Quaternion multiplication can be used to rotate vectors in three-dimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is well-known in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this monograph are to provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points; to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane; to derive the formula for quaternion multiplication from first principles; to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; to show how to apply sandwiching to compute perspective projections. Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography
Author(s): Ron Goldman, Brian Barsky
Series: Synthesis Lectures on Computer Graphics and Animation
Publisher: Morgan and Claypool Publishers
Year: 2010
Language: English
Pages: 176
Rethinking Quaternions......Page 1
Synthesis Lectures on Computer Graphics and Animation......Page 3
Keywords......Page 8
Contents......Page 11
Preface......Page 13
part I Theory......Page 19
chapter 1 Complex Numbers......Page 21
chapter 2 A Brief History of Number Systems and Multiplication......Page 29
2.1 MULTIPLICATION IN DIMENSIONS GREATER THAN TWO......Page 32
3.1 MASS-POINTS: A CLASSICAL MODEL FOR CONTEMPORARY COMPUTER GRAPHICS......Page 35
3.2 ARROWS IN FOUR DIMENSIONS......Page 39
3.3 MUTUALLY ORTHOGONAL PLANES IN FOUR DIMENSIONS......Page 40
chapter 4 The Algebra of Quaternion Multiplication......Page 45
chapter 5 the Geometry of Quaternion Multiplication......Page 55
chapter 6 Affine, Semi-Affine, and Projective Transformations in Three Dimensions......Page 65
6.1 ROTATION......Page 67
6.2 MIRROR IMAGE......Page 72
6.3 PERSPECTIVE PROJECTION......Page 77
6.3.1 Perspective Projection and Singular 4 × 4 Matrices......Page 78
6.3.2 Perspective Projection by Sandwiching with Quaternions......Page 80
6.4 ROTORPERSPECTIVES AND ROTOREFLECTIONS......Page 90
chapter 7 Recapitulation: Insights and Results......Page 95
part II Computation......Page 99
8.1 MATRIX REPRESENTATIONS FOR QUATERNION MULTIPLICATION......Page 101
8.2 MATRIX REPRESENTATIONS FOR ROTATIONS......Page 103
8.3 MATRIX REPRESENTATIONS FOR MIRROR IMAGES......Page 106
8.4 MATRIX REPRESENTATIONS FOR PERSPECTIVE PROJECTIONS......Page 108
9.1 EFFICIENCY: QUATERNIONS VERSUS MATRICES......Page 113
9.2 AVOIDING DISTORTION BY RENORMALIZATION......Page 114
9.3 KEY FRAME ANIMATION AND SPHERICAL LINEAR INTERPOLATION......Page 115
chapter 10 Summary—Formulas From Quaternion Algebra......Page 119
Bookmark 3......Page 125
chapter 11 Goals and Motivation......Page 127
chapter 12 Clifford Algebras and Quaternions......Page 129
chapter 13 Clifford Algebra for the Plane......Page 131
14.1 SCALARS, VECTORS, BIVECTORS, AND PSEUDOSCALARS......Page 135
14.2 WEDGE PRODUCT AND CROSS PRODUCT......Page 136
14.3 DUALITY......Page 137
14.4 BIVECTORS......Page 139
14.5 QUATERNIONS......Page 140
15.1 ODD ORDER: MASS-POINTS......Page 143
15.2 EVEN ORDER: QUATERNIONS......Page 145
chapter 16 Decomposing Mass-Points Into Two Mutually Orthogonal Planes......Page 147
16.1 ACTION OF q (b, q ) ON b ⊥......Page 148
16.2 ACTION OF q (b, q ) ON b||......Page 149
16.3 SANDWICHING......Page 152
chapter 17 Rotation, Reflection, and Perspective Projection......Page 155
17.1 ROTATION......Page 156
17.2 MIRROR IMAGE......Page 157
17.3 PERSPECTIVE PROJECTION......Page 159
chapter 18 Summary......Page 163
chapter 19 Some SImple Alternative Homogenous Models for Computer Graphics......Page 167
References......Page 171
Further Reading......Page 173
Author Biography......Page 175
