find a C2-continuous curve F , such that F (ui) = xi,for all i, 0 i N .

As stated above, the problem is actually underdetermined. Indeed, there are many dif-ferent types of curves that solve the above problem (defined by Fourier series, Lagrangeinterpolants, etc), and we need to be more specific as to what kind of curve we would like touse. In most cases, efficiency is the dominant factor, and it turns out that piecewise poly-nomial curves are usually the best choice. Even then, the problem is still underdetermined.However, the problem is no longer underdetermined if we specify some end conditions, forinstance the tangents at x0 and xN . In this case, it can be shown that there is a uniqueB-spline curve solving the above problem (see section 6.8). The next figure shows N +1 = 8data points, and a C2-continuous spline curve F passing through these points, for a uniformsequence of reals ui.

Other points d1, . . . , d8 are also shown. What happens is that the interpolating B-splinecurve is really determined by some sequence of points d1, . . . , dN+1 called de Boor controlpoints (with d1 = x0 and dN+1 = xN ). Instead of specifying the tangents at x0 and xN ,we can specify the control points d0 and dN . Then, it turns out that d1, . . . , dN1 can becomputed from x0, . . . , xN (and d0, dN) by solving a system of linear equations of the form

11 1 1

2 2 2 0. . .

0 N2 N2 N2N1 N1 N1

1

d0d1d2...

dN2dN1dN

=

r0r1r2...

rN2rN1rN

where r0 and rN may be chosen arbitrarily, the coefficients i, i, i are easily computed fromthe uj, and ri = (ui+1 ui1) xi for 1 i N 1 (see section 6.8).

10 CHAPTER 1. INTRODUCTION

b

bc

bc

bc

bc

bc

bc

bc

bc

b

b

b

b

b

b

b

x0 = d1

x1

x2

x3

x4

x5

x6

x7 = d8

d0

d1

d2

d3

d4

d5

d6

d7

Figure 1.1: A C2 interpolation spline curve passing through the points x0, x1, x2, x3, x4, x5,x6, x7

The previous example suggests that curves can be defined in terms of control points .Indeed, specifying curves and surfaces in terms of control points is one of the major techniquesused in geometric design. For example, in medical imaging, one may want to find the contourof some organ, say the heart, given some discrete data. One may do this by fitting a B-spline curve through the data points. In computer animation, one may want to have a personmove from one location to another, passing through some intermediate locations, in a smoothmanner. Again, this problem can be solved using B-splines. Many manufacturing problemsinvolve fitting a surface through some data points. Let us mention automobile design, planedesign, (wings, fuselage, etc), engine parts, ship hulls, ski boots, etc.

We could go on and on with many other examples, but it is now time to review somebasics of affine geometry!

Part IBasics of Affine Geometry

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12

Chapter 2

Basics of Affine Geometry

2.1 Affine Spaces

Geometrically, curves and surfaces are usually considered to be sets of points with somespecial properties, living in a space consisting of points. Typically, one is also interestedin geometric properties invariant under certain transformations, for example, translations,rotations, projections, etc. One could model the space of points as a vector space, but this isnot very satisfactory for a number of reasons. One reason is that the point corresponding tothe zero vector (0), called the origin, plays a special role, when there is really no reason to havea privileged origin. Another reason is that certain notions, such as parallelism, are handledin an akward manner. But the deeper reason is that vector spaces and affine spaces reallyhave different geometries. The geometric properties of a vector space are invariant underthe group of bijective linear maps, whereas the geometric properties of an affine space areinvariant under the group of bijective affine maps, and these two groups are not isomorphic.Roughly speaking, there are more affine maps than linear maps.

Affine spaces provide a better framework for doing geometry. In particular, it is possibleto deal with points, curves, surfaces, etc, in an intrinsic manner, that is, independentlyof any specific choice of a coordinate system. As in physics, this is highly desirable toreally understand whats going on. Of course, coordinate systems have to be chosen tofinally carry out computations, but one should learn to resist the temptation to resort tocoordinate systems until it is really necessary.

Affine spaces are the right framework for dealing with motions, trajectories, and physicalforces, among other things. Thus, affine geometry is crucial to a clean presentation ofkinematics, dynamics, and other parts of physics (for example, elasticity). After all, a rigidmotion is an affine map, but not a linear map in general. Also, given an m n matrix Aand a vector b Rm, the set U = {x Rn | Ax = b} of solutions of the system Ax = b is anaffine space, but not a vector space (linear space) in general.

13

14 CHAPTER 2. BASICS OF AFFINE GEOMETRY

Use coordinate systems only when needed!

This chapter proceeds as follows. We take advantage of the fact that almost every affineconcept is the counterpart of some concept in linear algebra. We begin by defining affinespaces, stressing the physical interpretation of the definition in terms of points (particles)and vectors (forces). Corresponding to linear combinations of vectors, we define affine com-binations of points (barycenters), realizing that we are forced to restrict our attention tofamilies of scalars adding up to 1. Corresponding to linear subspaces, we introduce affinesubspaces as subsets closed under affine combinations. Then, we characterize affine sub-spaces in terms of certain vector spaces called their directions. This allows us to define aclean notion of parallelism. Next, corresponding to linear independence and bases, we defineaffine independence and affine frames. We also define convexity. Corresponding to linearmaps, we define affine maps as maps preserving affine combinations. We show that everyaffine map is completely defined by the image of one point and a linear map. We investi-gate briefly some simple affine maps, the translations and the central dilatations. Certaintechnical proofs and some complementary material on affine geometry are relegated to anappendix (see Chapter B).

Our presentation of affine geometry is far from being comprehensive, and it is biasedtowards the algorithmic geometry of curves and surfaces. For more details, the reader isreferred to Pedoe [59], Snapper and Troyer [77], Berger [5, 6], Samuel [69], Tisseron [83], andHilbert and Cohn-Vossen [42].

Suppose we have a particle moving in 3-space and that we want to describe the trajectoryof this particle. If one looks up a good textbook on dynamics, such as Greenwood [40], onefinds out that the particle is modeled as a point, and that the position of this point xis determined with respect to a frame in R3 by a vector. Curiously, the notion of aframe is rarely defined precisely, but it is easy to infer that a frame is a pair (O, (e1 ,e2 ,e3 ))consisting of an origin O (which is a point) together with a basis of three vectors (e1 ,e2 ,

e3).

For example, the standard frame in R3 has origin O = (0, 0, 0) and the basis of three vectorse1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). The position of a point x is then defined bythe unique vector from O to x.

But wait a minute, this definition seems to be defining frames and the position of a pointwithout defining what a point is! Well, let us identify points with elements of R3. If so,given any two points a = (a1, a2, a3) and b = (b1, b2, b3), there is a unique free vector denotedab from a to b, the vector

ab = (b1 a1, b2 a2, b3 a3). Note that

b = a +ab,

addition being understood as addition in R3. Then, in the standard frame, given a point

x = (x1, x2, x3), the position of x is the vectorOx = (x1, x2, x3), which coincides with the

point itself. In the standard frame, points and vectors are identified.

2.1. AFFINE SPACES 15

bc

bc

bc

O

a

b

ab

Figure 2.1: Points and free vectors

What if we pick a frame with a different origin, say = (1, 2, 3), but the same basis

vectors (e1 ,e2 ,e3)? This time, the point x = (x1, x2, x3) is defined by two position vectors:

Ox = (x1, x2, x3) in the frame (O, (

e1 ,e2 ,e3 )), andx = (x1 1, x2 2, x3 3) in the frame (, (e1 ,e2 ,e3 )).This is because

Ox =O+

x and

O = (1, 2, 3).

We note that in the second frame (, (e1 ,e2 ,e3 )), points and position vectors are nolonger identified. This gives us evidence that points are not vectors. It may be computation-ally convenient to deal with points using position vectors, but such a treatment is not frameinvariant, which has undesirable effets. Inspired by physics, it is important to define pointsand properties of points that are frame invariant. An undesirable side-effect of the presentapproach shows up if we attempt to define linear combinations of points. First, let us reviewthe notion of linear combination of vectors. Given two vectors u and v of coordinates(u1, u2, u3) and (v1, v2, v3) with respect to the basis (

e1 ,e2 ,e3 ), for any two scalars , , wecan define the linear combination u + v as the vector of coordinates

(u1 + v1, u2 + v2, u3 + v3).

If we choose a different basis (e1 ,

e2 ,

e3 ) and if the matrix P expressing the vectors (

e1 ,

e2 ,

e3 ) over the basis (e1 ,e2 ,e3 ) is

P =

a1 b1 c1a2 b2 c2a3 b3 c3

,

16 CHAPTER 2. BASICS OF AFFINE GEOMETRY

which means that the columns of P are the coordinates of theej over the basis (

e1 ,e2 ,e3 ),since

u1e1 +u2e2 +u3e3 = u1

e1 +u

2

e2 +u

3

e3 and v1

e1 + v2e2 + v3e3 = v1e1 + v

2

e2 + v

3

e3 ,

it is easy to see that the coordinates (u1, u2, u3) and (v1, v2, v3) ofu and v with respect to

the basis (e1 ,e2 ,e3 ) are given in terms of the coordinates (u1, u2, u3) and (v1, v2, v3) of uand v with respect to the basis (e1 ,

e2 ,

e3 ) by the matrix equationsu1u2

u3

= Pu1u2u3

andv1v2v3

= Pv1v2v3

.From the above, we getu1u2

u3

= P1u1u2u3

andv1v2v3

= P1v1v2v3

,and by linearity, the coordinates

(u1 + v1, u

2 + v

2, u

3 + v

3)

of u + v with respect to the basis (e1 ,e2 ,

e3 ) are given byu1 + v1u2 + v2

u3 + v3

= P1u1u2u3

+ P1v1v2v3

= P1u1 + v1u2 + v2u3 + v3

.Everything worked out because the change of basis does not involve a change of origin.

On the other hand, if we consider the change of frame from the frame (O, (e1 ,e2 ,e3 )) tothe frame (, (e1 ,e2 ,e3 )), where O = (1, 2, 3), given two points a and b of coordinates(a1, a2, a3) and (b1, b2, b3) with respect to the frame (O, (

e1 ,e2 ,e3 )) and of coordinates(a1, a

2, a

3) and (b

1, b

2, b

3) of with respect to the frame (, (

e1 ,e2 ,e3 )), since(a1, a

2, a

3) = (a1 1, a2 2, a3 3) and (b1, b2, b3) = (b1 1, b2 2, b3 3),

the coordinates of a+ b with respect to the frame (O, (e1 ,e2 ,e3 )) are(a1 + b1, a2 + b2, a3 + b3),

but the coordinates(a1 + b

1, a

2 + b

2, a

3 + b

3)

2.1. AFFINE SPACES 17

of a+ b with respect to the frame (, (e1 ,e2 ,e3 )) are(a1 + b1 (+ )1, a2 + b2 (+ )2, a3 + b3 (+ )3)

which are different from

(a1 + b1 1, a2 + b2 2, a3 + b3 3),unless + = 1.

Thus, we discovered a major difference between vectors and points: the notion of linearcombination of vectors is basis independent, but the notion of linear combination of pointsis frame dependent. In order to salvage the notion of linear combination of points, somerestriction is needed: the scalar coefficients must add up to 1.

A clean way to handle the problem of frame invariance and to deal with points in a moreintrinsic manner is to make a clearer distinction between points and vectors. We duplicateR3 into two copies, the first copy corresponding to points, where we forget the vector spacestructure, and the second copy corresponding to free vectors, where the vector space structureis important. Furthermore, we make explicit the important fact that the vector space R3 actson the set of points R3: given any point a = (a1, a2, a3) and any vector

v = (v1, v2, v3),we obtain the point

a+v = (a1 + v1, a2 + v2, a3 + v3),which can be thought of as the result of translating a to b using the vector v . We canimagine that v is placed such that its origin coincides with a and that its tip coincides withb. This action +: R3 R3 R3 satisfies some crucial properties. For example,

a+0 = a,

(a+u ) +v = a+ (u +v ),

and for any two points a, b, there is a unique free vectorab such that

b = a +ab.

It turns out that the above properties, although trivial in the case of R3, are all that isneeded to define the abstract notion of affine space (or affine structure). The basic idea is

to consider two (distinct) sets E andE , where E is a set of points (with no structure) and

E is a vector space (of free vectors) acting on the set E. Intuitively, we can think of the

elements ofE as forces moving the points in E, considered as physical particles. The effect

of applying a force (free vector) u E to a point a E is a translation. By this, we meanthat for every force u E , the action of the force u is to move every point a E to thepoint a + u E obtained by the translation corresponding to u viewed as a vector. Sincetranslations can be composed, it is natural that

E is a vector space.

18 CHAPTER 2. BASICS OF AFFINE GEOMETRY

For simplicity, it is assumed that all vector spaces under consideration are defined overthe field R of real numbers. It is also assumed that all families of vectors and scalars arefinite. The formal definition of an affine space is as follows.

Did you say A fine space?

Definition 2.1.1. An affine space is either the empty set, or a triple E,E ,+ consistingof a nonempty set E (of points), a vector space

E (of translations, or free vectors), and an

action +: E E E, satisfying the following conditions.(AF1) a+ 0 = a, for every a E;

(AF2) (a+ u) + v = a+ (u+ v), for every a E, and every u, v E ;

(AF3) For any two points a, b E, there is a unique u E such that a+ u = b. The uniquevector u E such that a + u = b is denoted by ab, or sometimes by b a. Thus, wealso write

b = a+ab

(or even b = a+ (b a)).

The dimension of the affine space E,E ,+ is the dimension dim(E ) of the vector spaceE . For simplicity, it is denoted as dim(E).

Note that a(a + v) = v

for all a E and all v E , since a(a + v) is the unique vector such that a+v = a+a(a+ v).Thus, b = a + v is equivalent to

ab = v. The following diagram gives an intuitive picture

of an affine space. It is natural to think of all vectors as having the same origin, the nullvector.

The axioms defining an affine space E,E ,+ can be interpreted intuitively as sayingthat E and

E are two different ways of looking at the same object, but wearing different

sets of glasses, the second set of glasses depending on the choice of an origin in E. Indeed,we can choose to look at the points in E, forgetting that every pair (a, b) of points defines a

unique vector ab inE , or we can choose to look at the vectors u in

E , forgetting the points

in E. Furthermore, if we also pick any point a in E, point that can be viewed as an origin

in E, then we can recover all the points in E as the translated points a + u for all u E .This can be formalized by defining two maps between E and

E .

2.1. AFFINE SPACES 19

bc

bc

bc

EE

a

b = a+ u

c = a + wu

v

w

Figure 2.2: Intuitive picture of an affine space

For every a E, consider the mapping from E to E:

u 7 a + u,

where u E , and consider the mapping from E to E :

b 7 ab,

where b E. The composition of the first mapping with the second is

u 7 a+ u 7 a(a+ u),

which, in view of (AF3), yields u. The composition of the second with the first mapping is

b 7 ab 7 a +ab,

which, in view of (AF3), yields b. Thus, these compositions are the identity fromE to

E

and the identity from E to E, and the mappings are both bijections.

When we identify E toE via the mapping b 7 ab, we say that we consider E as the

vector space obtained by taking a as the origin in E, and we denote it as Ea. Thus, an

affine space E,E ,+ is a way of defining a vector space structure on a set of points E,without making a commitment to a fixed origin in E. Nevertheless, as soon as we committo an origin a in E, we can view E as the vector space Ea. However, we urge the reader to

think of E as a physical set of points and ofE as a set of forces acting on E, rather than

reducing E to some isomorphic copy of Rn. After all, points are points, and not vectors! For

notational simplicity, we will often denote an affine space E,E ,+ as (E,E ), or even asE. The vector space

E is called the vector space associated with E.

20 CHAPTER 2. BASICS OF AFFINE GEOMETRY

One should be careful about the overloading of the addition symbol +. Addition is well-defined on vectors, as in u+ v, the translate a + u of a point a E by a vector u E

is also well-defined, but addition of points a+ b does not make sense. In this respect, thenotation b a for the unique vector u such that b = a + u, is somewhat confusing, since itsuggests that points can be substracted (but not added!). Yet, we will see in section 10.1that it is possible to make sense of linear combinations of points, and even mixed linearcombinations of points and vectors.

Any vector spaceE has an affine space structure specified by choosing E =

E , and

letting + be addition in the vector spaceE . We will refer to this affine structure on a vector

space as the canonical (or natural) affine structure onE . In particular, the vector space Rn

can be viewed as an affine space denoted as An. In order to distinguish between the doublerole played by members of Rn, points and vectors, we will denote points as row vectors, andvectors as column vectors. Thus, the action of the vector space Rn over the set Rn simplyviewed as a set of points, is given by

(a1, . . . , an) +

u1...un

= (a1 + u1, . . . , an + un).The affine space An is called the real affine space of dimension n. In most cases, we willconsider n = 1, 2, 3.

2.2 Examples of Affine Spaces

Let us now give an example of an affine space which is not given as a vector space (atleast, not in an obvious fashion). Consider the subset L of A2 consisting of all points (x, y)satisfying the equation

x+ y 1 = 0.The set L is the line of slope 1 passing through the points (1, 0) and (0, 1).

The line L can be made into an official affine space by defining the action +: LR Lof R on L defined such that for every point (x, 1 x) on L and any u R,

(x, 1 x) + u = (x+ u, 1 x u).

It immediately verified that this action makes L into an affine space. For example, for anytwo points a = (a1, 1 a1) and b = (b1, 1 b1) on L, the unique (vector) u R such thatb = a + u is u = b1 a1. Note that the vector space R is isomorphic to the line of equationx+ y = 0 passing through the origin.

2.2. EXAMPLES OF AFFINE SPACES 21

bc

bc

L

Figure 2.3: An affine space: the line of equation x+ y 1 = 0

Similarly, consider the subset H of A3 consisting of all points (x, y, z) satisfying theequation

x+ y + z 1 = 0.The set H is the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1). The planeH can be made into an official affine space by defining the action +: H R2 H of R2 onH defined such that for every point (x, y, 1 x y) on H and any

(uv

) R2,

(x, y, 1 x y) +(uv

)= (x+ u, y + v, 1 x u y v).

For a slightly wilder example, consider the subset P of A3 consisting of all points (x, y, z)satisfying the equation

x2 + y2 z = 0.The set P is paraboloid of revolution, with axis Oz. The surface P can be made into anofficial affine space by defining the action +: P R2 P of R2 on P defined such that forevery point (x, y, x2 + y2) on P and any

(uv

) R2,

(x, y, x2 + y2) +

(uv

)= (x+ u, y + v, (x+ u)2 + (y + v)2).

This should dispell any idea that affine spaces are dull. Affine spaces not already equippedwith an obvious vector space structure arise in projective geometry.

22 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc

bc

bc

EE

a

b

c

ab

bc

ac

Figure 2.4: Points and corresponding vectors in affine geometry

2.3 Chasles Identity

Given any three points a, b, c E, since c = a+ac, b = a +ab, and c = b+bc , we get

c = b+bc = (a+

ab) +

bc = a + (

ab +

bc)

by (AF2), and thus, by (AF3), ab +

bc = ac,

which is known as Chasles identity .

Since a = a +aa and by (AF1), a = a + 0, by (AF3), we getaa = 0.

Thus, letting a = c in Chasles identity, we get

ba = ab.

Given any four points a, b, c, d E, since by Chasles identityab +

bc =

ad+

dc = ac,

we haveab =

dc iff

bc =

ad (the parallelogram law).

2.4 Affine Combinations, Barycenters

A fundamental concept in linear algebra is that of a linear combination. The correspondingconcept in affine geometry is that of an affine combination, also called a barycenter . However,there is a problem with the naive approach involving a coordinate system, as we saw in section2.1. Since this problem is the reason for introducing affine combinations, at the risk of boring

2.4. AFFINE COMBINATIONS, BARYCENTERS 23

bc

bc

bc

bc

O = d

c

a

b

Figure 2.5: Two coordinates systems in R2

certain readers, we give another example showing what goes wrong if we are not careful indefining linear combinations of points. Consider R2 as an affine space, under its natural

coordinate system with origin O = (0, 0) and basis vectors

(10

)and

(01

). Given any two

points a = (a1, a2) and b = (b1, b2), it is natural to define the affine combination a + b asthe point of coordinates (a1 + b1, a2 + b2).

Thus, when a = (1,1) and b = (2, 2), the point a+ b is the point c = (1, 1). However,let us now consider the new coordinate system with respect to the origin c = (1, 1) (and thesame basis vectors). This time, the coordinates of a are (2,2), and the coordinates of bare (1, 1), and the point a + b is the point d of coordinates (1,1). However, it is clearthat the point d is identical to the origin O = (0, 0) of the first coordinate system. Thus,a + b corresponds to two different points depending on which coordinate system is used forits computation!

Thus, some extra condition is needed in order for affine combinations to make sense. Itturns out that if the scalars sum up to 1, the definition is intrinsic, as the following lemmashows.

Lemma 2.4.1. Given an affine space E, let (ai)iI be a family of points in E, and let(i)iI be a family of scalars. For any two points a, b E, the following properties hold: (1)If

iI i = 1, then

a+iI

iaai = b+

iI

ibai.

(2) If

iI i = 0, then iI

iaai =

iI

ibai.

24 CHAPTER 2. BASICS OF AFFINE GEOMETRY

Proof. (1) By Chasles identity (see section 2.3), we have

a+iI

iaai = a+

iI

i(ab +

bai)

= a+ (iI

i)ab +

iI

ibai

= a+ab +

iI

ibai since

iI i = 1

= b+iI

ibai since b = a+

ab.

(2) We also have iI

iaai =

iI

i(ab +

bai)

= (iI

i)ab +

iI

ibai

=iI

ibai,

since

iI i = 0.

Thus, by lemma 2.4.1, for any family of points (ai)iI in E, for any family (i)iI ofscalars such that

iI i = 1, the point

x = a+iI

iaai

is independent of the choice of the origin a E. The unique point x is called the barycenter(or barycentric combination, or affine combination) of the points ai assigned the weights i.and it is denoted as

iIiai.

In dealing with barycenters, it is convenient to introduce the notion of a weighted point ,which is just a pair (a, ), where a E is a point, and R is a scalar. Then, given a familyof weighted points ((ai, i))iI , where

iI i = 1, we also say that the point

iI iai is

the barycenter of the family of weighted points ((ai, i))iI . Note that the barycenter x ofthe family of weighted points ((ai, i))iI is the unique point such that

ax =iI

iaai for every a E,

2.4. AFFINE COMBINATIONS, BARYCENTERS 25

and setting a = x, the point x is the unique point such thatiI

ixai = 0.

In physical terms, the barycenter is the center of mass of the family of weighted points((ai, i))iI (where the masses have been normalized, so that

iI i = 1, and negative

masses are allowed).

Remarks:

(1) For all m 2, it is easy to prove that the barycenter of m weighted points can beobtained by repeated computations of barycenters of two weighted points.

(2) When

iI i = 0, the vector

iI iaai does not depend on the point a, and we may

denote it as

iI iai. This observation will be used in Chapter 10.1 to define a vectorspace in which linear combinations of both points and vectors make sense, regardlessof the value of

iI i.

The figure below illustrates the geometric construction of the barycenters g1 and g2 ofthe weighted points

(a, 1

4

),(b, 1

4

), and

(c, 1

2

), and (a,1), (b, 1), and (c, 1).

The point g1 can be constructed geometrically as the middle of the segment joining c tothe middle 1

2a+ 1

2b of the segment (a, b), since

g1 =1

2

(1

2a+

1

2b

)+1

2c.

The point g2 can be constructed geometrically as the point such that the middle12b+ 1

2c of

the segment (b, c) is the middle of the segment (a, g2), since

g2 = a + 2(1

2b+

1

2c

).

Later on, we will see that a polynomial curve can be defined as a set of barycenters of afixed number of points. For example, let (a, b, c, d) be a sequence of points in A2. Observethat

(1 t)3 + 3t(1 t)2 + 3t2(1 t) + t3 = 1,since the sum on the left-hand side is obtained by expanding (t + (1 t))3 = 1 using thebinomial formula. Thus,

(1 t)3 a+ 3t(1 t)2 b+ 3t2(1 t) c+ t3 dis a well-defined affine combination. Then, we can define the curve F : A A2 such that

F (t) = (1 t)3 a+ 3t(1 t)2 b+ 3t2(1 t) c+ t3 d.Such a curve is called a Bezier curve, and (a, b, c, d) are called its control points . Note thatthe curve passes through a and d, but generally not through b and c. We will see in the nextchapter how any point F (t) on the curve can be constructed using an algorithm performingthree affine interpolation steps (the de Casteljau algorithm).

26 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc bc

bc

bc

bc

bc bc

bc

bc

bc

a b

c

g1

a b

cg2

Figure 2.6: Barycenters, g1 =14a+ 1

4b+ 1

2c, g2 = a + b+ c.

2.5. AFFINE SUBSPACES 27

2.5 Affine Subspaces

In linear algebra, a (linear) subspace can be characterized as a nonempty subset of a vectorspace closed under linear combinations. In affine spaces, the notion corresponding to thenotion of (linear) subspace is the notion of affine subspace. It is natural to define an affinesubspace as a subset of an affine space closed under affine combinations.

Definition 2.5.1. Given an affine space E,E ,+, a subset V of E is an affine subspace iffor every family of weighted points ((ai, i))iI in V such that

iI i = 1, the barycenter

iI iai belongs to V .

An affine subspace is also called a flat by some authors. According to definition 2.5.1,the empty set is trivially an affine subspace, and every intersection of affine subspaces is anaffine subspace.

As an example, consider the subset U of A2 defined by

U = {(x, y) R2 | ax+ by = c},i.e. the set of solutions of the equation

ax+ by = c,

where it is assumed that a 6= 0 or b 6= 0. Given any m points (xi, yi) U and any m scalarsi such that 1 + + m = 1, we claim that

mi=1

i(xi, yi) U.

Indeed, (xi, yi) U means thataxi + byi = c,

and if we multiply both sides of this equation by i and add up the resulting m equations,we get

mi=1

(iaxi + ibyi) =mi=1

ic,

and since 1 + + m = 1, we get

a

(mi=1

ixi

)+ b

(mi=1

iyi

)=

(mi=1

i

)c = c,

which shows that (mi=1

ixi,

mi=1

iyi

)=

mi=1

i(xi, yi) U.

28 CHAPTER 2. BASICS OF AFFINE GEOMETRY

Thus, U is an affine subspace of A2. In fact, it is just a usual line in A2.

It turns out that U is closely related to the subset of R2 defined by

U = {(x, y) R2 | ax+ by = 0},

i.e. the set of solution of the homogeneous equation

ax+ by = 0

obtained by setting the right-hand side of ax+ by = c to zero. Indeed, for any m scalars i,the same calculation as above yields that

mi=1

i(xi, yi) U ,

this time without any restriction on the i, since the right-hand side of the equation is

null. Thus,U is a subspace of R2. In fact,

U is one-dimensional, and it is just a usual line

in R2. This line can be identified with a line passing through the origin of A2, line which isparallel to the line U of equation ax + by = c. Now, if (x0, y0) is any point in U , we claimthat

U = (x0, y0) +U ,

where

(x0, y0) +U = {(x0 + u1, y0 + u2) | (u1, u2) U }.

First, (x0, y0) +U U , since ax0 + by0 = c and au1 + bu2 = 0 for all (u1, u2) U . Second,

if (x, y) U , then ax+ by = c, and since we also have ax0 + by0 = c, by subtraction, we geta(x x0) + b(y y0) = 0,

which shows that (x x0, y y0) U , and thus (x, y) (x0, y0) +U . Hence, we also haveU (x0, y0) +U , and U = (x0, y0) +U .

The above example shows that the affine line U defined by the equation

ax+ by = c

is obtained by translating the parallel lineU of equation

ax+ by = 0

passing through the origin. In fact, given any point (x0, y0) U ,

U = (x0, y0) +U .

2.5. AFFINE SUBSPACES 29

U

U

Figure 2.7: An affine line U and its direction

More generally, it is easy to prove the following fact. Given any mn matrix A and anyvector c Rm, the subset U of An defined by

U = {x Rn | Ax = c}

is an affine subspace of An. Furthermore, if we consider the corresponding homogeneousequation Ax = 0, the set

U = {x Rn | Ax = 0}

is a subspace of Rn, and for any x0 U , we have

U = x0 +U .

This is a general situation. Affine subspaces can be characterized in terms of subspaces ofE . Let V be a nonempty subset of E. For every family (a1, . . . , an) in V , for any family(1, . . . , n) of scalars, for every point a V , observe that for every x E,

x = a+ni=1

iaai

is the barycenter of the family of weighted points

((a1, 1), . . . , (an, n), (a, 1ni=1

i)),

30 CHAPTER 2. BASICS OF AFFINE GEOMETRY

sinceni=1

i + (1ni=1

i) = 1.

Given any point a E and any subspace V of E , let a+ V denote the following subsetof E:

a+ V = {a+ v | v V }.Lemma 2.5.2. Let E,E ,+ be an affine space. (1) A nonempty subset V of E is an affinesubspace iff, for every point a V , the set

Va = {ax | x V }

is a subspace ofE . Consequently, V = a + Va. Furthermore,

V = {xy | x, y V }

is a subspace ofE and Va = V for all a E. Thus, V = a+ V .

(2) For any subspace V ofE , for any a E, the set V = a+ V is an affine subspace.

Proof. (1) Clearly,0 V . If V is an affine subspace, then V is closed under barycentric

combinations, and by the remark before the lemma, for every x E,ax =

ni=1

iaai

iff x is the barycenterni=1

iai + (1ni=1

i)a

of the family of weighted points ((a1, 1), . . . , (an, n), (a, 1 n

i=1 i)). Then, it is clear

that Va is closed under linear combinations, and thus, it is a subspace ofE . Since V =

{xy | x, y V } and Va = {ax | x V }, where a V , it is clear that Va V . Conversely,since xy = ay ax,and since Va is a subspace of

E , we have V Va. Thus, V = Va, for every a V .

(2) If V = a+ V , where V is a subspace ofE , then, for every family of weighted points,

((a+ vi, i))1in, where vi V , and 1 + + n = 1, the barycenter x being given by

x = a +ni=1

ia(a + vi) = a +

ni=1

ivi

is in V , since V is a subspace ofE .

2.5. AFFINE SUBSPACES 31

bc

EE

a

V = a+ V

V

Figure 2.8: An affine subspace V and its directionV

In particular, when E is the natural affine space associated with a vector space E, lemma2.5.2 shows that every affine subspace of E is of the form u+U , for a subspace U of E. Thesubspaces of E are the affine subspaces that contain 0.

The subspace V associated with an affine subspace V is called the direction of V . It is

also clear that the map +: V V V induced by +: E E E confers to V, V,+ anaffine structure.

By the dimension of the subspace V , we mean the dimension of V . An affine subspaceof dimension 1 is called a line, and an affine subspace of dimension 2 is called a plane. Anaffine subspace of codimension 1 is called a hyperplane (recall that a subspace F of a vectorspace E has codimension 1 iff there is some subspace G of dimension 1 such that E = F G,the direct sum of F and G, see Appendix A). We say that two affine subspaces U and V areparallel if their directions are identical. Equivalently, since U = V , we have U = a+ U , andV = b+U , for any a U and any b V , and thus, V is obtained from U by the translationab.

By lemma 2.5.2, a line is specified by a point a E and a nonzero vector v E , i.e. aline is the set of all points of the form a + u, for R. We say that three points a, b, care collinear , if the vectors

ab and ac are linearly dependent. If two of the points a, b, c are

distinct, say a 6= b, then there is a unique R, such that ac = ab, and we define theratio

acab= .

A plane is specified by a point a E and two linearly independent vectors u, v E , i.e.a plane is the set of all points of the form a+ u+v, for , R. We say that four pointsa, b, c, d are coplanar , if the vectors

ab,ac, and ad, are linearly dependent. Hyperplanes will

be characterized a little later.

Lemma 2.5.3. Given an affine space E,E ,+, for any family (ai)iI of points in E, theset V of barycenters

iI iai (where

iI i = 1) is the smallest affine subspace containing

32 CHAPTER 2. BASICS OF AFFINE GEOMETRY

(ai)iI .

Proof. If (ai)iI is empty, then V = , because of the condition

iI i = 1. If (ai)iI isnonempty, then the smallest affine subspace containing (ai)iI must contain the set V ofbarycenters

iI iai, and thus, it is enough to show that V is closed under affine combina-

tions, which is immediately verified.

Given a nonempty subset S of E, the smallest affine subspace of E generated by S isoften denoted as S. For example, a line specified by two distinct points a and b is denotedas a, b, or even (a, b), and similarly for planes, etc.

Remark: Since it can be shown that the barycenter of n weighted points can be obtainedby repeated computations of barycenters of two weighted points, a nonempty subset V ofE is an affine subspace iff for every two points a, b V , the set V contains all barycentriccombinations of a and b. If V contains at least two points, V is an affine subspace iff for anytwo distinct points a, b V , the set V contains the line determined by a and b, that is, theset of all points (1 )a+ b, R.

2.6 Affine Independence and Affine Frames

Corresponding to the notion of linear independence in vector spaces, we have the notion ofaffine independence. Given a family (ai)iI of points in an affine space E, we will reduce thenotion of (affine) independence of these points to the (linear) independence of the families(aiaj)j(I{i}) of vectors obtained by chosing any ai as an origin. First, the following lemmashows that it sufficient to consider only one of these families.

Lemma 2.6.1. Given an affine space E,E ,+, let (ai)iI be a family of points in E. Ifthe family (aiaj)j(I{i}) is linearly independent for some i I, then (aiaj)j(I{i}) is linearlyindependent for every i I.

Proof. Assume that the family (aiaj)j(I{i}) is linearly independent for some specific i I.Let k I with k 6= i, and assume that there are some scalars (j)j(I{k}) such that

j(I{k})

jakaj = 0 .

Sinceakaj = akai +aiaj ,

2.6. AFFINE INDEPENDENCE AND AFFINE FRAMES 33

we have j(I{k})

jakaj =

j(I{k})

jakai +

j(I{k})

jaiaj ,

=

j(I{k})jakai +

j(I{i,k})

jaiaj ,

=

j(I{i,k})jaiaj

j(I{k})

j

aiak,and thus

j(I{i,k})jaiaj

j(I{k})

j

aiak = 0 .Since the family (aiaj)j(I{i}) is linearly independent, we must have j = 0 for all j (I {i, k}) and j(I{k}) j = 0, which implies that j = 0 for all j (I {k}).

We define affine independence as follows.

Definition 2.6.2. Given an affine space E,E ,+, a family (ai)iI of points in E is affinelyindependent if the family (aiaj)j(I{i}) is linearly independent for some i I.

Definition 2.6.2 is reasonable, since by Lemma 2.6.1, the independence of the family(aiaj)j(I{i}) does not depend on the choice of ai. A crucial property of linearly independentvectors (u1 , . . . ,um) is that if a vector v is a linear combination

v =mi=1

iui

of the ui , then the i are unique. A similar result holds for affinely independent points.Lemma 2.6.3. Given an affine space E,E ,+, let (a0, . . . , am) be a family of m+1 pointsin E. Let x E, and assume that x = mi=0 iai, where mi=0 i = 1. Then, the family(0, . . . , m) such that x =

mi=0 iai is unique iff the family (

a0a1, . . . ,a0am) is linearlyindependent.

Proof. Recall that

x =mi=0

iai iffa0x =

mi=1

ia0ai,

wherem

i=0 i = 1. However, it is a well known result of linear algebra that the family(1, . . . , m) such that

a0x =mi=1

ia0ai

34 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc

bc

bc

EE

a0 a1

a2

a0a1

a0a2

Figure 2.9: Affine independence and linear independence

is unique iff (a0a1, . . . ,a0am) is linearly independent (for a proof, see Chapter A, LemmaA.1.10). Thus, if (a0a1, . . . ,a0am) is linearly independent, by lemma A.1.10, (1, . . . , m)is unique, and since 0 = 1

mi=1 i, 0 is also unique. Conversely, the uniqueness of

(0, . . . , m) such that x =m

i=0 iai implies the uniqueness of (1, . . . , m) such that

a0x =mi=1

ia0ai,

and by lemma A.1.10 again, (a0a1, . . . ,a0am) is linearly independent.

Lemma 2.6.3 suggests the notion of affine frame. Affine frames are the affine analogs

of bases in vector spaces. Let E,E ,+ be a nonempty affine space, and let (a0, . . . , am)be a family of m + 1 points in E. The family (a0, . . . , am) determines the family of m

vectors (a0a1, . . . ,a0am) in E . Conversely, given a point a0 in E and a family of m vectors(u1, . . . , um) in

E , we obtain the family of m+1 points (a0, . . . , am) in E, where ai = a0+ui,

1 i m.Thus, for any m 1, it is equivalent to consider a family of m+ 1 points (a0, . . . , am) in

E, and a pair (a0, (u1, . . . , um)), where the ui are vectors inE .

When (a0a1, . . . ,a0am) is a basis of E , then, for every x E, since x = a0 +a0x, thereis a unique family (x1, . . . , xm) of scalars, such that

x = a0 + x1a0a1 + + xma0am.

The scalars (x1, . . . , xm) are coordinates with respect to (a0, (a0a1, . . . ,a0am)). Since

x = a0 +

mi=1

xia0ai iff x = (1

mi=1

xi)a0 +

mi=1

xiai,

2.6. AFFINE INDEPENDENCE AND AFFINE FRAMES 35

x E can also be expressed uniquely as

x =mi=0

iai

withm

i=0 i = 1, and where 0 = 1 m

i=1 xi, and i = xi for 1 i m. The scalars(0, . . . , m) are also certain kinds of coordinates with respect to (a0, . . . , am). All this issummarized in the following definition.

Definition 2.6.4. Given an affine space E,E ,+, an affine frame with origin a0 is afamily (a0, . . . , am) of m+1 points in E such that (

a0a1, . . . ,a0am) is a basis of E . The pair(a0, (

a0a1, . . . ,a0am)) is also called an affine frame with origin a0. Then, every x E can beexpressed as

x = a0 + x1a0a1 + + xma0am

for a unique family (x1, . . . , xm) of scalars, called the coordinates of x w.r.t. the affine frame(a0, (

a0a1, . . . ,a0am)). Furthermore, every x E can be written as

x = 0a0 + + mamfor some unique family (0, . . . , m) of scalars such that 0+ +m = 1 called the barycentriccoordinates of x with respect to the affine frame (a0, . . . , am).

The coordinates (x1, . . . , xm) and the barycentric coordinates (0, . . . , m) are related bythe equations 0 = 1

mi=1 xi and i = xi, for 1 i m. An affine frame is called an

affine basis by some authors. The figure below shows affine frames for |I| = 0, 1, 2, 3.

A family of two points (a, b) in E is affinely independent iffab 6= 0, iff a 6= b. If a 6= b, the

affine subspace generated by a and b is the set of all points (1)a+b, which is the uniqueline passing through a and b. A family of three points (a, b, c) in E is affinely independent

iffab and ac are linearly independent, which means that a, b, and c are not on a same line

(they are not collinear). In this case, the affine subspace generated by (a, b, c) is the set of allpoints (1 )a+ b+ c, which is the unique plane containing a, b, and c. A family offour points (a, b, c, d) in E is affinely independent iff

ab, ac, and ad are linearly independent,

which means that a, b, c, and d are not in a same plane (they are not coplanar). In thiscase, a, b, c, and d, are the vertices of a tetrahedron.

Given n+1 affinely independent points (a0, . . . , an) in E, we can consider the set of points0a0+ +nan, where 0+ +n = 1 and i 0 (i R). Such affine combinations arecalled convex combinations . This set is called the convex hull of (a0, . . . , an) (or n-simplexspanned by (a0, . . . , an)). When n = 1, we get the line segment between a0 and a1, includinga0 and a1. When n = 2, we get the interior of the triangle whose vertices are a0, a1, a2,

36 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc

bc bc

bc bc

bc

bc

bc

bc

bc

a0

a0 a1

a0 a1

a2

a0

a3

a2

a1

Figure 2.10: Examples of affine frames.

bc bc

bc bc

a0 a1

da2

Figure 2.11: A paralellotope

including boundary points (the edges). When n = 3, we get the interior of the tetrahedronwhose vertices are a0, a1, a2, a3, including boundary points (faces and edges). The set

{a0 + 1a0a1 + + na0an | where 0 i 1 (i R)},

is called the parallelotope spanned by (a0, . . . , an). When E has dimension 2, a parallelotopeis also called a parallelogram, and when E has dimension 3, a parallelepiped . A parallelotopeis shown in figure 2.11: it consists of the points inside of the parallelogram (a0, a1, a2, d),including its boundary.

More generally, we say that a subset V of E is convex if for any two points a, b V , wehave c V for every point c = (1 )a+ b, with 0 1 ( R).

Points are not vectors!The following example illustrates why treating points as vectors may cause problems.

Let a, b, c be three affinely independent points in A3. Any point x in the plane (a, b, c) can

2.7. AFFINE MAPS 37

be expressed asx = 0a+ 1b+ 2c,

where 0 + 1 + 2 = 1. How can we compute 0, 1, 2? Letting a = (a1, a2, a3), b =(b1, b2, b3), c = (c1, c2, c3), and x = (x1, x2, x3) be the coordinates of a, b, c, x in the standardframe of A3, it is tempting to solve the system of equationsa1 b1 c1a2 b2 c2

a3 b3 c3

012

=x1x2x3

.However, there is a problem when the origin of the coordinate system belongs to the

plane (a, b, c), since in this case, the matrix is not invertible! What we should really be doingis to solve the system

0Oa+ 1

Ob+ 2

Oc =

Ox,

where O is any point not in the plane (a, b, c). An alternative is to use certain well chosencross-products. It can be shown that barycentric coordinates correspond to various ratios ofareas and volumes, see the Problems.

2.7 Affine Maps

Corresponding to linear maps, we have the notion of an affine map. An affine map is definedas a map preserving affine combinations.

Definition 2.7.1. Given two affine spaces E,E ,+ and E ,E ,+, a function f : E E is an affine map iff for every family ((ai, i))iI of weighted points in E such that

iI i = 1,

we havef(iI

iai) =iI

if(ai).

In other words, f preserves barycenters.

Affine maps can be obtained from linear maps as follows. For simplicity of notation, thesame symbol + is used for both affine spaces (instead of using both + and +).

Given any point a E, any point b E , and any linear map h : E E , we claim thatthe map f : E E defined such that

f(a+ v) = b+ h(v)

is an affine map. Indeed, for any family (i)iI of scalars such that

iI i = 1, for anyfamily (vi )iI , since

iIi(a + vi) = a+

iI

ia(a+ vi) = a+

iI

ivi,

38 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc bc

bc bc

bc bc

bc bc

a b

cd

a b

cd

Figure 2.12: The effect of a shear

and iI

i(b+ h(vi)) = b+iI

ib(b+ h(vi)) = b+

iI

ih(vi),

we have

f(iI

i(a+ vi)) = f(a+iI

ivi)

= b+ h(iI

ivi)

= b+iI

ih(vi)

=iI

i(b+ h(vi))

=iI

if(a+ vi).

Note that the condition

iI i = 1 was implicitly used (in a hidden call to Lemma2.4.1) in deriving that

iIi(a+ vi) = a +

iI

ivi andiI

i(b+ h(vi)) = b+iI

ih(vi).

As a more concrete example, the map(x1x2

)7(1 20 1

)(x1x2

)+

(31

)defines an affine map in A2. It is a shear followed by a translation. The effect of this shearon the square (a, b, c, d) is shown in figure 2.12. The image of the square (a, b, c, d) is theparallelogram (a, b, c, d).

Let us consider one more example. The map(x1x2

)7(1 11 3

)(x1x2

)+

(30

)

2.7. AFFINE MAPS 39

bc bc

bc bc

bc

bc

bc

bc

a b

cd

a

b

c

d

Figure 2.13: The effect of an affine map

is an affine map. Since we can write(1 11 3

)=2

(22

22

22

22

)(1 20 1

),

this affine map is the composition of a shear, followed by a rotation of angle /4, followed bya magnification of ratio

2, followed by a translation. The effect of this map on the square

(a, b, c, d) is shown in figure 2.13. The image of the square (a, b, c, d) is the parallelogram(a, b, c, d).

The following lemma shows the converse of what we just showed. Every affine map isdetermined by the image of any point and a linear map.

Lemma 2.7.2. Given an affine map f : E E , there is a unique linear map f : E E ,such that

f(a+ v) = f(a) + f(v),

for every a E and every v E .Proof. The proof is not difficult, but very technical. It can be found in Appendix B, Section

B.1. We simply sketch the main ideas. Let a be any point in E. If a linear map f :E E

satisfying the condition of the lemma exists, this map must satisfy the equation

f(v) =f(a)f(a+ v)

for every v E . We then have to check that the map defined in such a way is linear andthat its definition does not depend on the choice of a E.

The unique linear map f :E E given by lemma 2.7.2 is called the linear map asso-

ciated with the affine map f .

40 CHAPTER 2. BASICS OF AFFINE GEOMETRY

bc

bc

bc

bc

EE

E E

a

f(a)

a+ v

f(a) + f(v)

= f(a+ v)

v

f(v)

f f

Figure 2.14: An affine map f and its associated linear mapf

Note that the conditionf(a+ v) = f(a) + f(v),

for every a E and every v E can be stated equivalently asf(x) = f(a) + f(ax), or f(a)f(x) = f(ax),

for all a, x E. Lemma 2.7.2 shows that for any affine map f : E E , there are pointsa E, b E , and a unique linear map f : E E , such that

f(a+ v) = b+ f(v),

for all v E (just let b = f(a), for any a E). Affine maps for which f is the identitymap are called translations . Indeed, if

f = id,

f(x) = f(a)+f (ax) = f(a)+ax = x+xa+af(a)+ax = x+xa+af(a)xa = x+af(a),

and so xf(x) =

af(a),

which shows that f is the translation induced by the vectoraf(a) (which does not depend

on a).

2.7. AFFINE MAPS 41

bc bc

bc

bc

bc

bc bc

bc

a0 a1

a2

0a0 + 1a1 + 2a2

b0

b1 b2

0b0 + 1b1 + 2b2

Figure 2.15: An affine map mapping a0, a1, a2 to b0, b1, b2.

Since an affine map preserves barycenters, and since an affine subspace V is closed underbarycentric combinations, the image f(V ) of V is an affine subspace in E . So, for example,the image of a line is a point or a line, the image of a plane is either a point, a line, or aplane.

It is easily verified that the composition of two affine maps is an affine map. Also, givenaffine maps f : E E and g : E E , we have

g(f(a+ v)) = g(f(a) + f(v)) = g(f(a)) + g(f(v)),

which shows that (g f) = g f . It is easy to show that an affine map f : E E is injectiveiff f :

E E is injective, and that f : E E is surjective iff f : E E is surjective.

An affine map f : E E is constant iff f : E E is the null (constant) linear map equalto 0 for all v E .

If E is an affine space of dimension m and (a0, a1, . . . , am) is an affine frame for E, forany other affine space F , for any sequence (b0, b1, . . . , bm) of m + 1 points in F , there is aunique affine map f : E F such that f(ai) = bi, for 0 i m. Indeed, f must be suchthat

f(0a0 + + mam) = 0b0 + + mbm,where 0 + + m = 1, and this defines a unique affine map on the entire E, since(a0, a1, . . . , am) is an affine frame for E. The following diagram illustrates the above resultwhen m = 2.

Using affine frames, affine maps can be represented in terms of matrices. We explain howan affine map f : E E is represented with respect to a frame (a0, . . . , an) in E, the more

42 CHAPTER 2. BASICS OF AFFINE GEOMETRY

general case where an affine map f : E F is represented with respect to two affine frames(a0, . . . , an) in E and (b0, . . . , bm) in F being analogous. Since

f(a0 +x ) = f(a0) +f (x )

for all x E , we have a0f(a0 +

x ) = a0f(a0) +f (x ).

Since x , a0f(a0), anda0f(a0 +

x ), can be expressed asx = x1a0a1 + + xna0an,

a0f(a0) = b1a0a1 + + bna0an,

a0f(a0 +x ) = y1a0a1 + + yna0an,

if A = (ai j) is the nn-matrix of the linear map f over the basis (a0a1, . . . ,a0an), letting x,y, and b denote the column vectors of components (x1, . . . , xn), (y1, . . . , yn), and (b1, . . . , bn),

a0f(a0 +

x ) = a0f(a0) +f (x )

is equivalent toy = Ax+ b.

Note that b 6= 0 unless f(a0) = a0. Thus, f is generally not a linear transformation, unless ithas a fixed point , i.e., there is a point a0 such that f(a0) = a0. The vector b is the translationpart of the affine map. Affine maps do not always have a fixed point. Obviously, nonnulltranslations have no fixed point. A less trivial example is given by the affine map(

x1x2

)7(1 00 1

)(x1x2

)+

(10

).

This map is a reflection about the x-axis followed by a translation along the x-axis. Theaffine map (

x1x2

)7(1 334

14

)(x1x2

)+

(11

)can also be written as (

x1x2

)7(2 00 1

2

)( 12

32

32

12

)(x1x2

)+

(11

)which shows that it is the composition of a rotation of angle /3, followed by a stretch(by a factor of 2 along the x-axis, and by a factor of 1/2 along the y-axis), followed by a

2.7. AFFINE MAPS 43

translation. It is easy to show that this affine map has a unique fixed point. On the otherhand, the affine map (

x1x2

)7(

85

65

310

25

)(x1x2

)+

(11

)has no fixed point, even though(

85

65

310

25

)=

(2 00 1

2

)(453

5

35

45

),

and the second matrix is a rotation of angle such that cos = 45and sin = 3

5. For more

on fixed points of affine maps, see the problems.

There is a useful trick to convert the equation y = Ax + b into what looks like a linearequation. The trick is to consider an (n + 1) (n + 1)-matrix. We add 1 as the (n + 1)thcomponent to the vectors x, y, and b, and form the (n+ 1) (n + 1)-matrix(

A b0 1

)so that y = Ax+ b is equivalent to(

y1

)=

(A b0 1

)(x1

).

This trick is very useful in kinematics and dynamics, where A is a rotation matrix. Suchaffine maps are called rigid motions .

If f : E E is a bijective affine map, given any three collinear points a, b, c in E,with a 6= b, where say, c = (1 )a + b, since f preserves barycenters, we have f(c) =(1)f(a)+f(b), which shows that f(a), f(b), f(c) are collinear in E . There is a converseto this property, which is simpler to state when the ground field is K = R. The conversestates that given any bijective function f : E E between two real affine spaces of thesame dimension n 2, if f maps any three collinear points to collinear points, then f isaffine. The proof is rather long (see Berger [5] or Samuel [69]).

Given three collinear points where a, b, c, where a 6= c, we have b = (1 )a + c forsome unique , and we define the ratio of the sequence a, b, c, as

ratio(a, b, c) =

(1 ) =abbc,

provided that 6= 1, i.e. that b 6= c. When b = c, we agree that ratio(a, b, c) =. We warnour readers that other authors define the ratio of a, b, c as ratio(a, b, c) = ba

bc. Since affine

maps preserves barycenters, it is clear that affine maps preserve the ratio of three points.

44 CHAPTER 2. BASICS OF AFFINE GEOMETRY

2.8 Affine Groups

We now take a quick look at the bijective affine maps. Given an affine space E, the set ofaffine bijections f : E E is clearly a group, called the affine group of E, and denoted asGA(E). Recall that the group of bijective linear maps of the vector space E is denoted asGL(E). Then, the map f 7 f defines a group homomorphism L : GA(E) GL(E). Thekernel of this map is the set of translations on E.

The subset of all linear maps of the form idE, where R {0}, is a subgroup ofGL(E), and is denoted as RidE (where idE(u) = u, and R = R {0}). The subgroupDIL(E) = L1(RidE) of GA(E) is particularly interesting. It turns out that it is thedisjoint union of the translations and of the dilatations of ratio 6= 1.

The elements of DIL(E) are called affine dilatations . Given any point a E, and anyscalar R, a dilatation (or central dilatation or homothety) of center a and ratio is amap Ha, defined such that

Ha,(x) = a+ ax,

for every x E.

Remark: The terminology does not seem to be universally agreed upon. The terms affinedilatation and central dilatation are used by Pedoe [59]. Snapper and Troyer use the termdilation for an affine dilatation and magnification for a central dilatation [77]. Samuel useshomothety for a central dilatation, a direct translation of the French homothetie [69]. Sincedilation is shorter than dilatation and somewhat easier to pronounce, perhaps we use usethat!

Observe that Ha,(a) = a, and when 6= 0 and x 6= a, Ha,(x) is on the line defined bya and x, and is obtained by scaling ax by .

Figure 2.16 shows the effect of a central dilatation of center d. The triangle (a, b, c) ismagnified to the triangle (a, b, c). Note how every line is mapped to a parallel line.

When = 1, Ha,1 is the identity. Note that Ha, = idE . When 6= 0, it is clear thatHa, is an affine bijection. It is immediately verified that

Ha, Ha, = Ha,.

We have the following useful result.

Lemma 2.8.1. Given any affine space E, for any affine bijection f GA(E), if f = idE,for some R with 6= 1, then there is a unique point c E such that f = Hc,.Proof. Choose some origin a in E. Since f is affine, we have

f(b) = f(a+ab) = f(a) +

ab

2.8. AFFINE GROUPS 45

bc

bc

bc

bc

bc

bc

bc

d

a

b

c

a

b

c

Figure 2.16: The effect of a central dilatation

for all b E. Thus, there is a point c E such that f(c) = c iff

c = f(a) + ac,

iffac = af(a) + ac,

iff

(1 )ac = af(a),iff

ac = 11

af(a),

that is,

c =1

1 f(a)

1 a.

Taking this unique point c as the origin, we have f(b) = c + cb for all b E, which shows

that f = Hc,.

Clearly, if f = idE , the affine map f is a translation. Thus, the group of affine dilatationsDIL(E) is the disjoint union of the translations and of the dilatations of ratio 6= 0, 1.Affine dilatations can be given a purely geometric characterization.

Another point worth mentioning is that affine bijections preserve the ratio of volumes of

parallelotopes. Indeed, given any basis B = (u 1, . . . ,u m) of the vector space E associatedwith the affine space E, given any m + 1 affinely independent points (a0, . . . , am), we can

46 CHAPTER 2. BASICS OF AFFINE GEOMETRY

compute the determinant detB(a0a1, . . . ,a0am) w.r.t. the basis B. For any bijective affine

map f : E E, since

detB(f (a0a1), . . . ,f (a0am)) = det(f )detB(a0a1, . . . ,a0am)

and the determinant of a linear map is intrinsic (i.e., only depends onf , and not on the

particular basis B), we conclude that the ratio

detB(f (a0a1), . . . ,f (a0am))

detB(a0a1, . . . ,a0am) = det(

f )

is independent of the basis B. Since detB(a0a1, . . . ,a0am) is the volume of the parallelotope

spanned by (a0, . . . , am), where the parallelotope spanned by any point a and the vectors

(u 1, . . . ,u m) has unit volume (see Berger [5], Section 9.12), we see that affine bijectionspreserve the ratio of volumes of parallelotopes. In fact, this ratio is independent of the choiceof the parallelotopes of unit volume. In particular, the affine bijections f GA(E) suchthat det(

f ) = 1 preserve volumes. These affine maps form a subgroup SA(E) of GA(E)

called the special affine group of E.

2.9 Affine Hyperplanes

We now consider affine forms and affine hyperplanes. In section 2.5, we observed that theset L of solutions of an equation

ax+ by = c

is an affine subspace of A2 of dimension 1, in fact a line (provided that a and b are not bothnull). It would be equally easy to show that the set P of solutions of an equation

ax+ by + cz = d

is an affine subspace of A3 of dimension 2, in fact a plane (provided that a, b, c are not allnull). More generally, the set H of solutions of an equation

1x1 + + mxm = is an affine subspace of Am, and if 1, . . . , m are not all null, it turns out that it is a subspaceof dimension m 1 called a hyperplane.

We can interpret the equation

1x1 + + mxm = in terms of the map f : Rm R defined such that

f(x1, . . . , xm) = 1x1 + + mxm

2.9. AFFINE HYPERPLANES 47

for all (x1, . . . , xm) Rm. It is immediately verified that this map is affine, and the set H ofsolutions of the equation

1x1 + + mxm = is the null set, or kernel, of the affine map f : Am R, in the sense that

H = f1(0) = {x Am | f(x) = 0},

where x = (x1, . . . , xm).

Thus, it is interesting to consider affine forms , which are just affine maps f : E Rfrom an affine space to R. Unlike linear forms f , for which Ker f is never empty (since italways contains the vector 0), it is possible that f1(0) = , for an affine form f . Given anaffine map f : E R, we also denote f1(0) as Ker f , and we call it the kernel of f . Recallthat an (affine) hyperplane is an affine subspace of codimension 1. The relationship betweenaffine hyperplanes a