Vector Analysis for Computer Graphics

1846288037......Page 1
Vector Analysis for Computer Graphics......Page 3
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