Geometric algebra for computer graphics

Title:

Personal Author:

Vince, John (John A.)

Publication Information:

London : Springer-Verlag, 2008

Physical Description:

xvi, 252 p. : ill. ; 25 cm.

ISBN:

9781846289965

Subject Term:

Computer graphics -- Mathematics

Clifford Algebras

Geometry -- Data processing

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010177504	T385 V562 2008	Open Access Book	Book	Searching... Unknown

Geometric algebra (a Clifford Algebra) has been applied to different branches of physics for a long time but is now being adopted by the computer graphics community and is providing exciting new ways of solving 3D geometric problems.

John Vince (author of numerous books including 'Geometry for Computer Graphics' and 'Vector Analysis for Computer Graphics') has tackled this complex subject in his usual inimitable style, and provided an accessible and very readable introduction.

As well as putting geometric algebra into its historical context, John tackles complex numbers and quaternions; the nature of wedge product and geometric product; reflections and rotations (showing how geometric algebra can offer a powerful way of describing orientations of objects and virtual cameras); and how to implement lines, planes, volumes and intersections. Introductory chapters also look at algebraic axioms, vector algebra and geometric conventions and the book closes with a chapter on how the algebra is applied to computer graphics.

Preface	p. vii
1 Introduction	p. 1
1.1 Aims and objectives of this book	p. 1
1.2 Mathematics for CGI software	p. 1
1.3 The book's structure	p. 2
2 Elementary Algebra	p. 5
2.1 Introduction	p. 5
2.2 Numbers, variables and arithmetic operators	p. 6
2.3 Closure	p. 6
2.4 Identity element	p. 6
2.5 Inverse element	p. 7
2.6 The associative law	p. 8
2.7 The commutative law	p. 8
2.8 The distributive law	p. 8
2.9 Summary	p. 9
3 Complex Algebra	p. 11
3.1 Introduction	p. 11
3.2 Complex numbers	p. 11
3.3 Complex arithmetic	p. 12
3.4 The complex plane	p. 16
3.5 i as a rotor	p. 17
3.6 The product of two complex numbers	p. 18
3.7 Powers of complex numbers	p. 19
3.8 e, i, sin and cos	p. 19
3.9 Logarithm of a complex number	p. 21
3.10 Summary	p. 22
4 Vector Algebra	p. 23
4.1 Introduction	p. 23
4.2 Vector quantities and their graphical representation	p. 24
4.3 Vector spaces	p. 25
4.4 Linear combinations	p. 27
4.5 Spanning sets	p. 28
4.6 Linear independence and dependence	p. 29
4.7 Standard bases	p. 31
4.8 Orthogonal bases	p. 31
4.9 Dimension	p. 32
4.10 Subspaces	p. 32
4.11 Scalar product	p. 33
4.12 Vector product	p. 35
4.13 Summary	p. 37
5 Quaternion Algebra	p. 39
5.1 Introduction	p. 39
5.2 Adding quaternions	p. 41
5.3 The quaternion product	p. 42
5.4 The magnitude of a quaternion	p. 43
5.5 The unit quaternion	p. 43
5.6 The pure quaternion	p. 43
5.7 The conjugate of a quaternion	p. 44
5.8 The inverse quaternion	p. 44
5.9 Quaternion algebra	p. 45
5.10 Rotating vectors using quaternions	p. 46
5.11 Summary	p. 48
6 Geometric Conventions	p. 49
6.1 Introduction	p. 49
6.2 Clockwise and anticlockwise	p. 49
6.3 Left and right-handed axial systems	p. 52
6.4 Summary	p. 54
7 Geometric Algebra	p. 55
7.1 Introduction	p. 55
7.2 Foundations of geometric algebra	p. 56
7.3 Introduction to geometric algebra	p. 56
7.3.1 Length, area and volume	p. 56
7.4 The outer product	p. 58
7.4.1 Some algebraic properties	p. 59
7.4.2 Visualizing the outer product	p. 59
7.4.3 Orthogonal bases	p. 60
7.5 The outer product in action	p. 69
7.5.1 Area of a triangle	p. 70
7.5.2 The sine rule	p. 72
7.5.3 Intersection of two lines	p. 73
7.6 Summary	p. 77
8 The Geometric Product	p. 79
8.1 Introduction	p. 79
8.2 Clifford's definition of the geometric product	p. 80
8.2.1 Orthogonal vectors	p. 81
8.2.2 Linearly dependent vectors	p. 81
8.2.3 Linearly independent vectors	p. 82
8.2.4 The product of identical basis vectors	p. 84
8.2.5 The product of orthogonal basis vectors	p. 84
8.2.6 The imaginary properties of the outer product	p. 85
8.3 The unit bivector pseudoscalar	p. 85
8.3.1 The rotational properties of the pseudoscalar	p. 85
8.4 Summary of the products	p. 87
8.5 Multivectors in R[superscript 2]	p. 87
8.6 The relationship between bivectors, complex numbers and vectors	p. 89
8.7 Reversion	p. 90
8.8 Rotations in R[superscript 2]	p. 90
8.9 The vector-bivector product in R[superscript 2]	p. 91
8.10 Volumes and the trivector	p. 92
8.11 The unit trivector pseudoscalar	p. 94
8.12 The product of the unit basis vectors in R[superscript 3]	p. 96
8.12.1 The product of identical basis vectors	p. 96
8.12.2 The product of orthogonal basis vectors	p. 96
8.12.3 The imaginary properties of the unit bivectors	p. 97
8.13 The vector-unit bivector product in R[superscript 3]	p. 97
8.14 The vector-bivector product in R[superscript 3]	p. 100
8.15 Unit bivector-bivector products in R[superscript 3]	p. 106
8.16 Unit vector-trivector product in R[superscript 3]	p. 106
8.17 Unit bivector-trivector product in R[superscript 3]	p. 107
8.18 Unit trivector-trivector product in R[superscript 3]	p. 107
8.19 Higher products in R[superscript 3]	p. 108
8.20 Blades	p. 108
8.21 Duality transformation	p. 109
8.22 Summary of products in R[superscript 3]	p. 110
8.23 Multivectors in R[superscript 3]	p. 111
8.24 Relationship between vector algebra and geometric algebra	p. 113
8.25 Relationship between the outer product and the cross product	p. 114
8.26 Relationship between geometric algebra and quaternions	p. 116
8.27 Inverse of a vector	p. 118
8.28 The meet operation	p. 120
8.29 Summary	p. 122
9 Reflections and Rotations	p. 125
9.1 Introduction	p. 125
9.2 Reflections	p. 127
9.2.1 Reflecting vectors	p. 127
9.2.2 Reflecting bivectors	p. 129
9.2.3 Reflecting trivectors	p. 132
9.3 Rotations	p. 133
9.3.1 Rotating by double reflecting	p. 133
9.3.2 Rotors	p. 138
9.3.3 Rotor matrix	p. 141
9.3.4 Building rotors	p. 146
9.3.5 Interpolating rotors	p. 150
9.4 Summary	p. 153
10 Geometric Algebra and Geometry	p. 155
10.1 Introduction	p. 155
10.2 Point inside a triangle	p. 155
10.2.1 Point inside a 2D triangle	p. 155
10.2.2 Point inside a 3D triangle	p. 158
10.3 The relationship between bivectors and direction cosines	p. 162
10.4 Lines and planes	p. 166
10.4.1 Relative orientation of a point and a line	p. 166
10.4.2 Relative orientation of a point and a plane	p. 169
10.4.3 Shortest distance from a point to a plane	p. 171
10.4.4 A line intersecting a plane	p. 175
10.5 Perspective projection	p. 179
10.6 Back-face removal	p. 183
10.7 Homogeneous coordinates	p. 184
10.7.1 Introduction	p. 184
10.7.2 Representing 2D lines in 3D homogeneous space	p. 186
10.7.3 Intersection of two lines in R[superscript 2]	p. 192
10.7.4 Representing 3D lines in 4D homogeneous space	p. 195
10.7.5 Representing lines and planes in 4D homogeneous space	p. 196
10.8 Summary	p. 197
11 Conformal Geometry	p. 199
11.1 Introduction	p. 199
11.1.1 Spatial dimension	p. 199
11.1.2 Algebraic underpinning	p. 199
11.1.3 Mathematical language and notation	p. 200
11.1.4 Protection	p. 200
11.2 Stereographic projection	p. 201
11.3 Signatures and null vectors	p. 204
11.4 The basis blades for the conformal model	p. 209
11.5 Representing geometric objects	p. 210
11.5.1 Points	p. 210
11.5.2 Point pair	p. 211
11.5.3 Lines	p. 212
11.5.4 Circles	p. 213
11.5.5 Planes	p. 216
11.5.6 Spheres	p. 217
11.6 Conformal transformations	p. 218
11.6.1 Translations	p. 218
11.6.2 Rotations	p. 221
11.6.3 Dilations	p. 224
11.6.4 Reflections	p. 226
11.6.5 Intersections	p. 230
11.7 Summary	p. 230
12 Applications of Geometric Algebra	p. 231
12.1 Introduction	p. 231
12.2 3D Linear transformations	p. 231
12.2.1 Scale transform	p. 234
12.2.2 Refraction transform	p. 235
12.3 Rigid-body pose control	p. 238
12.4 Ray tracing	p. 240
12.5 Summary	p. 240
13 Programming Tools for Geometric Algebra	p. 241
13.1 Introduction	p. 241
13.2 Programming implications	p. 241
13.3 Programming tools	p. 242
13.4 Summary	p. 242
14 Conclusion	p. 243
References	p. 245
Index	p. 249

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