Vector Analysis for Computer Graphics

Description

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 Preface......Page 5 Contents......Page 7 Representing vector quantities......Page 12 Non-collinear vectors......Page 18 Cartesian coordinates......Page 22 Length of a vector......Page 24 Vector algebra......Page 25 Unit vectors......Page 28 Rectangular unit vectors......Page 29 Problem 1......Page 30 Problem 2......Page 31 Problem 3......Page 32 Scalar product......Page 33 Vector product......Page 38 Surface normals......Page 44 The algebra of vector products......Page 45 Scalar triple product......Page 47 The vector triple product......Page 52 Perpendicular vectors......Page 55 Linear interpolation......Page 59 Spherical interpolation......Page 60 Direction cosines......Page 63 Change of axial system......Page 65 Summary......Page 68 The parametric form of the line equation......Page 71 The Cartesian form of the line equation......Page 75 The general form of the line equation......Page 78 2D space partitioning......Page 79 A line perpendicular to a vector......Page 84 The position and distance of a point on a line perpendicular to the origin......Page 87 The Cartesian form of the line equation......Page 88 The parametric form of the line equation......Page 90 The position and distance of the nearest point on a line to a point......Page 91 The Cartesian form of the line equation......Page 92 The parametric form of the line equation......Page 94 The Cartesian form of the line equation......Page 96 The parametric form of the line equation......Page 98 A line perpendicular to a line through a point......Page 99 The parametric form of the line equation......Page 100 A line equidistant from two points......Page 103 The intersection of two straight lines......Page 104 The point of intersection of two 2D line segments......Page 107 The Cartesian form of the plane equation......Page 110 The parametric form of the plane equation......Page 112 A plane equation from three points......Page 113 A plane perpendicular to a line and passing through a point......Page 115 A plane through two points and parallel to a line......Page 117 3D space partitioning......Page 119 The angle between two planes......Page 122 The angle between a line and a plane......Page 123 The position and distance of the nearest point on a plane to a point......Page 124 The reflection of a point in a plane......Page 127 A plane between two points......Page 129 A line reflecting off a line......Page 131 A line reflecting off a plane......Page 134 Introduction......Page 136 Two intersecting lines in R2......Page 137 A line intersecting a circle in R2......Page 140 A line intersecting an ellipse in R2......Page 145 The shortest distance between two skew lines in R3......Page 147 Two intersecting lines in R3......Page 149 A line intersecting a plane......Page 151 A line intersecting a sphere......Page 153 A line intersecting an ellipsoid......Page 156 A line intersecting a cylinder......Page 159 A line intersecting a cone......Page 165 A line intersecting a triangle......Page 167 A point inside a triangle......Page 173 A sphere intersecting a plane......Page 174 A sphere touching a triangle......Page 179 Two intersecting planes......Page 182 Rotating a vector about an arbitrary axis......Page 185 Complex numbers......Page 188 Complex number operations......Page 189 The complex conjugate......Page 190 i as a rotator......Page 191 Unifying e, i, sin, and cos......Page 192 Complex numbers as rotators......Page 193 Quaternions......Page 194 Quaternions as rotators......Page 196 The complex conjugate of a quaternion......Page 198 The norm of a quaternion......Page 199 Rotating vectors using quaternions......Page 201 Representing a quaternion as a matrix......Page 203 The derivative of a vector......Page 207 The normal vector to a planar curve......Page 210 The normal vector to a surface......Page 212 Perspective transform......Page 218 Horizontally oblique projection plane......Page 219 Vertically oblique projection plane......Page 222 Arbitrary orientation of the projection plane......Page 224 Pseudo fish-eye projection......Page 228 Light sources......Page 230 Local reflection models......Page 232 Shading......Page 235 Bump mapping......Page 236 Close encounters of the first kind......Page 245 Close encounters of the second kind......Page 247 Appendix A......Page 250 Appendix B......Page 251 References......Page 255 Further Reading......Page 256 Index......Page 257