A complete overview of the geometry associated with computer graphics that provides everything a reader needs to understand the topic. Includes a summary hundreds of formulae used to solve 2D and 3D geometric problems; worked examples; proofs; mathematical strategies for solving geometric problems; a glossary of terms used in geometry.
Author(s): John Vince
Edition: 1st Edition.
Year: 2004
Language: English
Pages: 364
Contents......Page 9
Preface......Page 6
1 Geometry......Page 21
1.1.2 Angles......Page 24
1.1.3 Trigonometry......Page 25
1.2.1 Properties of circles......Page 29
1.2.2 Ellipses......Page 30
1.3.2 Similar triangles......Page 31
1.3.4 Theorem of Pythagoras......Page 32
1.3.7 Area of a triangle......Page 33
1.3.8 Inscribed and circumscribed circles......Page 34
1.3.10 Spherical trigonometry......Page 35
1.4 Quadrilaterals......Page 36
1.5.3 Area of a regular polygon......Page 39
1.6.2 Pyramids......Page 41
1.6.5 Spheres......Page 42
1.6.7 Platonic solids......Page 43
1.7.2 Cartesian coordinates in R[sup(3)]......Page 46
1.7.4 Cylindrical coordinates......Page 47
1.7.5 Spherical coordinates......Page 48
1.8.5 Algebraic notation for a vector......Page 49
1.8.11 Scalar (dot) product......Page 50
1.8.15 Scalar triple product......Page 51
1.8.18 Area of a triangle......Page 52
1.9.4 Quaternion multiplication......Page 53
1.9.8 Quaternion as a matrix......Page 54
1.10.4 Rotation about the origin in R[sup(2)]......Page 55
1.10.9 Reflection about the yaxis in R[sup(2)]......Page 56
1.10.14 The identity matrix in R[sup(2)]......Page 57
1.10.18 Rotation about the xaxis in R[sup(3)]......Page 58
1.10.23 Reflection about the zxplane in R[sup(3)]......Page 59
1.10.28 Translated change of axes in R[sup(3)]......Page 60
1.10.30 The identity matrix in R[sup(3)]......Page 61
1.11.4 Parametric form of the straight line equation......Page 62
1.11.6 Straight-line equation from two points......Page 63
1.11.7 Point of intersection of two straight lines......Page 64
1.11.9 Three points lie on a straight line......Page 65
1.11.11 Position and distance of a point on a line perpendicular to the origin......Page 66
1.11.13 Position of a point reflected in a line......Page 67
1.11.15 Line equidistant from two points......Page 68
1.11.16 Two-dimensional line segment......Page 69
1.12.2 Touching and intersecting circles......Page 71
1.13.2 Ellipse......Page 73
1.13.4 Hyperbola......Page 74
1.14.4 Three points lie on a straight line......Page 75
1.14.8 Shortest distance between two skew lines......Page 76
1.14.10 Normal to a line through a point......Page 77
1.15.3 Hessian normal form of the plane equation......Page 78
1.15.6 Plane equation from three points......Page 79
1.15.9 Intersection of two planes......Page 80
1.15.11 Angle between two planes......Page 81
1.15.14 Position and distance of the nearest point on a plane to a point......Page 82
1.15.17 Reflected ray on a surface......Page 83
1.16.3 Touching spheres......Page 84
1.17.2 Unknown coordinate value inside a triangle......Page 86
1.18.3 Planar patch......Page 87
1.18.7 Quadratic Bézier patch......Page 88
1.18.8 Cubic Bézier patch......Page 89
1.19 Second degree surfaces in standard form......Page 90
2 Examples......Page 92
2.1 Trigonometry......Page 94
2.2 Circles......Page 97
2.3.2 Checking for congruent triangles......Page 98
2.3.3 Solving the angles and sides of a triangle......Page 99
2.3.4 Calculating the area of a triangle......Page 100
2.3.5 The center and radius of the inscribed and circumscribed circles for a triangle......Page 101
2.4 Quadrilaterals......Page 103
2.5 Polygons......Page 105
2.6.2 Conical frustum, spherical segment and torus......Page 107
2.6.3 Tetrahedron......Page 108
2.7.3 Polar coordinates......Page 109
2.7.4 Cylindrical coordinates......Page 110
2.7.5 Spherical coordinates......Page 111
2.8.6 Vector addition/subtraction......Page 113
2.8.10 Vector (cross) product......Page 114
2.8.13 Area of a triangle......Page 115
2.9.5 Rotating a vector......Page 116
2.9.6 Quaternion as a matrix......Page 117
2.10.3 Translation in R[sup(2)]......Page 118
2.10.6 Shearing along the x-axis in R[sup(2)]......Page 119
2.10.9 Reflection about the yaxis in R[sup(2)]......Page 120
2.10.12 Translated change of axes in R[sup(2)]......Page 121
2.10.15 Scaling relative to the origin in R[sup(3)]......Page 122
2.10.18 Rotation about the xaxis in R[sup(3)]......Page 123
2.10.21 Rotation about an arbitrary axis in R[sup(3)]......Page 124
2.10.24 Reflection about the xyplane in R[sup(3)]......Page 125
2.10.27 Reflection about a plane parallel with the xyplane in R[sup(3)]......Page 126
2.10.30 The identity matrix in R[sup(3)]......Page 127
2.11.2 Derive the unit normal vector and perpendicular from the origin to the line for the line equation......Page 128
2.11.3 Derive the straightline equation from two points......Page 129
2.11.4 Point of intersection of two straight lines......Page 130
2.11.5 Calculate the angle between two straight lines......Page 131
2.11.6 Test if three points lie on a straight line......Page 132
2.11.7 Test for parallel and perpendicular lines......Page 133
2.11.8 Find the position and distance of the nearest point on a line to the origin......Page 134
2.11.9 Find the position and distance of the nearest point on a line to a point......Page 135
2.11.10 Find the reflection of a point in a line passing through the origin......Page 136
2.11.11 Find the reflection of a point in a line......Page 137
2.11.12 Find the normal to a line through a point......Page 138
2.11.13 Find the line equidistant from two points......Page 139
2.11.15 Intersecting two line segments......Page 140
2.12.1 Line intersecting a circle......Page 142
2.12.2 Touching and intersecting circles......Page 145
2.13.3 Parabola......Page 147
2.13.4 Hyperbola......Page 148
2.14.2 Intersection of two straight lines......Page 149
2.14.4 Test if three points lie on a straight line......Page 150
2.14.7 Find the position and distance of the nearest point on a line to a point......Page 151
2.14.9 Find the normal to a line through a point......Page 152
2.14.10 Find the shortest distance between two skew lines......Page 153
2.15.3 Hessian normal form of the plane equation......Page 154
2.15.5 Converting a plane equation from parametric form to general form......Page 155
2.15.6 Plane equation from three points......Page 156
2.15.8 Plane through two points and parallel to a line......Page 157
2.15.9 Intersection of two planes......Page 158
2.15.10 Intersection of three planes......Page 160
2.15.12 Angle between a line and a plane......Page 162
2.15.14 Position and distance of the nearest point on a plane to a point......Page 163
2.15.16 Plane equidistant from two points......Page 164
2.15.17 Reflected ray on a surface......Page 165
2.16.1 Line intersecting a sphere......Page 167
2.16.2 Sphere touching a plane......Page 168
2.16.3 Touching spheres......Page 169
2.17.1 Coordinates of a point inside a triangle......Page 170
2.17.2 Unknown coordinate value inside a triangle......Page 171
2.18.1 Parametric curves in R[sup(2)]......Page 173
2.18.2 Parametric curves in R[sup(3)]......Page 177
2.18.3 Planar patch......Page 181
2.18.4 Parametric surfaces in R[sup(3)]......Page 182
2.18.6 Cubic Bézier curve......Page 184
2.18.7 Quadratic Bézier patch......Page 185
2.18.8 Cubic Bézier patch......Page 186
2.19 Second degree surfaces in standard form......Page 187
3 Proofs......Page 188
3.1.3 Pythagorean identities......Page 190
3.1.4 Useful trigonometric values......Page 191
3.1.5 Compound angle identities......Page 192
3.1.7 Multipleangle identities......Page 194
3.1.8 Functions of the halfangle......Page 195
3.1.9 Functions of the halfangle using the perimeter of a triangle......Page 196
3.1.10 Functions converting to the halfangle tangent form......Page 197
3.1.11 Relationships between sums of functions......Page 199
3.1.12 Inverse trigonometric functions......Page 201
3.2.2 Alternate segment theorem......Page 202
3.2.3 Area of a circle, sector and segment......Page 203
3.2.6 Secanttangent theorem......Page 205
3.2.7 Area of an ellipse......Page 206
3.3.2 Properties of triangles......Page 208
3.3.3 Altitude theorem......Page 211
3.3.4 Area of a triangle......Page 212
3.3.6 The medians of a triangle are concurrent at its centroid......Page 215
3.3.7 Radius and center of the inscribed circle for a triangle......Page 217
3.3.8 Radius and center of the circumscribed circle for a triangle......Page 220
3.4.1 Properties of quadrilaterals......Page 226
3.4.3 The diagonals of a parallelogram bisect each other......Page 229
3.4.4 The diagonals of a square are equal, intersect at right angles and bisect the opposite angles......Page 230
3.4.6 Area of a quadrilateral......Page 231
3.4.7 Area of a general quadrilateral using Heron’s formula......Page 233
3.4.8 Area of a trapezoid......Page 235
3.4.9 Radius and center of the circumscribed circle for a rectangle......Page 236
3.5.2 The external angles of a polygon......Page 237
3.5.3 Alternate internal angles of a cyclic polygon......Page 238
3.5.4 Area of a regular polygon......Page 239
3.5.5 Area of a polygon......Page 240
3.5.6 Properties of regular polygons......Page 241
3.6.1 Volume of a prism......Page 243
3.6.2 Surface area of a rectangular pyramid......Page 244
3.6.3 Volume of a rectangular pyramid......Page 245
3.6.5 Volume of a triangular pyramid......Page 246
3.6.7 Surface area of a right conical frustum......Page 247
3.6.8 Volume of a cone......Page 248
3.6.10 Surface area of a sphere......Page 249
3.6.11 Volume of a sphere......Page 250
3.6.13 Radii of the spheres associated with the Platonic solids......Page 252
3.6.14 Inner and outer radii for the Platonic solids......Page 257
3.6.15 Dihedral angles for the Platonic solids......Page 261
3.6.16 Surface area and volume of the Platonic solids......Page 265
3.7.2 Polar coordinates......Page 268
3.7.4 Spherical coordinates......Page 269
3.8.3 Scalar (dot) product......Page 271
3.8.6 Angle between two vectors......Page 272
3.8.8 The non-commutative law of the vector product......Page 273
3.8.10 Scalar triple product......Page 274
3.9.1 Definition of a quaternion......Page 275
3.10.1 Scaling in R[sup(2)]......Page 279
3.10.3 Rotation in R[sup(2)]......Page 280
3.10.4 Shearing in R[sup(2)]......Page 281
3.10.5 Reflection in R[sup(2)]......Page 282
3.10.6 Change of axes in R[sup(2)]......Page 283
3.10.8 Scaling in R[sup(3)]......Page 284
3.10.10 Rotation in R[sup(3)]......Page 285
3.10.11 Reflection in R[sup(3)]......Page 287
3.10.12 Change of axes in R[sup(3)]......Page 289
3.10.13 Identity matrix in R[sup(3)]......Page 290
3.11.1 Cartesian form of the line equation......Page 291
3.11.3 Equation of a line from two points......Page 292
3.11.4 Point of intersection of two straight lines......Page 294
3.11.5 Angle between two straight lines......Page 295
3.11.6 Three points lie on a straight line......Page 296
3.11.7 Parallel and perpendicular straight lines......Page 297
3.11.9 Position and distance of a point on a line perpendicular to the origin......Page 298
3.11.10 Position and distance of the nearest point on a line to a point......Page 299
3.11.11 Position of a point reflected in a line......Page 300
3.11.12 Normal to a line through a point......Page 302
3.11.13 Line equidistant from two points......Page 303
3.11.14 Equation of two-dimensional line segment......Page 304
3.11.15 Point of intersection of two two-dimensional line segments......Page 305
3.12.1 Line and a circle......Page 307
3.12.2 Touching and intersecting circles......Page 309
3.13.2 Ellipse......Page 312
3.13.3 Parabola......Page 314
3.13.4 Hyperbola......Page 315
3.14.2 Intersection of two straight lines......Page 316
3.14.5 Parallel and perpendicular straight lines......Page 317
3.14.7 Position and distance of the nearest point on a line to a point......Page 318
3.14.8 Position of a point reflected in a line......Page 319
3.14.9 Normal to a line through a point......Page 320
3.14.10 Shortest distance between two skew lines......Page 321
3.15.1 Equation to a plane......Page 322
3.15.2 Plane equation from three points......Page 325
3.15.5 Intersection of two planes......Page 327
3.15.6 Intersection of three planes......Page 329
3.15.9 Intersection of a line and a plane......Page 330
3.15.10 Position and distance of the nearest point on a plane to a point......Page 331
3.15.12 Plane equidistant from two points......Page 332
3.15.13 Reflected ray on a surface......Page 333
3.16.1 Line intersecting a sphere......Page 334
3.16.3 Touching spheres......Page 335
3.17.2 Unknown coordinate value inside a triangle......Page 337
3.18.2 Bézier curves in R[sup(2)] and R[sup(3)]......Page 338
3.18.3 Bézier surface patch in R[sup(3)]......Page 340
A......Page 343
D......Page 344
H......Page 346
O......Page 347
R......Page 348
T......Page 349
Z......Page 350
5 Bibliography......Page 351
A......Page 353
C......Page 354
H......Page 355
O......Page 356
Q......Page 357
S......Page 358
T......Page 359
V......Page 360
