Visualizing Quaternions (eBook)

Front Cover 1
Visualizing Quaternions 8
Copyright Page 9
Contents 12
About the Author 11
Foreword 24
Preface 26
Acknowledgments 32
Part I: Elements of Quaternions 34
Chapter 1. The Discovery of Quaternions 38
1.1 Hamilton's Walk 38
1.2 Then Came Octonions 41
1.3 The Quaternion Revival 42
Chapter 2. Folklore of Rotations 46
2.1 The Belt Trick 47
2.2 The Rolling Ball 47
2.3 The Apollo 10 Gimbal-lock Incident 52
2.4 3D Game Developer's Nightmare 59
2.5 The Urban Legend of the Upside-down F16 60
2.6 Quaternions to the Rescue 62
Chapter 3. Basic Notation 64
3.1 Vectors 64
3.2 Length of a Vector 65
3.3 3D Dot Product 65
3.4 3D Cross Product 66
3.5 Unit Vectors 66
3.6 Spheres 66
3.7 Matrices 66
3.8 Complex Numbers 67
Chapter 4. What are Quaternions? 68
Chapter 5. Road Map to Quaternion Visualization 72
5.1 The Complex Number Connection 72
5.2 The Cornerstones of Quaternion Visualization 72
Chapter 6. Fundamentals of Rotations 76
6.1 2D Rotations 76
6.2 Quaternions and 3D Rotations 79
6.3 Recovering . and n 84
6.4 Euler Angles and Quaternions 85
6.5 † Optional Remarks 87
6.6 Conclusion 89
Chapter 7. Visualizing Algebraic Structure 90
7.1 Algebra of Complex Numbers 90
7.2 Quaternion Algebra 96
Chapter 8. Visualizing Spheres 102
8.1 2D: Visualizing an Edge-On Circle 103
8.2 The Square Root Method 107
8.3 3D: Visualizing a Balloon 109
8.4 4D: Visualizing Quaternion Geometry on S3 113
Chapter 9. Visualizing Logarithms and Exponentials 120
9.1 Complex Numbers 120
9.2 Quaternions 124
Chapter 10. Visualizing Interpolation Methods 126
10.1 Basics of Interpolation 126
10.2 Quaternion Interpolation 134
10.3 Equivalent 3 x 3 Matrix Method 137
Chapter 11. Looking at Elementary Quaternion Frames 138
11.1 A Single Quaternion Frame 138
11.2 Several Isolated Frames 139
11.3 A Rotating Frame Sequence 140
11.4 Synopsis 143
Chapter 12. Quaternions and the Belt Trick: Connecting to the Identity 144
12.1 Very Interesting, but Why? 146
12.2 The Details 147
12.3 Frame-sequence Visualization Methods 151
Chapter 13. Quaternions and the Rolling Ball: Exploiting Order Dependence 156
13.1 Order Dependence 156
13.2 The Rolling Ball Controller 158
13.3 Rolling Ball Quaternions 161
13.4 Commutators 163
13.5 Three degrees of freedom from two 164
Chapter 14. Quaternions and Gimbal Lock: Limiting the Available Space 166
14.1 Guidance System Suspension 166
14.2 Mathematical Interpolation Singularities 167
14.3 Quaternion Viewpoint 167
Part II: Advanced Quaternion Topics 170
Chapter 15. Alternative Ways of Writing Quaternions 174
15.1 Hamilton's Generalization of Complex Numbers 175
15.2 Pauli Matrices 176
15.3 Other Matrix Forms 177
Chapter 16. Efficiency and Complexity Issues 180
16.1 Extracting a Quaternion 181
16.2 Efficiency of Vector Operations 183
Chapter 17. Advanced Sphere Visualization 186
17.1 Projective Method 186
17.2 Distance-preserving Flattening Methods 189
Chapter 18. More on Logarithms and Exponentials 198
18.1 2D Rotations 198
18.2 3D Rotations 200
18.3 Using Logarithms for Quaternion Calculus 204
18.4 Quaternion Interpolations Versus Log 204
Chapter 19. Two-Dimensional Curves 206
19.1 Orientation Frames for 2D Space Curves 206
19.2 What Is a Map? 209
19.3 Tangent and Normal Maps 210
19.4 Square Root Form 212
Chapter 20. Three-Dimensional Curves 214
20.1 Introduction to 3D Space Curves 214
20.2 General Curve Framings in 3D 216
20.3 Tubing 219
20.4 Classical Frames 219
20.5 Mapping the Curvature and Torsion 227
20.6 Theory of Quaternion Frames 229
20.7 Assigning Smooth Quaternion Frames 235
20.8 Examples: Torus Knot and Helix Quaternion Frames 242
20.9 Comparison of Quaternion Frame Curve Lengths 243
Chapter 21. 3D Surfaces 246
21.1 Introduction to 3D Surfaces 246
21.2 Quaternion Weingarten Equations 251
21.3 Quaternion Gauss Map 254
21.4 Example: The Sphere 256
21.5 Examples: Minimal Surface Quaternion Maps 261
Chapter 22. Optimal Quaternion Frames 266
22.1 Background 266
22.2 Motivation 267
22.3 Methodology 269
22.4 The Space of Frames 272
22.5 Choosing Paths in Quaternion Space 281
22.6 Examples 284
Chapter 23. Quaternion Volumes 290
23.1 Three-degree-of-freedom Orientation Domains 292
23.2 Application to the Shoulder Joint 295
23.3 Data Acquisition and the Double-covering Problem 297
23.4 Application Data 301
Chapter 24. Quaternion Maps of Streamlines 304
24.1 Visualization Methods 304
24.2 3D Flow Data Visualizations 307
24.3 Brushing: Clusters and Inverse Clusters 308
24.4 Advanced Visualization Approaches 308
Chapter 25. Quaternion Interpolation 316
25.1 Concepts of Euclidean Linear Interpolation 317
25.2 The Double Quad 325
25.3 Direct Interpolation of 3D Rotations 327
25.4 Quaternion Splines 337
25.5 Quaternion de Casteljau Splines 341
25.6 Equivalent Anchor Points 348
25.7 Angular Velocity Control 352
25.8 Exponential-map Quaternion Interpolation 354
25.9 Global Minimal Acceleration Method 359
Chapter 26. Quaternion Rotator Dynamics 362
26.1 Static Frame 363
26.2 Torque 367
26.3 Quaternion Angular Momentum 368
Chapter 27. Concepts of the Rotation Group 372
27.1 Brief Introduction to Group Representations 372
27.2 Basic Properties of Spherical Harmonics 377
Chapter 28. Spherical Riemannian Geometry 384
28.1 Induced Metric on the Sphere 384
28.2 Induced Metrics of Spheres 386
28.3 Elements of Riemannian Geometry 395
28.4 Riemann Curvature of Spheres 396
28.5 Geodesics and Parallel Transport on the Sphere 399
28.6 Embedded-vector Viewpoint of the Geodesics 401
Part III: Beyond Quaternions 406
Chapter 29. The Relationship of 4D Rotations to Quaternions 410
29.1 What Happened in Three Dimensions 410
29.2 Quaternions and Four Dimensions 411
Chapter 30. Quaternions and the Four Division Algebras 414
30.1 Division Algebras 414
30.2 Relation to Fiber Bundles 419
30.3 Constructing the Hopf Fibrations 420
Chapter 31. Clifford Algebras 426
31.1 Introduction to Clifford Algebras 427
31.2 Foundations 428
31.3 Examples of Clifford Algebras 435
31.4 Higher Dimensions 441
31.5 Pin(N), Spin(N), O(N), SO(N), and all that 443
Chapter 32. Conclusions 446
Appendices 448
A. Notation 452
A.1 Vectors 452
A.2 Length of a Vector 453
A.3 Unit Vectors 454
A.4 Polar Coordinates 454
A.5 Spheres 455
A.6 Matrix Transformations 455
A.7 Features of Square Matrices 456
A.8 Orthogonal Matrices 457
A.9 Vector Products 457
A.10 Complex Variables 459
B. 2D Complex Frames 462
C. 3D Quaternion Frames 466
C.1 Unit Norm 466
C.2 Multiplication Rule 466
C.3 Mapping to 3D rotations 468
C.4 Rotation Correspondence 470
C.5 Quaternion Exponential Form 470
D. Frame and Surface Evolution 472
D.1 Quaternion Frame Evolution 472
D.2 Quaternion Surface Evolution 474
E. Quaternion Survival Kit 476
F. Quaternion Methods 484
F.1 Quaternion Logarithms and Exponentials 484
F.2 The Quaternion Square Root Trick 485
F.3 The â.b formula simplified 486
F.4 Gram-Schmidt Spherical Interpolation 487
F.5 Direct Solution for Spherical Interpolation 488
F.6 Converting Linear Algebra to Quaternion Algebra 490
F.7 Useful Tensor Methods and Identities 490
G. Quaternion Path Optimization Using Surface Evolver 494
H. Quaternion Frame Integration 496
I. Hyperspherical Geometry 500
I.1 Definitions 500
I.2 Metric Properties 501
References 504
Index 520