Dorst L., Fontijne D., Mann S. Geometric Algebra for Computer Science. An Object Oriented Approach to Geometry
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INDEX 625
containment in, 76
dual representation, 83, 190
empty, 45
graded algebra of, 44–50
Grassmann space of, 23
homogeneous, 24, 42–44, 134
ladder of, 45–46
in linear algebra, 10
linear transformations of, 99–123
metric products of, 65–98
offset, 136, 280, 288
as operators, 188–91
operators on, 141
oriented, spanning, 23–64
orthogonal projection, 83–86
projection as sandwiching, 190
projection to, 155–58
reflections by, 188–90
reflections of, 168–69
rotations of, 169–76
sizing up, 66–71
squared norm, 67–68
union of, 127
Support points, 281, 284
Support vectors, 281, 284, 291, 316, 317
Surrounds, 446–47
of dual round, 446
of round, 446
Symbolic matrices, 525, 526
Symmetry
geometric product for vectors, 143
subspace products from, 152–53
T
Tangent bundle, 406
Tangent(s)
2-blade, 404–5
circle, 223–224, 386
flat, 445
plane, 86, 411
properties, 407
space, 233, 236
vector, 405, 428, 451–54
vectors, rays as, 573–74
direct, 405, 461
dual, 406
of flats, 445
as intersections of touching rounds,
404–9
of rounds, 445
without differentiating, 445
Tilting a mirror application, 228–30
Transformational changes, 215–20
change of transformation, 220
commutator product, 215–17
rotor-induced, 217–19
transformation of change, 219–20
Transformations
affine, 299, 306–8, 334
of change, 219–20
change of, 220–21
concatenated, 191
Euclidean, 356, 364–70
of flats in homogeneous model,
335
loading into OpenGL, 348–49
as objects, 190–91
perturbations, 217
primitives, with OpenGL matrices,
349–51
projective, 299, 308–9
of rotors, 194
scene, 566–72
of standard blades (conformal), 477
nested, 191
See also Rotation(s), Reflection(s),
Scaling, Translation(s), Versors
Translate-rotate-scale (TRS) versor,
471–74, 488, 489, 491
Translation(s)
algebraic properties, 380–81
as versors, 380
camera, 570–71
covariance, 312
dual formula, 304
Euclidean k-blade, 372
Euclidean transformations, 365–66
of flats, 303
homogeneous model, 303–4
hyperbolic geometry, 481
interpolation, 266
on locations, 303–4
object, 568–69
optimal, 262, 263
rotations, 380–81
spherical geometry, 482–83
swapping law, 472
translation invariant, 380
Transversions, 475–77
closed-form solution, 477
defined, 475
Trigonometry
planar, 249-251
spherical, 251–54
Tr iangles
barycentric coordinates, 316
circumcenter of, 457, 459
line intersection, 298, 333
Pascal’s, 45
spherical, 179–80, 252–54
oriented area, 44, 251, 284, 316, 427
planar, 249–51, 284
Tr ifocal constraint, 346
Tr ifocal tensor
defined, 345
as line-parameterized homography,
346
Tr igonometric functions, 184–85, 531
Trivectors
addition of, 62
visualization of, 36
Two lines in a plane, 292–94
coincident lines, 294
finite intersection point, 293–94
parallel lines, 294
Two skew lines in space, 295–96
meet, 295
relative orientation, 298
U
Underscore constructors in Gaigen2,
54–55
Undualization, 81
Uniform scaling
as outermorphism example, 103
point-reflection into origin, 109
Union
of blades, 126, 127
directions, 248
encoding, 135
magnitude, ambiguity, 126
See also join
of subspaces, 127
Venn diagram, 536
Unit points, 274, 277
Unit quaternions, 181–82
Universality, 491
Universal orthogonal transformations,
12–14, 191–96
626 INDEX
V
Vector differentiation, 230–35
adjoint as derivative, 233–34
chain rule, 234–35
product rule, 234–35
framework, 230
of identity function, 232
of inner product, 232–33
of norm, 233
of outer product, 233
properties, 234–35
See also Differentiation
Vector fields
example, 60
singularities in, 60–62
Vec tor s
in angular relationships, 251, 252
basis, 89–90, 144–45
in conformal model, 451–55
decomposition, 156
defined, 24
dividing by, 145–46
duality, 81
free, 451
general, as dual planes and spheres,
361–64
geometric product for, 142–47
in homogeneous model, 10, 272–78
inverse, 79, 113–14, 145
line, 453
linear transformations of, 100–101
nonorthonormal, 94
normal, 86, 122–23, 453
null, 66, 359–61, 587
orthogonal projection of, 84
points as, 274–78
position, 451, 454
ratios, as operators, 146–47
reflecting in, 204
squared norm of subspace, 67–8
support, 281, 284, 291, 316, 317
tangent, 453–54
unit, ratio, 172
in vector space model, 10
visualizing, 26–27
with zero norms, 145
Vector space model, 8, 247–70
3-D rotor computations, 256–60
angular relationships, 248–56
estimation in, 260–63
planar reflection, 378
Vector spaces, 23–25
defined, 24
as modeling tools, 8–9, 245
See also Subspaces
Vector transformations, 114–15
Velocities, 86, 88, 276
Versor inverse method, 530
Versor product, 192–93
blades in, 195
grade-preserving, 194, 368
orthogonal transformations as,
193–95
sandwiching, 13, 190–92
structure preservation, 368
summary, 197
See also Products
Versors, 191–96
of composition of operators, 193
computation, 220
conversiontohomogeneous4x4
matrices, 486–93
defined, 195
Euclidean, 364–65
Euclidean transformations as, 364–70
even, 193
first-order perturbations, 386
inverse, 529–30
test, 532–33
translation, 365–66
logarithm of, 532
odd, 193
orthogonal transformation
generation, 191–96
spherical geometry, 482–83
structure-preserving properties,
194, 364
TRS, 491
Vertices, 561–62
storage, 561
with tangent 2-blades, 562
Visualization
bivectors, 31-34
conformal model, 410–417
contraction, 75–77
outerproduct,10
quaternions, 181–82
rotations, 179–80
trivectors, 236
vectors, 26–27
Volumes
attitude, 33
elements, 33–36
orientation, 35
of simplex, 44, 427
properties, 33–35
scalar weig ht, 35
Voronoi diagrams, 415–17
cell analysis, 455–60
defined, 415
determining, 416–17
programming example, 428–31
illustrated, 416
W
Weight
blades, 408–9
defined, 43
directions, 248
lines, 26
points, 38, 275
preservation, 369
volumes, 35
World points, 262–63