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Geometric Algebra for Computer Science
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Geometric Algebra for Computer Science - An Object-Oriented Approach to Geometry (Morgan Kaufmann, 2009
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Morgan Kaufmann Publishers is an imprint of Elsevier
30 Corporate Drive, Suite 400, Burlington, MA 01803, USA
Copyright
c
2007 by Elsevier Inc. All rights reserved.
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product names appear in initial capital or all capital letters. Readers, however, should contact the
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permission of the publisher.
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Library of Congress Cataloging-in-Publication Data
Application submitted
ISBN: 978-0-12-374942-0
For information on all Morgan Kaufmann publications,
visit our Web site at www.mkp.com or www.books.elsevier.com
Printed in China
091011 5432
List of Figures
1.1 Example of the use of geometric algebra 2
1.2 Code to generate Figure 1.1 5
1.3 Example of the use of geometric algebra 6
1.4 The outer product and its interpretations 11
2.1 Spanning homogeneous subspaces in a 3-D vector space 25
2.2 Imagining vector addition 27
2.3 Bivector representations 32
2.4 Imagining bivector addition in 2-D space 33
2.5 Bivector addition in 3-D space 34
2.6 The associativity of the outer product 35
2.7 Solving linear equations with bivectors 40
2.8 Intersecting lines in the plane 41
2.9 Code for drawing bivectors 58
2.10 Drawing bivectors screenshot (Example 1) 59
2.11 The orientation of front- and back-facing polygons 59
2.12 A wire-frame torus with and without backface culling 60
2.13 The code that renders a model from its 2-D vertices (Exercise 2) 61
2.14 Sampling a vector field and summing trivectors 62
2.15 Code to test for singularity (Example 3) 63
2.16 A helix-shaped singularity, as detected by Example 3 64
3.1 Computing the scalar product of 2-blades 70
3.2 From scalar product to contraction 72
3.3 Thecontractionofavectorontoa2-blade 76
3.4 Duality of vectors in 2-D 81
xx
LIST OF FIGURES xxi
3.5 Duality of vectors and bivectors in 3-D 82
3.6 Projection onto a subspace 84
3.7 Three uses of the cross product 87
3.8 Duality and the cross product 89
3.9 Orthonormalization code (Example 1) 93
3.10 Orthonormalization 94
3.11 Reciprocal frame code 96
3.12 Color space conversion code (Example 4) 97
3.13 Color space conversion screenshot 98
4.1 The defining properties of a linear transformation 100
4.2 Projection onto a line a in the b-direction 104
4.3 A rotation around the origin of unit vectors in the plane 105
4.4 Projectionofavectorontoabivector 121
4.5 Matrix representation of projection code 122
4.6 Transforming normals vector 123
5.1 The ambiguity of the magnitude of the intersection of two planes 126
5.2 The
meet of two oriented planes 130
5.3 A line meeting a plane in the origin 131
5.4 When the
join of two (near-)parallel vectors becomes a 2-blade (Example 3) 140
6.1 Non-invertibility of the subspace products 142
6.2 Ratios of vectors 146
6.3 Projection and rejection of a vector 156
6.4 Reflecting a vector in a line 158
6.5 Gram-Schmidt orthogonalization 163
6.6 Gram-Schmidt orthogonalization code (Example 2) 164
7.1 Line and plane reflection 169
7.2 A rotation in a plane parallel to I is two reflections in vectors in that plane 170
7.3 A rotor in action 171
7.4 Sense of rotation 175
7.5 The unique rotor-based rotations in the range
= [0, 4π) 176
7.6 (a) A spherical triangle. (b) Composition of rotations through concatenation
of rotor arcs 180
7.7 Areflectorinaction 189
7.8 The rotor product in Euclidean spaces as a Taylor series 197
xxii LIST OF FIGURES
7.9 Interactive version of Figure 7.2 205
7.10 Rotation matrix to rotor conversion 207
7.11 2-D Julia fractal code 210
7.12 A 2-D Julia fractal, computed using the geometric product of real vectors 211
7.13 3-D Julia fractal 212
8.1 Directional differentiation of a vector inversion 227
8.2 Changes in reflection of a rotating mirror 229
8.3 The directional derivative of the spherical projection 241
10.1 A triang le a + b + c = 0 in a directed plane I 249
10.2 The angle between a vector and a bivector (see text) 252
10.3 A spherical triangle 253
10.4 Interpolation of rotations 259
10.5 Interpolation of rotations (Example 1) 266
10.6 Crystallography (Example 2) 267
10.7 External camera calibration (Example 3) 268
11.1 The extra dimension of the homogeneous representation space 274
11.2 Representing offset subspaces in
R
n+1
280
11.3 Defining offset subspaces fully in the base space 288
11.4 The dual hyperplane representation in
R
2
and R
1
290
11.5 The intersection of two offset lines L and M to produce a point 293
11.6 The
meet of two skew lines 295
11.7 The relative orientation of oriented flats 296
11.8 The combinations of four points taken in the cross ratio 300
11.9 The combinations of four lines taken in the cross ratio 301
11.10 Conics in the homogeneous model 308
11.11 Finding a line through a point, perpendicular to a given line 310
11.12 The orthogonal projection in the homogeneous model (see text) 315
11.13 The beginning of a row of equidistant telegraph poles 319
11.14 Example 2 in action 323
11.15 Perspective projection (Example 4) 325
12.1 Pl
¨
ucker coordinates of a line in 3-D 329
12.2 A pinhole camera 337
12.3 The epipolar constraint 342
12.4 The plane of rays generated by a line observation L 343
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