Vector Analysis for Computer Graphics (Second Edition)

Thefirst edition ofVector Analysis for Computer Graphicswas published in 2007, and after submitting the manuscript to Springer, I came across geometric algebra. Since then, I have thought about it intensely, and written two books on the subject. Meanwhile, what I call traditional vector analysis, still exists, and continues to be taught throughout the world. So, I decided to write a 2nd edition, and here it is.The book’s main objective is to introduce the subject of vector analysis to readers studying computer graphics, although it will be of interest to a wider readership. It includes most of the original topics, but I have extended various areas, and it still comprises eleven chapters. Chap.1covers the history of vector analysis, which answers most of the questions relating to the existence of the subject. I do believe that the stories behind the emergence of vectors are very relevant to the subject. Chap.2introduces dependent and independent equations, which are so important to vector analysis.Chap. 3covers the ideas behind vectors, building upon the concepts of linear equations, while Chap.4introduces the two products associated with vectors: the scalar and the vector product. The next two chapters are new: Chap.5introduces vector-valued functions and how they are differentiated and integrated; whilst Chap.6deals with vector differential operators: grad, div and curl. Chap.7 develops the use of the grad differential operator by showing how tangent and normal vectors are computed for various curves and surfaces. Chapters8–10cover applications to lines, planes and intersections. Chap.11 examines three ways of rotating vectors inR2andR3,finishing with quaternions. I have thought very carefully about what to leave out, and what to include, and really hope that you are happy with the result.Breinton, UK. John Vince Preface 6 Contents 7 1 History of Vector Analysis 12 1.1 Introduction 12 1.2 Vector 12 1.3 Quaternion 13 1.4 Some History 14 1.5 Summary 18 References 18 2 Linear Equations 19 2.1 Introduction 19 2.2 Systems of Linear Equations 19 2.3 Independent Linear Equations in mathbbR2 20 2.4 Dependent Linear Equations in mathbbR2 21 2.5 Consistent and Inconsistent Linear Equations 22 2.6 Independent Linear Equations in mathbbR3 22 2.7 Dependent Linear Equations in mathbbR3 23 2.8 Matrix Notation 24 2.8.1 Linear Equations in mathbbR2 24 2.8.2 Second-Order Determinant 24 2.8.3 Third-Order Determinant 26 2.9 Solving Linear Equations 27 2.9.1 Solving Linear Equations in mathbbR2 28 2.9.2 Solving Linear Equations in mathbbR3 29 2.10 Summary 30 References 30 3 Vector Algebra 31 3.1 Introduction 31 3.2 Groups, Rings and Fields 31 3.2.1 Groups 31 3.2.2 Abelian Group 33 3.2.3 Rings 33 3.2.4 Fields 33 3.2.5 Division Ring 34 3.3 Space 34 3.3.1 Euclidean Space 34 3.3.2 Vector Space 35 3.3.3 Normed Vector Space 36 3.4 Vector Definition 36 3.4.1 Geometric Vector 36 3.4.2 Euclidean Vector 36 3.4.3 Position Vector 37 3.4.4 n-tuple 37 3.4.5 Vector Notation 37 3.4.6 Column and Row Vectors and the Transpose Operation 38 3.4.7 Vector Magnitude 39 3.4.8 Vector Addition 39 3.4.9 Unit Vector 39 3.4.10 Null Vector 40 3.4.11 Vector Direction 40 3.5 Linearly Dependent and Independent Vectors 40 3.6 Span of a Set of Vectors 43 3.7 Basis and Dimension 45 3.7.1 Cartesian Vectors 46 3.8 Summary 47 References 47 4 Products of Vectors 48 4.1 Introduction 48 4.2 Products 48 4.3 Scalar Product 50 4.3.1 Algebraic Definition 50 4.3.2 Axioms 50 4.3.3 Geometric Definition 51 4.4 Vector Product 52 4.4.1 Geometric Definition 55 4.4.2 The Right-Hand Rule 55 4.5 Triple Products 56 4.5.1 Scalar Triple Product 56 4.5.2 Vector Triple Product 58 4.6 Summary 59 References 60 5 Differentiating Vector-Valued Functions 61 5.1 Introduction 61 5.2 Vector-Valued Functions 61 5.3 Differentiating Vector-Valued Functions 63 5.3.1 Velocity and Speed 63 5.3.2 Acceleration 66 5.3.3 Rules for Differentiating Vector-Valued Functions 69 5.4 Integrating Vector-Valued Functions 69 5.4.1 Velocity of a Falling Object 70 5.4.2 Position of a Moving Object 71 5.5 Summary 72 5.5.1 Summary of Formulae 72 References 73 6 Vector Differential Operators 74 6.1 Introduction 74 6.2 Scalar Fields 74 6.3 Vector Fields 76 6.4 The Gradient of a Scalar Field 77 6.4.1 Gradient of a Scalar Field in mathbbR2 78 6.4.2 Gradient of a Scalar Field in mathbbR3 80 6.4.3 Surface Normal Vectors 81 6.5 The Divergence of a Vector Field 84 6.6 Curl of a Vector Field 87 6.7 Worked Examples 90 6.7.1 Gradient of a Scalar Field 90 6.7.2 Normal Vector to an Ellipse 90 6.7.3 Divergence of a Vector Field 92 6.7.4 Curl of a Vector Field 92 6.8 Summary 93 6.8.1 Summary of Formulae 93 References 94 7 Tangent and Normal Vectors 95 7.1 Introduction 95 7.2 Tangent Vector to a Curve 95 7.3 Normal Vector to a Curve 97 7.3.1 Unit Tangent and Normal Vectors to a Line 99 7.3.2 Unit Tangent and Normal Vectors to a Parabola 101 7.3.3 Unit Tangent and Normal Vectors to a Circle 103 7.3.4 Unit Tangent and Normal Vectors to an Ellipse 105 7.3.5 Unit Tangent and Normal Vectors to a Sine Curve 107 7.3.6 Unit Tangent and Normal Vectors to a cosh Curve 108 7.3.7 Unit Tangent and Normal Vectors to a Helix 110 7.3.8 Unit Tangent and Normal Vectors to a Quadratic Bézier Curve 112 7.4 Unit Tangent and Normal Vectors to a Surface 114 7.4.1 Unit Normal Vectors to a Bilinear Patch 114 7.4.2 Unit Normal Vectors to a Quadratic Bézier Patch 115 7.4.3 Unit Tangent and Normal Vectors to a Sphere 118 7.4.4 Unit Tangent and Normal Vectors to a Torus 119 7.5 Summary 121 7.5.1 Summary of Formulae 122 8 Straight Lines 125 8.1 Introduction 125 8.2 Line Equations 125 8.2.1 The Parametric Form of the Line Equation 126 8.2.2 The Cartesian Form of the Line Equation 128 8.2.3 The General Form of the Line Equation 132 8.3 2-D Space Partitioning 132 8.4 Perpendicular Vectors 137 8.5 A Line Perpendicular to a Vector 139 8.6 The Position of a Point Reflected in a Line 140 8.6.1 The Cartesian Form of the Line Equation 141 8.6.2 The Parametric Form of the Line Equation 143 8.7 The Equation of a Line Segment 145 8.8 The Intersection of Two Straight Lines 147 8.9 The Point of Intersection of Two Line Segments in mathbbR2 150 8.10 Summary 154 9 The Plane 155 9.1 Introduction 155 9.2 The Cartesian Form of the Plane Equation 155 9.3 The Parametric Form of the Plane Equation 157 9.4 A Plane Equation from Three Points 160 9.5 3-D Space Partitioning 162 9.6 The Angle Between Two Planes 166 9.7 The Position and Distance of the Nearest Point on a Plane to a Point 167 9.8 The Reflection of a Point in a Plane 169 9.9 A Plane Between Two Points 172 9.10 Summary 174 10 Intersections 175 10.1 Introduction 175 10.2 Two Intersecting Lines in mathbbR2 176 10.2.1 Parametric Line Equations 176 10.2.2 Cartesian Line Equations 179 10.3 A Line Intersecting a Circle in mathbbR2 181 10.4 A Line Intersecting an Ellipse in mathbbR2 185 10.5 The Shortest Distance Between Two Skew Lines in mathbbR3 189 10.6 Two Intersecting Lines in mathbbR3 192 10.7 A Line Intersecting a Plane 195 10.8 A Line Intersecting a Sphere 197 10.9 A Line Intersecting a Cylinder 201 10.10 A Line Intersecting a Cone 207 10.11 A Line Intersecting a Triangle 209 10.12 A Point Inside a Triangle 217 10.13 A Sphere Intersecting a Plane 220 10.14 A Sphere Touching a Triangle 224 10.15 Two Intersecting Planes 227 10.16 Summary 230 Reference 230 11 Rotating Vectors 231 11.1 Introduction 231 11.2 Rotating Vectors in mathbbR2 About the Origin 231 11.3 Rotating Vectors in mathbbR3 About an Axis 232 11.3.1 Euler Angles 233 11.3.2 Rodrigues' Rotation Formula 235 11.3.3 Quaternions 237 11.3.4 Adding and Subtracting Quaternions 238 11.3.5 Multiplying Quaternions 238 11.3.6 Pure Quaternion 239 11.3.7 The Inverse Quaternion 239 11.3.8 Unit Quaternion 240 11.3.9 Rotating Vectors About an Axis 240 11.3.10 The Double Angle 243 11.3.11 Quaternion as a Matrix 245 11.4 Summary 248 References 248 Index 249 In my last book, Geometry for Computer Graphics, I employed a mixture of algebra and vector analysis to prove many of the equations used in computer graphics. At the time, I did not make any distinction between the two methodologies, but slowly it dawned upon me that I had had to discover, for the first time, how to use vector analysis and associated strategies for solving geometric problems. I suppose that mathematicians are taught this as part of their formal mathematical training, but then, I am not a mathematician! After some deliberation, I decided to write a book that would introduce the beginner to the world of vectors and their application to the geometric problems encountered in computer graphics. I accepted the fact that there would be some duplication of formulas between this and my last book; however, this time I would concentrate on explaining how problems are solved. The book contains eleven chapters: The first chapter distinguishes between scalar and vector quantities, which is reasonably straightforward. The second chapter introduces vector repres- tation, starting with Cartesian coordinates and concluding with the role of direction cosines in changes in axial systems. The third chapter explores how the line equation has a natural vector interpretation and how vector analysis is used to resolve a variety of line-related, geometric problems. Chapter 4 repeats Chapter 3 in the context of the plane. "This second edition has been completely restructured, resulting in a compelling description of vector analysis from its first appearance as a byproduct of Hamilton's quaternions to the use of vectors in solving geometric problems. The result provides readers from different backgrounds with a complete introduction to vector analysis. The author shows why vectors are so useful and how it is possible to develop analytical skills in manipulating vector algebra. Using over 150 full-colour illustrations, the author demonstrates in worked examples how this relatively young branch of mathematics has become a powerful and central tool in describing and solving a wide range of geometric problems. These may be in the form of lines, surfaces and volumes, which may touch, collide, intersect, or create shadows upon complex surfaces. The book is divided into eleven chapters covering the history of vector analysis, linear equations, vector algebra, vector products, differentiating vector-valued functions, vector differential operators, tangent and normal vectors, straight lines, planes, intersections and rotating vectors. The new chapters are about the history, differentiating vector-valued functions, differential operators and tangent and normal vectors. The original chapters have been reworked and illustrated."--Page 4 of cover This book is a complete introduction to vector analysis, especially within the context of computer graphics. The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to, among others, lines, planes, intersections, rotating vectors, and vector differentiation.