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E-grāmata: Geometric Algebra: An Algebraic System for Computer Games and Animation

4.00/5 (2 ratings by Goodreads)
  • Formāts: PDF+DRM
  • Izdošanas datums: 20-May-2009
  • Izdevniecība: Springer London Ltd
  • Valoda: eng
  • ISBN-13: 9781848823792
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Geometric Algebra: An Algebraic System for Computer Games and AnimationPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 20-May-2009
  • Izdevniecība: Springer London Ltd
  • Valoda: eng
  • ISBN-13: 9781848823792
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Geometric algebra is still treated as an obscure branch of algebra and most books have been written by competent mathematicians in a very abstract style. This restricts the readership of such books especially by programmers working in computer graphics, who simply want guidance on algorithm design.



Geometric algebra provides a unified algebraic system for solving a wide variety of geometric problems. John Vince reveals the beauty of this algebraic framework and communicates to the reader new and unusual mathematical concepts using colour illustrations, tabulations, and easy-to-follow algebraic proofs.



The book includes many worked examples to show how the algebra works in practice and is essential reading for anyone involved in designing 3D geometric algorithms.

Recenzijas

From the reviews:

Geometric algebra (GA), a truly fascinating area of mathematics, provides a powerful, unified language of exceptional clarity and generality to describe one-, two-, three-, and higher-dimensional geometries. This books outstanding feature is the use of tables and colors to develop some arithmetical details. The book is better suited for self-study than for the classroom. I recommend it for upper-level undergraduates, graduate students, teachers, researchers, and technical libraries. (Edgar R. Chavez, ACM Computing Reviews, December, 2009)

In the current volume, the author simplifies the presentation based on some of his new ideas on the subject. The volume is self-contained and can be used by students and computer graphics professionals. a good course in linear algebra and some mathematical maturity. Summing Up: Recommended. Computer graphics, computer animation, and computer games collections for upper-division undergraduates, graduate students, and professionals. (D. Z. Spicer, Choice, Vol. 47 (7), March, 2010)

Geometric algebra is a topic of current interest in mathematical research and in applications in physics, engineering, and computer science . the book is directed to a computer programming audience. this accessible, introductory book may convince some computer graphics programmers of the usefulness of geometric algebra . (Adam Coffman, Mathematical Reviews, Issue 2011 i)

The books true value lies in describing important geometric transformations like reflection and rotation in a systematic way, and in listing many geometric primitives . For people working in computer graphics or in game design, these topics could be of considerable value, and they certainly justify the books title. (Rolf Klein, Zentralblatt MATH, Vol. 1226, 2012)

Preface vii
Symbols and notation xvii
Introduction
1(4)
Sense and nonsense
1(1)
Geometric algebra
2(3)
Products
5(8)
Introduction
5(1)
Real products
5(2)
Complex products
7(1)
Quaternion products
8(3)
Summary
11(2)
Vector Products
13(20)
Introduction
13(1)
The scalar product
13(1)
The vector product
14(2)
Dyadics
16(4)
The outer product
20(12)
Origins of the outer product
20(1)
The geometric meaning of the outer product in 2D
21(4)
The geometric meaning of the outer product in 3D
25(7)
Summary
32(1)
The Geometric Product
33(16)
Introduction
33(1)
Axioms
33(4)
Redefining the inner and outer products
37(4)
Blades
41(1)
The geometric product of different vectors
42(5)
Orthogonal vectors
43(1)
Parallel vectors
44(1)
Linearly independent vectors
44(3)
Summary
47(2)
Geometric Algebra
49(16)
Introduction
49(1)
Grades and pseudoscalars
49(1)
Multivectors
50(3)
Reversion
53(1)
The inverse of a multivector
54(4)
The imaginary properties of the outer product
58(1)
The rotational properties of the 2D unit bivector
59(1)
The imaginary properties of the 3D unit bivector and the trivector
60(1)
Duality
61(3)
Summary
64(1)
Products in 2D
65(10)
Introduction
65(1)
The scalar-vector product
66(1)
The scalar-bivector product
67(1)
The vector-vector products
67(4)
The inner product
67(2)
The outer product
69(1)
The geometric product
70(1)
The vector-bivector product
71(1)
The bivector-bivector product
72(1)
Summary
73(2)
Products in 3D
75(28)
Introduction
75(1)
The scalar-vector product
75(1)
The scalar-bivector product
76(1)
The scalar-trivector product
77(1)
The vector-vector products
77(4)
The inner product
77(2)
The outer product
79(1)
The geometric product
80(1)
The vector-bivector product
81(8)
The dyadic approach
81(3)
The geometric approach
84(5)
The vector-trivector products
89(1)
The inner product
89(1)
The outer product
89(1)
The geometric product
89(1)
The bivector-bivector products
90(3)
The inner product
91(1)
The outer products
91(1)
The geometric products
92(1)
The bivector-trivector products
93(1)
The inner product
93(1)
The trivector-trivector product
94(1)
Product summary
94(3)
Geometric algebra and the vector product
97(1)
The relationship between the vector product and outer product
97(1)
The relationship between quaternions and bivectors
98(1)
The meet operation
99(2)
Summary
101(2)
Reflections and Rotations
103(36)
Introduction
103(1)
Reflections using normal vectors
103(5)
Reflecting a vector in a mirror
108(1)
Reflecting a bivector in a mirror
109(2)
Reflecting a trivector in a mirror
111(1)
Reflecting scalars
112(1)
Rotations by reflections
112(5)
Rotors
117(4)
Rotating bivectors
121(2)
Rotating trivectors
123(1)
Rotating scalars
124(1)
Rotors in exponential form
124(1)
A rotor matrix
125(3)
Building rotors
128(7)
Interpolating rotors
135(3)
Summary
138(1)
Applied Geometric Algebra
139(44)
Introduction
139(1)
Sine rule
139(1)
Cosine rule
140(1)
Inscribed circle of a triangle
141(2)
Circumscribed circle of a triangle
143(4)
A point perpendicular to a point on a line
147(1)
A point rotated an angle relative to a point on a line
148(2)
Reflecting a vector about another vector
150(2)
The position and distance of the nearest point on a line to a point
152(3)
A line equidistant from two points
155(1)
Intersection of two 2D lines
156(4)
Intersection of two 2D lines using homogeneous coordinates
160(2)
Orientation of a point and a 2D line
162(1)
Intersection of a line and a circle
163(2)
Orientation of a point with a plane
165(1)
Plane equation
166(3)
Orientation of a point with a convex object
169(3)
Angle between a vector and a bivector
172(1)
Angle between two bivectors
173(2)
Intersection of a line and a plane
175(3)
Intersection of a line and a sphere
178(2)
Ray intersection and reflection
180(2)
Summary
182(1)
Conclusion
183(2)
Appendix A 185(2)
Appendix B 187(4)
Bibliography 191(2)
Index 193
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