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Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics [Pehme köide]

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(Okayama University, Japan)
  • Formaat: Paperback / softback, 208 pages, kõrgus x laius: 254x178 mm, kaal: 362 g
  • Ilmumisaeg: 30-Jun-2020
  • Kirjastus: CRC Press
  • ISBN-10: 0367575825
  • ISBN-13: 9780367575823
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Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and GraphicsSuurem pilt
  • Formaat: Paperback / softback, 208 pages, kõrgus x laius: 254x178 mm, kaal: 362 g
  • Ilmumisaeg: 30-Jun-2020
  • Kirjastus: CRC Press
  • ISBN-10: 0367575825
  • ISBN-13: 9780367575823
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Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision.





Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres.





With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today’s geometric algebra.

Arvustused

"Several software tools are available for executing geometric algebra, but the purpose of the book is to bring about a deeper insight and interest in the theory on which these tools are based." Zentralblatt MATH 1319

List of Figures
vii
List of Tables
xiii
Preface xv
Chapter 1 Introduction
1(4)
1.1 Purpose Of This Book
1(1)
1.2 Organization Of This Book
2(1)
1.3 Other Features
3(2)
Chapter 2 3D Euclidean Geometry
5(24)
2.1 Vectors
5(1)
2.2 Basis And Components
6(1)
2.3 Inner Product And Norm
7(2)
2.4 Vector Products
9(2)
2.5 Scalar Triple Product
11(3)
2.6 Projection, Rejection, And Reflection
14(2)
2.7 Rotation
16(1)
2.8 Planes
17(3)
2.9 Lines
20(3)
2.10 Planes And Lines
23(4)
2.10.1 Plane Through A Point And A Line
23(1)
2.10.2 Intersection Of A Plane And A Line
24(2)
2.10.3 Intersection Of Two Planes
26(1)
2.11 Supplemental Note
27(1)
2.12 Exercises
27(2)
Chapter 3 Oblique Coordinate Systems
29(14)
3.1 Reciprocal Basis
29(2)
3.2 Reciprocal Components
31(1)
3.3 Inner, Vector, And Scalar Triple Products
32(1)
3.4 Metric Tensor
33(1)
3.5 Reciprocity Of Expressions
34(2)
3.6 Coordinate Transformations
36(4)
3.7 Supplemental Note
40(1)
3.8 Exercises
40(3)
Chapter 4 Hamilton's Quaternion Algebra
43(12)
4.1 Quaternions
43(1)
4.2 Algebra Of Quaternions
44(1)
4.3 Conjugate, Norm, And Inverse
45(1)
4.4 Representation Of Rotation By Quaternion
46(6)
4.5 Supplemental Note
52(1)
4.6 Exercises
52(3)
Chapter 5 Grassmann's Outer Product Algebra
55(22)
5.1 Subspaces
55(4)
5.1.1 Lines
55(1)
5.1.2 Planes
56(1)
5.1.3 Spaces
57(2)
5.1.4 Origin
59(1)
5.2 Outer Product Algebra
59(2)
5.2.1 Axioms Of Outer Product
59(1)
5.2.2 Basis Expressions
60(1)
5.3 Contraction
61(5)
5.3.1 Contraction Of A Line
61(1)
5.3.2 Contraction Of A Plane
61(1)
5.3.3 Contraction Of A Space
62(2)
5.3.4 Summary Of Contraction
64(2)
5.4 Norm
66(2)
5.5 Duality
68(4)
5.5.1 Orthogonal Complements
68(2)
5.5.2 Basis Expression
70(2)
5.6 Direct And Dual Representations
72(1)
5.7 Supplemental Note
73(2)
5.8 Exercises
75(2)
Chapter 6 Geometric Product And Clifford Algebra
77(18)
6.1 Grassmann Algebra Of Multivectors
77(2)
6.2 Clifford Algebra
79(1)
6.3 Parity Of Multivectors
80(1)
6.4 Grassmann Algebra In The Clifford Algebra
81(1)
6.5 Properties Of The Geometric Product
82(4)
6.5.1 Geometric Product And Outer Product
82(1)
6.5.2 Inverse
83(3)
6.6 Projection, Rejection, And Reflection
86(1)
6.7 Rotation And Geometric Product
87(3)
6.7.1 Representation By Reflections
87(1)
6.7.2 Representation By Surface Element
88(1)
6.7.3 Exponential Expression Of Rotors
89(1)
6.8 Versors
90(2)
6.9 Supplemental Note
92(1)
6.10 Exercises
92(3)
Chapter 7 Homogeneous Space And Grassmann-Cayley Algebra
95(22)
7.1 Homogeneous Space
95(1)
7.2 Points At Infinity
96(2)
7.3 Plucker Coordinates Of Lines
98(3)
7.3.1 Representation Of A Line
99(1)
7.3.2 Equation Of A Line
99(1)
7.3.3 Computation Of A Line
100(1)
7.4 Plucker Coordinates Of Planes
101(3)
7.4.1 Representation Of A Plane
101(1)
7.4.2 Equation Of A Plane
102(1)
7.4.3 Computation Of A Plane
102(2)
7.5 Dual Representation
104(2)
7.5.1 Dual Representation Of Lines
105(1)
7.5.2 Dual Representation Of Planes
105(1)
7.5.3 Dual Representation Of Points
105(1)
7.6 Duality Theorem
106(6)
7.6.1 Dual Points, Dual Lines, And Dual Planes
106(1)
7.6.2 Join And Meet
107(1)
7.6.3 Join Of Two Points And Meet Of A Plane With A Line
108(1)
7.6.4 Join Of Two Points And Meet Of Two Planes
109(1)
7.6.5 Join Of Three Points And Meet Of Three Planes
109(3)
7.7 Supplemental Note
112(2)
7.8 Exercises
114(3)
Chapter 8 Conformal Space And Conformal Geometry: Geometric Algebra
117(28)
8.1 Conformal Space And Inner Product
117(2)
8.2 Representation Of Points, Planes, And Spheres
119(3)
8.2.1 Representation Of Points
119(1)
8.2.2 Representation Of Planes
120(1)
8.2.3 Representation Of Spheres
121(1)
8.3 Grassmann Algebra In Conformal Space
122(3)
8.3.1 Direct Representations Of Lines
122(1)
8.3.2 Direct Representation Of Planes
123(1)
8.3.3 Direct Representation Of Spheres
123(1)
8.3.4 Direct Representation Of Circles And Point Pairs
124(1)
8.4 Dual Representation
125(3)
8.4.1 Dual Representation For Planes
125(1)
8.4.2 Dual Representation For Lines
126(1)
8.4.3 Dual Representation Of Circles, Point Pairs, And Flat Points
127(1)
8.5 Clifford Algebra In The Conformal Space
128(4)
8.5.1 Inner, Outer, And Geometric Products
128(1)
8.5.2 Translator
129(2)
8.5.3 Rotor And Motor
131(1)
8.6 Conformal Geometry
132(9)
8.6.1 Conformal Transformations And Versors
132(1)
8.6.2 Reflectors
133(2)
8.6.3 Invertors
135(2)
8.6.4 Dilator
137(2)
8.6.5 Versors And Conformal Transformations
139(2)
8.7 Supplemental Note
141(1)
8.8 Exercises
142(3)
Chapter 9 Camera Imaging And Conformal Transformations
145(18)
9.1 Perspective Cameras
145(3)
9.2 Fisheye Lens Cameras
148(2)
9.3 Omnidirectional Cameras
150(2)
9.4 3D Analysis Of Omnidirectional Images
152(3)
9.5 Omnidirectional Cameras With Hyperbolic And Elliptic Mirrors
155(4)
9.6 Supplemental Note
159(1)
9.7 Exercises
159(4)
Answers 163(24)
Bibliography 187(2)
Index 189
Kenichi Kanatani is a professor emeritus at Okayama University. A fellow of IEICE and IEEE, Dr. Kanatani is the author of numerous books on computer vision and applied mathematics. He is also a board member of several journals and conferences.