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.
Author(s): John Vince
Publisher: Springer
Year: 2007
Language: English
Pages: 259
City: London
Tags: Математика;Линейная алгебра и аналитическая геометрия;Линейная алгебра;Векторная алгебра;
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
