|
|
| |
|
| |
|
 |
|
| |
John Vince
Geometric Algebra for Computer Graphics
erschienen April 2008
256 Seiten, 125 schw.-w. Abb., 125 schw.-w. Zeichn., 24 schw.-w. Tabellen, Gebunden
Springer-Verlag GmbH | ISBN: 1846289963
| |  | 82.34 EUR |  | | |
|
|
|
|
| |
Innerhalb 24 Stunden versandfertig. Expressversand: In Deutschland versandkostenfrei | Österreich: 4 € | Schweiz: ab 4 € | Europaweit ab 6 €. Versandkostenübersicht weltweit. Alle Preise inkl. MwSt. |
|
|
Ähnliche Bücher anzeigen
|
|
|
| |
| |
| VORWORT | öffnen |
|
PrefaceIn December 2006 I posted my manuscript Vector Analysis for Computer Graphics to Springer and looked forward to a short rest before embarking upon another book. But whilst surfing the Internet, and probably before my manuscript had reached its destination, I discovered a strange topic called geometric algebra. Advocates of geometric algebra (GA) were claiming that a revolution was coming and that the cross product was dead. I couldn't believe my eyes. I had just written a book about vecto...
[weiter lesen]
|
|
|
| KLAPPENTEXT | öffnen |
|
Geometric Algebra for Computer Graphics Since its invention, geometric algebra has been applied to various branches of physics such as cosmology and electrodynamics, and is now being embraced by the computer graphics community where it is providing new ways of solving geometric problems. It took over two thousand years to discover this algebra, which uses a simple and consistent notation to describe vectors and their products. John Vince (best-selling author of a number of books includi... [weiter lesen] |
|
|
| INHALTSVERZEICHNIS | öffnen |
Contents
Prefacevii
1 Introduction 1
1.1 Aims and objectives of this book 1
1.2 Mathematics for CGI software 1
1.3 The book's structure 2
2 Elementary Algebra 5
2.1 Introduction 5
2.2 Numbers, variables and arithmetic operators 6
2.3 Closure 6
2.4 Identity element 6
2.5 Inverse element 7
2.6 The associative law 8
2.7 The commutative law 8
2.8 The distributive law 8
2.9 Summary 9
3 Complex Algebra 11
3.1 Introduction 11
3.2 Complex numbers 11
3.3 Complex arithmetic 12
3.4 The complex plane 16
3.5 i as a rotor 17
3.6 The product of two complex numbers 18
3.7 Powers of complex numbers 19
3.8 e, i, sin and cos 19
3.9 Logarithm of a complex number 21
3.10 Summary 22
4 Vector Algebra 23
4.1 Introduction 23
4.2 Vector quantities and their graphical representation 24
4.3 Vector spaces 25
4.4 Linear combinations 27
4.5 Spanning sets 28
4.6 Linear independence and dependence 29
4.7 Standard bases 31
4.8 Orthogonal bases 31
4.9 Dimension 32
4.10 Subspaces 32
4.11 Scalar product 33
4.12 Vector product 35
4.13 Summary 37
5 Quaternion Algebra 39
5.1 Introduction 39
5.2 Adding quaternions 41
5.3 The quaternion product 42
5.4 The magnitude of a quaternion 43
5.5 The unit quaternion 43
5.6 The pure quaternion 43
5.7 The conjugate of a quaternion 44
5.8 The inverse quaternion 44
5.9 Quaternion algebra 45
5.10 Rotating vectors using quaternions 46
5.11 Summary 48
6 Geometric Conventions 49
6.1 Introduction 49
6.2 Clockwise and anticlockwise 49
6.3 Left and right-handed axial systems 52
6.4 Summary 54
7 Geometric Algebra 55
7.1 Introduction 55
7.2 Foundations of geometric algebra 56
7.3 Introduction to geometric algebra 56
7.3.1 Length, area and volume 56
7.4 The outer product 58
7.4.1 Some algebraic properties 59
7.4.2 Visualizing the outer product 59
7.4.3 Orthogonal bases 60
7.5 The outer product in action 69
7.5.1 Area of a triangle 70
7.5.2 The sine rule 72
7.5.3 Intersection of two lines 73
7.6 Summary 77
8 The Geometric Product 79
8.1 Introduction 79
8.2 Clifford's definition of the geometric product 80
[weiter lesen] |
|
|
|
|
| REGISTER | öffnen |
Index
Aaddition
- quaternions, 41
- vectors, 25
algebra
- complex numbers, 15
- geometric, 55
- origin, 5
- quaternion, 45
- vector, 23
anticlockwise, 49
area, 49
- signed, 65
- triangle, 70
Argand diagram, 16
associative law
- algebra, 8
- complex algebra, 15
- geometric algebra, 80
- quaternion algebra, 46
- vector algebra, 26
axial system
- change of, 163
- left-handed, 52
- right-handed, 52, 92
Bback-facing polygon, 183
basis blades for conformal model, 209
- orthogonal, 31, 60
- standard, 31
- vectors, 84
binary operators, 6
bivector, 58, 89
- inverse, 174
- products, 97, 106
- reflections, 129
blades, 108
CClifford, William, 56, 80
clockwise, 49
closure
- algebra, 6
- complex algebra, 15
- vector algebra, 26
- quaternion algebra, 42, 45
commutative law
- algebra, 8
- complex algebra, 15
- quaternion algebra, 46
- vector algebra, 27, 34, 36
complex number
- algebra, 11
- argument, 17
- arithmetic, 12
- as a rotator, 17
conjugate, 12
- graphical interpretation, 17
- logarithm, 21
- modulus, 16
- plane, 16
- power of, 19
- product, 18
conformal circles, 213
- dilations, 224
- geometry, 199
- intersections, 230
- lines, 212
- model, 199
- patent, 200
- planes, 216
- point pair, 211
- points, 210
- reflections, 226
- rotations, 221
- spheres, 217
- transformations, 218
- translation, 218
Cramer's rule, 192
cross product, 35, 114
cross-ratio, 201
Ddeterminant, 13
dilations, 224
dimension, 26, 32, 199
direction cosines, 162
distributive law
- algebra, 8
- complex algebra, 15
- geometric algebra, 80
- quaternion algebra, 46
- vector algebra, 27, 34, 36
dot product, 34
duality transform, 109, 187, 189
Eexponential, 19, 139, 219
GGAIGEN, 242
geometric algebra, 55, 79
- applications, 231
- programming implications, 241
- programming tools, 242
geometric conventions, 50
geometric product, 79
Gibbs, Josiah, 23
grade, 85, 103
- lowering, 105
- raising, 105
Grassmann, Hermann, 55
HHamilton, William, 23, 39, 55
Hestenes, David, 108, 200
homogeneous coordinates, 184, 193
Iidentity element algebra, 6
- complex algebra, 15
- quaternion algebra, 45
- vector algebra, 26
inside
- a 2 D triangle, 155
- a 3 D triangle, 158
interpolating linear, 239
- quadratic, 239
- rotors, 150
- scalars, 150
- vectors, 150
intersection/ intersections lines, 73, 230
- line and a plane, 197
inverse bivector, 174
- vector, 118
inverse element algebra, 7
- complex number, 15
- quaternion, 46
LLi, Hongbo, 200
line/ lines
- in 3 D homogeneous space, 186
- in 4 D homogeneous space, 186
- intersecting, 192
- intersecting a plane, 175
- parametric equation, 177
linear interpolation, 239
Mmagnitude vector, 24
meet operation, 120
Minkowski, Hermann, 204
Minkowski space, 204
Möbius, 23, 201
modulus, 16, 80, 83
multivector, 79, 83, 87, 111
- homogeneous, 113
Nnull vector, 204
Oouter product, 57, 65, 69, 114
Pparallelogram area, 13
perspective projection, 179
plane/ planes intersecting, 120
point/ points at infinity, 203
- betweeness, 201
- cross-ratio, 201
- inside a 2 D triangle, 155
- inside a 3 D triangle, 158
point on a line, 166
point on a plane, 170
- shortest distance, 171, 191
point pair, 211
product/ products antisymmetric, 63
- basis vectors, 84
- bivector, 91
- geometric, 79
- outer, 57
- scalar, 33, 41
- vector, 22, 35
- wedge, 50
programming tools, 242
projection perspective, 179
- stereographic, 201
pseudoscalar, 85, 94
- rotational properties, 85
Qquaternions addition, 41
- algebra, 45
- as rotators, 126
- complex conjugate, 44
- definition, 40
- geometric algebra, 116
- Hamilton's rules, 41
- interpolation, 150
- inverse, 44
- magnitude, 43
- matrix, 126
- product, 41
- pure, 43
- rotating a vector, 46
- subtracting, 41
- unit, 43
Rray tracing, 240
reflection/ reflections, 125, 127
- conformai, 226
- double, 133
- line off a plane, 229
- point in a plane, 229, 232
refraction, 235
reversion, 90
Riemann, Georg, 202
Riemann sphere, 202
rigid-body pose control, 238
Rockwood, Alyn, 200
rotating a vector, 46
rotations, 90, 125, 133
rotors, 138, 220
- building, 146
[weiter lesen] |
|
|
|
|
|
|