Visualizing quaternions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier [u.a.]
2006
|
| Series: | The Morgan Kaufmann series in interactive 3D technology
|
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014898137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XXXI, 498 S. Ill., graph. Darst. |
| ISBN: | 0120884003 9780120884001 |
Staff View
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| 100 | 1 | |a Hanson, Andrew J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Visualizing quaternions |c Andrew J. Hanson |
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier [u.a.] |c 2006 | |
| 300 | |a XXXI, 498 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a The Morgan Kaufmann series in interactive 3D technology | |
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| adam_text | Contents
ABOUT THE AUTHOR X
FOREWORD XXIII
PREFACE XXV
ACKNOWLEDGMENTS XXXI
PART I ELEMENTS OF QUATERNIONS 1
01 THE DISCOVERY OF QUATERNIONS 5
1.1 Hamilton s Walk 5
1.2 Then Came Octonions 8
1.3 The Quaternion Revival 9
02 FOLKLORE OF ROTATIONS 13
2.1 The Belt Trick 14
2.2 The Rolling Ball 14
2.3 The Apollo 10 Gimbal lock Incident 19
2.4 3D Game Developer s Nightmare 26
2.5 The Urban Legend of the Upside down Fl 6 27
2.6 Quaternions to the Rescue 29
03 BASIC NOTATION 31
3.1 Vectors 31
xi
Xjj CONTENTS
3.2 Length of a Vector 32
3.3 3D Dot Product 32
3.4 3D Cross Product 33
3.5 Unit Vectors 33
3.6 Spheres 33
3.7 Matrices 33
3.8 Complex Numbers 34
04 WHAT ARE QUATERNIONS? 35
05 ROAD MAP TO QUATERNION
VISUALIZATION 39
5.1 The Complex Number Connection 39
5.2 The Cornerstones of Quaternion Visualization 39
06 FUNDAMENTALS OF ROTATIONS 43
6.1 2D Rotations 43
6.1.1 Relation to Complex Numbers 44
6.1.2 The Half angle Form 44
6.1.3 Complex Exponential Version 45
6.2 Quaternions and 3D Rotations 46
6.2.1 Construction 46
6.2.2 Quaternions and Half Angles 49
6.2.3 Double Values 51
6.3 Recovering 0 and n 51
6.4 Euler Angles and Quaternions 52
*6.5 t Optional Remarks 54
6.5.1 t Connections to Group Theory 54
6.5.2 t Pure Quaternion Derivation 55
6.5.3 t Quaternion Exponential Version 55
6.6 Conclusion 56
07 VISUALIZING ALGEBRAIC STRUCTURE 57
7.1 Algebra of Complex Numbers 57
* A dagger (t) denotes a section with advanced content that can be skipped on a first reading.
CONTENTS xiii
7.1.1 Complex Numbers 58
7.1.2 Abstract View of Complex Multiplication 59
7.1.3 Restriction to Unit length Case 61
7.2 Quaternion Algebra 63
7.2.1 The Multiplication Rule 63
7.2.2 Scalar Product 65
7.2.3 Modulus of the Quaternion Product 65
7.2.4 Preservation of the Unit Quaternions 66
08 VISUALIZING SPHERES 69
8.1 2D: Visualizing an Edge on Circle 70
8.1.1 Trigonometric Function Method 71
8.1.2 Complex Variable Method 72
8.1.3 Square Root Method 73
8.2 The Square Root Method 74
8.3 3D: Visualizing a Balloon 76
8.3.1 Trigonometric Function Method 76
8.3.2 Square Root Method 77
8.4 4D: Visualizing Quaternion Geometry on S 80
8.4.1 Seeing the Parameters of a Single Quaternion 82
8.4.2 Hemispheres in S3 83
09 VISUALIZING LOGARITHMS AND
EXPONENTIALS 87
9.1 Complex Numbers 87
9.2 Quaternions 91
10 VISUALIZING INTERPOLATION METHODS 93
I O.I Basics of Interpolation 93
10.1.1 Interpolation Issues 93
10.1.2 Gram Schmidt Derivation of the SLERP 97
I O.I. 3 t Alternative Derivation 99
10.2 Quaternion Interpolation 101
10.3 Equivalent 3x3 Matrix Method 104
xiv CONTENTS
11 LOOKING AT ELEMENTARY QUATERNION
FRAMES 105
11.1 A Single Quaternion Frame 105
11.2 Several Isolated Frames 106
11.3 A Rotating Frame Sequence 107
11.4 Synopsis 110
12 QUATERNIONS AND THE BELT TRICK:
CONNECTING TO THE IDENTITY 111
12.1 Very Interesting, but Why? 113
12.1.1 The Intuitive Answer 113
12.1.2 | The Technical Answer 113
12.2 The Details 114
12.3 Frame sequence Visualization Methods 118
12.3.1 One Rotation 120
12.3.2 Two Rotations 121
12.3.3 Synopsis 122
13 QUATERNIONS AND THE ROLLING BALL:
EXPLOITING ORDER DEPENDENCE 123
13.1 Order Dependence 123
13.2 The Rolling Ball Controller 125
13.3 Rolling Ball Quaternions 128
13.4 t Commutators 130
13.5 Three Degrees of Freedom From Two 131
14 QUATERNIONS AND GIMBAL LOCK:
LIMITING THE AVAILABLE SPACE 133
14.1 Guidance System Suspension 133
14.2 Mathematical Interpolation Singularities 134
14.3 Quaternion Viewpoint 134
PART II ADVANCED QUATERNION TOPICS 137
CONTENTS xv
15 ALTERNATIVE WAYS OF WRITING
QUATERNIONS 141
15.1 Hamilton s Generalization of Complex Numbers 142
15.2 Pauli Matrices 143
15.3 Other Matrix Forms 144
16 EFFICIENCY AND COMPLEXITY ISSUES 147
16.1 Extracting a Quaternion 148
16.1.1 Positive Trace R 149
16.1.2 Nonpositive Trace R 149
16.2 Efficiency of Vector Operations 150
17 ADVANCED SPHERE VISUALIZATION 153
17.1 Projective Method 153
17.1.1 The CircleS1 153
17.1.2 General SN Polar Projection 155
17.2 Distance preserving Flattening Methods 156
17.2.1 Unroll and Flatten S 157
17.2.2 S2 Flattened Equal area Method 157
17.2.3 S3 Flattened Equal volume Method 159
18 MORE ON LOGARITHMS AND
EXPONENTIALS 165
18.1 2D Rotations 165
18.2 3D Rotations 167
18.3 Using Logarithms for Quaternion Calculus 171
18.4 Quaternion Interpolations Versus Log 171
19 TWO DIMENSIONAL CURVES 173
19.1 Orientation Frames for 2D Space Curves 173
19.1.1 2D Rotation Matrices 174
19.1.2 The Frame Matrix in 2D 175
19.1.3 Frame Evolution in 2D 176
19.2 What Is a Map? 176
19.3 Tangent and Normal Maps 177
19.4 Square Root Form 179
xvj CONTENTS
19.4.1 Frame Evolution in (a, b) 179
19.4.2 Simplifying the Frame Equations 179
20 THREE DIMENSIONAL CURVES 181
20.1 Introduction to 3D Space Curves 181
20.2 General Curve Framings in 3D 183
20.3 Tubing 186
20.4 Classical Frames 186
20.4.1 Frenet Serret Frame 186
20.4.2 Parallel Transport Frame 190
20.4.3 Geodesic Reference Frame 193
20.4.4 General Frames 193
20.5 Mapping the Curvature and Torsion 194
20.6 Theory of Quaternion Frames 196
2 0.6.1 Generic Quaternion Frame Equations 197
20.6.2 Quaternion Frenet Frames 200
20.6.3 Quaternion Parallel Transport Frames 202
20.7 Assigning Smooth Quaternion Frames 202
20.7.1 Assigning Quaternions to Frenet Frames 202
20.7.2 Assigning Quaternions to Parallel Transport
Frames 204
20.8 Examples: Torus Knot and Helix Quaternion Frames 209
20.9 Comparison of Quaternion Frame Curve Lengths 210
21 3D SURFACES 213
21.1 Introduction to 3D Surfaces 213
21.1.1 Classical Gauss Map 214
21.1.2 Surface Frame Evolution 215
21.1.3 Examples of Surface Framings 217
21.2 Quaternion Weingarten Equations 218
21.2.1 Quaternion Frame Equations 218
21.2.2 Quaternion Surface Equations (Weingarten
Equations) 220
21.3 Quaternion Gauss Map 221
21.4 Example: The Sphere 223
CONTENTS xvii
21.4.1 Quaternion Maps of Alternative Sphere Frames 223
21.4.2 Covering the Sphere and the Geodesic Reference
Frame South Pole Singularity 223
21.5 Examples: Minimal Surface Quaternion Maps 228
22 OPTIMAL QUATERNION FRAMES 233
22.1 Background 233
22.2 Motivation 234
22.3 Methodology 236
22.3.1 The Space of Possible Frames 237
22.3.2 Parallel Transport and Minimal Measure 238
22.4 The Space of Frames 239
22.4.1 Full Space of Curve Frames 242
22.4.2 Full Space of Surface Maps 243
22.5 Choosing Paths in Quaternion Space 248
22.5.1 Optimal Path Choice Strategies 249
22.5.2 General Remarks on Optimization in Quaternion
Space 250
22.6 Examples 251
22.6.1 Minimal Quaternion Frames for Space Curves 251
22.6.2 Minimal quaternion area Surface Patch Framings 256
23 QUATERNION VOLUMES 257
23.1 Three degree of freedom Orientation Domains 259
23.2 Application to the Shoulder Joint 262
23.3 Data Acquisition and the Double covering Problem 264
23.3.1 Sequential Data 264
23.3.2 The Sequential Nearest neighbor Algorithm 265
23.3.3 The Surface based Nearest neighbor Algorithm 265
23.3.4 The Volume based Nearest neighbor Algorithm 267
23.4 Application Data 268
24 QUATERNION MAPS OF STREAMLINES 271
24.1 Visualization Methods 271
24.1.1 Direct Plot of Quaternion Frame Fields 272
24.1.2 Similarity Measures for Quaternion Frames 273
xviii CONTENTS
24.1.3 Exploiting or Ignoring Double Points 273
24.2 3D Flow Data Visualizations 274
24.2.1 AVS Streamline Example 275
24.2.2 Deforming Solid Example 275
24.3 Brushing: Clusters and Inverse Clusters 275
24.4 Advanced Visualization Approaches 275
24.4.1 3D Rotations of Quaternion Displays 279
24.4.2 Probing Quaternion Frames with 4D light 281
25 QUATERNION INTERPOLATION 283
25.1 Concepts of Euclidean Linear Interpolation 284
25.1.1 Constructing Higher order Polynomial Splines 285
25.1.2 Matching 285
25.1.3 Schlag s Method 289
25.1.4 Control point Method 290
25.2 The Double Quad 292
25.3 Direct Interpolation of 3D Rotations 294
25.3.1 Relation to Quaternions 295
25.3.2 Method for Arbitrary Origin 296
25.3.3 Exponential Version 298
25.3.4 Special Vector Vector Case 299
25.3.5 Multiple level Interpolation Matrices 301
25.3.6 Equivalence of Quaternion and Matrix Forms 303
25.4 Quaternion Splines 304
25.5 Quaternion de Casteljau Splines 308
25.6 Equivalent Anchor Points 315
25.7 Angular Velocity Control 319
25.8 Exponential map Quaternion Interpolation 321
25.9 Global Minimal Acceleration Method 326
25.9.1 Why a Cubic? 326
25.9.2 Extension to Quaternion Form 327
26 QUATERNION ROTATOR DYNAMICS 329
26.1 Static Frame 330
26.2 Torque 334
CONTENTS xix
26.3 Quaternion Angular Momentum 335
27 CONCEPTS OF THE ROTATION GROUP 339
27.1 Brief Introduction to Group Representations 339
27.1.1 Complex Versus Real 341
27.1.2 What Is a Representation? 342
27.2 Basic Properties of Spherical Harmonics 344
27.2.1 Representations and Rotation invariant Properties 346
11 . L.I Properties of Expansion Coefficients Under
Rotations 348
28 SPHERICAL RIEMANNIAN GEOMETRY 351
2 8.1 Induced Metric on the Sphere 351
28.2 Induced Metrics of Spheres 353
2 8.2.1 S Induced Metrics 356
28.2.2 S2 Induced Metrics 357
28.2.3 S3 Induced Metrics 358
28.2.4 Toroidal Coordinates on S3 360
28.2.5 Axis angle Coordinates on S 361
28.2.6 General Form for the Square root Induced Metric 361
28.3 Elements of Riemannian Geometry 362
28.4 Riemann Curvature of Spheres 363
28.4.1 S1 364
28.4.2 S2 364
28.4.3 S3 365
28.5 Geodesies and Parallel Transport on the Sphere 366
28.6 Embedded vector Viewpoint of the Geodesies 368
PART III BEYOND QUATERNIONS 373
29 THE RELATIONSHIP OF 40 ROTATIONS
TO QUATERNIONS 377
29.1 What Happened in Three Dimensions 377
29.2 Quaternions and Four Dimensions 378
xx CONTENTS
30 QUATERNIONS AND THE FOUR DIVISION
ALGEBRAS 381
3 0.1 Division Algebras 381
30.1.1 The Number Systems with Dimensions 1, 2, 4,
and 8 382
30.1.2 Parallelizable Spheres 385
30.2 Relation to Fiber Bundles 386
30.3 Constructing the Hopf Fibrations 387
30.3.1 Real: S° fiber+ S1 base = S bundle 387
30.3.2 Complex: S1 fiber + S2 base = S3 bundle 389
30.3.3 Quaternion: S3 fiber + S4 base = S7 bundle 390
30.3.4 Octonion: S7 fiber + S8 base = S15 bundle 391
31 CLIFFORD ALGEBRAS 393
31.1 Introduction to Clifford Algebras 394
31.2 Foundations 395
31.2.1 Clifford Algebras and Rotations 397
31.2.2 Higher dimensional Clifford Algebra Rotations 400
31.3 Examples of Clifford Algebras 402
31.3.1 ID Clifford Algebra 402
31.3.2 2D Clifford Algebra 403
31.3.3 2D Rotations Done Right 404
31.3.4 3D Clifford Algebra 406
31.3.5 Clifford Implementation of 3D Rotations 407
31.4 Higher Dimensions 408
31.5 Pin(N),Spin(N),O(N),SO(N), and All That... 410
32 CONCLUSIONS 413
APPENDICES 415
A NOTATION 419
A. I Vectors 419
A.2 Length of a Vector 420
A.3 Unit Vectors 421
CONTENTS xxi
A.4 Polar Coordinates 421
A. 5 Spheres 422
A.6 Matrix Transformations 422
A. 7 Features of Square Matrices 423
A.8 Orthogonal Matrices 424
A. 9 Vector Products 424
A.9.1 2D Dot Product 424
A.9.2 2D Cross Product 425
A.9.3 3D Dot Product 425
A.9.4 3D Cross Product 425
A. 10 Complex Variables 426
B 2D COMPLEX FRAMES 429
C 3D QUATERNION FRAMES 433
C.I Unit Norm 433
C.2 Multiplication Rule 433
C.3 Mapping to 3D rotations 435
C.4 Rotation Correspondence 437
C.5 Quaternion Exponential Form 437
D FRAME AND SURFACE EVOLUTION 439
D.I Quaternion Frame Evolution 439
D.2 Quaternion Surface Evolution 441
E QUATERNION SURVIVAL KIT 443
F QUATERNION METHODS 451
F. I Quaternion Logarithms and Exponentials 451
F.2 The Quaternion Square Root Trick 452
F.3 The a —* b formula simplified 453
F.4 Gram Schmidt Spherical Interpolation 454
F.5 Direct Solution for Spherical Interpolation 455
F.6 Converting Linear Algebra to Quaternion Algebra 457
F.7 Useful Tensor Methods and Identities 457
F.7.1 Einstein Summation Convention 457
xxii CONTENTS
¥.7.2 Kronecker Delta 458
F.7.3 Levi Civita Symbol 458
G QUATERNION PATH OPTIMIZATION USING
SURFACE EVOLVER 461
H QUATERNION FRAME INTEGRATION 463
I HYPERSPHERICAL GEOMETRY 467
1.1 Definitions 467
1.2 Metric Properties 468
REFERENCES 471
INDEX 487
|
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| id | DE-604.BV021683956 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:39:20Z |
| institution | BVB |
| isbn | 0120884003 9780120884001 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014898137 |
| oclc_num | 634697950 |
| open_access_boolean | |
| owner | DE-29T DE-91G DE-BY-TUM DE-523 DE-19 DE-BY-UBM DE-B768 |
| owner_facet | DE-29T DE-91G DE-BY-TUM DE-523 DE-19 DE-BY-UBM DE-B768 |
| physical | XXXI, 498 S. Ill., graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Elsevier [u.a.] |
| record_format | marc |
| series2 | The Morgan Kaufmann series in interactive 3D technology |
| spellingShingle | Hanson, Andrew J. Visualizing quaternions Visualisierung (DE-588)4188417-6 gnd Quaternion (DE-588)4176653-2 gnd |
| subject_GND | (DE-588)4188417-6 (DE-588)4176653-2 |
| title | Visualizing quaternions |
| title_auth | Visualizing quaternions |
| title_exact_search | Visualizing quaternions |
| title_full | Visualizing quaternions Andrew J. Hanson |
| title_fullStr | Visualizing quaternions Andrew J. Hanson |
| title_full_unstemmed | Visualizing quaternions Andrew J. Hanson |
| title_short | Visualizing quaternions |
| title_sort | visualizing quaternions |
| topic | Visualisierung (DE-588)4188417-6 gnd Quaternion (DE-588)4176653-2 gnd |
| topic_facet | Visualisierung Quaternion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014898137&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hansonandrewj visualizingquaternions |
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