Vector Analysis for Computer Graphics

This book is a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating vector algebra. Even though vector analysis is a relatively recent development in the history of mathematics, it has become a powerful and central tool in describing and solving a wide range of geometric problems. 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. Vector Analysis for Computer Graphics......Page 1 Vector Analysis for Computer Graphics......Page 2 Preface......Page 6 Contents......Page 8 Representing vector quantities......Page 13 Non-collinear vectors......Page 19 Cartesian coordinates......Page 23 Length of a vector......Page 25 Vector algebra......Page 26 Unit vectors......Page 29 Rectangular unit vectors......Page 30 Problem 1......Page 31 Problem 2......Page 32 Problem 3......Page 33 Scalar product......Page 34 Vector product......Page 39 Surface normals......Page 45 The algebra of vector products......Page 46 Scalar triple product......Page 48 The vector triple product......Page 53 Perpendicular vectors......Page 56 Linear interpolation......Page 60 Spherical interpolation......Page 61 Direction cosines......Page 64 Change of axial system......Page 66 Summary......Page 69 The parametric form of the line equation......Page 72 The Cartesian form of the line equation......Page 76 The general form of the line equation......Page 79 2D space partitioning......Page 80 A line perpendicular to a vector......Page 85 The position and distance of a point on a line perpendicular to the origin......Page 88 The Cartesian form of the line equation......Page 89 The parametric form of the line equation......Page 91 The position and distance of the nearest point on a line to a point......Page 92 The Cartesian form of the line equation......Page 93 The parametric form of the line equation......Page 95 The Cartesian form of the line equation......Page 97 The parametric form of the line equation......Page 99 A line perpendicular to a line through a point......Page 100 The parametric form of the line equation......Page 101 A line equidistant from two points......Page 104 The intersection of two straight lines......Page 105 The point of intersection of two 2D line segments......Page 108 The Cartesian form of the plane equation......Page 111 The parametric form of the plane equation......Page 113 A plane equation from three points......Page 114 A plane perpendicular to a line and passing through a point......Page 116 A plane through two points and parallel to a line......Page 118 3D space partitioning......Page 120 The angle between two planes......Page 123 The angle between a line and a plane......Page 124 The position and distance of the nearest point on a plane to a point......Page 125 The reflection of a point in a plane......Page 128 A plane between two points......Page 130 A line reflecting off a line......Page 132 A line reflecting off a plane......Page 135 Introduction......Page 137 Two intersecting lines in R2......Page 138 A line intersecting a circle in R2......Page 141 A line intersecting an ellipse in R2......Page 146 The shortest distance between two skew lines in R3......Page 148 Two intersecting lines in R3......Page 150 A line intersecting a plane......Page 152 A line intersecting a sphere......Page 154 A line intersecting an ellipsoid......Page 157 A line intersecting a cylinder......Page 160 A line intersecting a cone......Page 166 A line intersecting a triangle......Page 168 A point inside a triangle......Page 174 A sphere intersecting a plane......Page 175 A sphere touching a triangle......Page 180 Two intersecting planes......Page 183 Rotating a vector about an arbitrary axis......Page 186 Complex numbers......Page 189 Complex number operations......Page 190 The complex conjugate......Page 191 i as a rotator......Page 192 Unifying e, i, sin, and cos......Page 193 Complex numbers as rotators......Page 194 Quaternions......Page 195 Quaternions as rotators......Page 197 The complex conjugate of a quaternion......Page 199 The norm of a quaternion......Page 200 Rotating vectors using quaternions......Page 202 Representing a quaternion as a matrix......Page 204 The derivative of a vector......Page 208 The normal vector to a planar curve......Page 211 The normal vector to a surface......Page 213 Perspective transform......Page 219 Horizontally oblique projection plane......Page 220 Vertically oblique projection plane......Page 223 Arbitrary orientation of the projection plane......Page 225 Pseudo fish-eye projection......Page 229 Light sources......Page 231 Local reflection models......Page 233 Shading......Page 236 Bump mapping......Page 237 Close encounters of the first kind......Page 246 Close encounters of the second kind......Page 248 Appendix A......Page 251 Appendix B......Page 252 References......Page 256 Further Reading......Page 257 Index......Page 258 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.

Vector analysis is relatively young in the history of mathematics, however, in the short period of its existence it has become a powerful and central tool in describing and solving a wide range of geometric problems, many, of which, arise in computer graphics. These may be in the form of describing lines, surfaces and volumes, which may touch, collide, intersect, or create shadows upon complex surfaces.

Vector Analysis for Computer Graphics provides a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating the vector algebra. Each topic covered is placed in the context of a practical application within computer graphics.

The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to lines, planes, intersections, rotating vectors, vector differentiation, projections, rendering and motion.

Vector analysis is relatively young in the history of mathematics, however, in the short period of its existence it has become a powerful and central tool in describing and solving a wide range of geometric problems, many, of which, arise in computer graphics. These may be in the form of describing lines, surfaces and volumes, which may touch, collide, intersect, or create shadows upon complex surfaces. Vector Analysis for Computer Graphics provides a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful and how it is possible to develop analytical skills in manipulating the vector algebra. Each topic covered is placed in the context of a practical application within computer graphics. The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to lines, planes, intersections, rotating vectors, vector differentiation, projections, rendering and motion "Vector Analysis for Computer Graphics provides a complete introduction to vector analysis, especially within the context of computer graphics. The author shows why vectors are useful, and how it is possible to develop analytical skills in manipulating the vector algebra. Each topic covered is placed in the context of a practical application within computer graphics."--Jacket