VISUALIZING QUATERNIONS

Transcription

1 THE MORGAN KAUFMANN SERIES IN INTERACTIVE 3D TECHNOLOGY VISUALIZING QUATERNIONS ANDREW J. HANSON «WW m.-:ki -. " ;. *' AMSTERDAM BOSTON HEIDELBERG ^ M Ä V l LONDON NEW YORK OXFORD <* И PARIS * SAN DIEGO * SAN FRANCISCO,. ~ ' *-ШЯь^ SINGAPORE SYDNEY TOKYO ELSEVIER Morgan Kaufmann is an imprint of Elsevier MORGAN KAUFMANN PUBLISHERS

2 Contents ABOUT THE AUTHOR FOREWORD PREFACE ACKNOWLEDGMENTS X XXIII XXV XXXI PART I ELEMENTS OF QUATERNIONS 1 01 THE DISCOVERY OF QUATERNIONS 5 l. I Hamilton's Walk S l. 2 Then Came Octonions 8 l. 3 The Quaternion Revival 9 02 FOLKLORE OF ROTATIONS The Belt Trick The Rolling Ball The Apollo 10 Gimbal-lock Incident D Game Developer's Nightmare The Urban Legend of the Upside-down F Quaternions to the Rescue BASIC NOTATION Vectors 31

3 XII CONTENTS 3.2 Length of a Vector D Dot Product D Cross Product Unit Vectors Spheres Matrices Complex Numbers WHAT ARE QUATERNIONS? ROAD MAP TO QUATERNION VISUALIZATION The Complex Number Connection The Cornerstones of Quaternion Visualization FUNDAMENTALS OF ROTATIONS D Rotations Relation to Complex Numbers The Half-angle Form Complex Exponential Version Quaternions and 3D Rotations Construction Quaternions and Half Angles Double Values Recovering в and n Euler Angles and Quaternions 52 * 6.5 f Optional Remarks f Connections to Group Theory f "Pure" Quaternion Derivation f Quaternion Exponential Version Conclusion VISUALIZING ALGEBRAIC STRUCTURE Algebra of Complex Numbers 5 7 * A dagger (t) denotes a section with advanced content that can be skipped on a first reading.

4 7.1.1 Complex Numbers Abstract View of Complex Multiplication Restriction to Unit-length Case Quaternion Algebra The Multiplication Rule Scalar Product Modulus of the Quaternion Product Preservation of the Unit Quaternions VISUALIZING SPHERES D: Visualizing an Edge-on Circle Trigonometric Function Method Complex Variable Method Square Root Method The Square Root Method D: Visualizing a Balloon Trigonometric Function Method Square Root Method D: Visualizing Quaternion Geometry on S Seeing the Parameters of a Single Quaternion Hemispheres in S VISUALIZING LOGARITHMS AIMD EXPONENTIALS Complex Numbers Quaternions VISUALIZING INTERPOLATION METHODS 93 I O.I Basics of Interpolation 93 I Interpolation Issues Gram-Schmidt Derivation of the SLERP 97 Ю.1.3 t Alternative Derivation Quaternion Interpolation lol l 0.3 Equivalent 3x3 Matrix Method 104

5 11 LOOKING AT ELEMENTARY QUATERNION FRAMES A Single Quaternion Frame Several Isolated Frames A Rotating Frame Sequence Synopsis l l 0 12 QUATERNIONS AND THE BELT TRICK: CONNECTING TO THE IDENTITY Very Interesting, but Why? I l The Intuitive Answer f The Technical Answer The Details Frame-sequence Visualization Methods One Rotation Two Rotations Synopsis QUATERNIONS AND THE ROLLING BALL: EXPLOITING ORDER DEPENDENCE Order Dependence l The Rolling Ball Controller I Rolling Ball Quaternions l f Commutators S Three Degrees of Freedom From Two l 3 l 14 QUATERNIONS AND GIMBAL LOCK: LIMITING THE AVAILABLE SPACE Guidance System Suspension Mathematical Interpolation Singularities Quaternion Viewpoint 134 PART II ADVANCED QUATERNION TOPICS 137

6 xv 15 ALTERNATIVE WAYS OF WRITING QUATERNIONS Hamilton's Generalization of Complex Numbers Pauli Matrices Other Matrix Forms EFFICIENCY AND COMPLEXITY ISSUES Extracting a Quaternion Positive Trace R Nonpositive Trace R Efficiency of Vector Operations ADVANCED SPHERE VISUALIZATION 153 l 7.1 Proj ective Method The CircleS 1 1S General S Polar Projection lss 17.2 Distance-preserving Flattening Methods Unroll-and-Flatten S S 2 Flattened Equal -area Method S 3 Flattened Equal-volume Method MORE ON LOGARITHMS AND EXPONENTIALS D Rotations D Rotations Using Logarithms for Quaternion Calculus Quaternion Interpolations Versus Log 171 1Э TWO-DIMENSIONAL CURVES Orientation Frames for 2D Space Curves D Rotation Matrices The Frame Matrix in 2D Frame Evolution in 2D What Is a Map? Tangent and Normal Maps Square Root Form 179

7 XVI CONTENTS Frame Evolution in (a. b) Simplifying the Frame Equations l THREE-DIMENSIONAL CURVES Introduction to 3D Space Curves l 8 l 20.2 General Curve Framings in 3D l Tubing Classical Frames Frenet-Serret Frame Parallel Transport Frame Geodesic Reference Frame General Frames Mapping the Curvature and Torsion l Theory of Quaternion Frames Generic Quaternion Frame Equations Quaternion Frenet Frames Quaternion Parallel Transport Frames Assigning Smooth Quaternion Frames Assigning Quaternions to Frenet Frames Assigning Quaternions to Parallel Transport Frames Examples: Torus Knot and Helix Quaternion Frames Comparison of Quaternion Frame Curve Lengths D SURFACES Introduction to 3D Surfaces Classical Gauss Map Surface Frame Evolution 2 I Examples of Surface Framings Quaternion Weingarten Equations Quaternion Frame Equations Quaternion Surface Equations (Weingarten Equations) Quaternion Gauss Map Example: The Sphere 223

8 Quaternion Maps of Alternative Sphere Frames Covering the Sphere and the Geodesic Reference Frame South Pole Singularity Examples: Minimal Surface Quaternion Maps OPTIMAL QUATERNION FRAMES Background Motivation Methodology The Space of Possible Frames Parallel Transport and Minimal Measure The Space of Frames Full Space of Curve Frames Full Space of Surface Maps Choosing Paths in Quaternion Space Optimal Path Choice Strategies General Remarks on Optimization in Quaternion Space Examples Minimal Quaternion Frames for Space Curves Minimal-quaternion-area Surface Patch Framings QUATERNION VOLUMES Three-degree-of-freedom Orientation Domains Application to the Shoulder Joint Data Acquisition and the Double-covering Problem Sequential Data The Sequential Nearest-neighbor Algorithm The Surface-based Nearest-neighbor Algorithm The Volume-based Nearest-neighbor Algorithm Application Data QUATERNION MAPS OF STREAMLINES Visualization Methods Direct Plot of Quaternion Frame Fields Similarity Measures for Quaternion Frames 273

9 XVIII CONTENTS Exploiting or Ignoring Double Points D Flow Data Visualizations AVS Streamline Example Deforming Solid Example Brushing: Clusters and Inverse Clusters Advanced Visualization Approaches D Rotations of Quaternion Displays Probing Quaternion Frames with 4D Light QUATERNION INTERPOLATION Concepts of Euclidean Linear Interpolation Constructing Higher-order Polynomial Splines Matching Schlag 's Method Control-point Method The Double Quad Direct Interpolation of 3D Rotations Relation to Quaternions Method for Arbitrary Origin Exponential Version Special Vector Vector Case Multiple-level Interpolation Matrices Equivalence of Quaternion and Matrix Forms Quaternion Splines Quaternion de Casteljau Splines Equivalent Anchor Points Angular Velocity Control Exponential-map Quaternion Interpolation Global Minimal Acceleration Method Why a Cubic? Extension to Quaternion Form 327 2G QUATERNION ROTATOR DYNAMICS Static Frame Torque 334

10 XIX 26.3 Quaternion Angular Momentum CONCEPTS OF THE ROTATION GROUP Brief Introduction to Group Representations Complex Versus Real What Is a Representation? Basic Properties of Spherical Harmonics Representations and Rotation-invariant Properties Properties of Expansion Coefficients Under Rotations SPHERICAL RIEMANNIAN GEOMETRY Induced Metric on the Sphere 35 l 28.2 Induced Metrics of Spheres S 1 Induced Metrics S 2 Induced Metrics S 3 Induced Metrics Toroidal Coordinates on S Axis-angle Coordinates on S General Form for the Square-root Induced Metric Elements of Riemannian Geometry Riemann Curvature of Spheres S S S Geodesies and Parallel Transport on the Sphere Embedded-vector Viewpoint of the Geodesies 368 PART III BEYOND QUATERNIONS THE RELATIONSHIP OF 4D ROTATIONS TO QUATERNIONS What Happened in Three Dimensions Quaternions and Four Dimensions 378

11 30 QUATERNIONS AND THE FOUR DIVISION ALGEBRAS Division Algebras The Number Systems with Dimensions l, 2, 4, and Parallelizable Spheres Relation to Fiber Bundles Constructing the Hopf Fibrations Real: S fiber + S 1 base = S' bundle Complex: S' fiber + S 2 base = S 3 bundle Quaternion: S 3 fiber + S 4 base = S 7 bundle Octonion: S 7 fiber + S 8 base = S 15 bundle CLIFFORD ALGEBRAS Introduction to Clifford Algebras Foundations Clifford Algebras and Rotations Higher-dimensional Clifford Algebra Rotations Examples of Clifford Algebras ID Clifford Algebra D Clifford Algebra D Rotations Done Right D Clifford Algebra Clifford Implementation of 3D Rotations Higher Dimensions Pin(N), Spin(/V),0(W),SO(W), and All That CONCLUSIONS 413 APPENDICES 415 A NOTATION 419 A. I Vectors 419 A.2 Length of a Vector 420 A.3 Unit Vectors 421

12 xxi A.4 Polar Coordinates 421 A.S 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 В 2D COMPLEX FRAMES 429 С 3D QUATERNION FRAMES 433 C.l Unit Norm 433 С 2 Multiplication Rule 433 C.3 Mapping to 3D rotations 435 C.4 Rotation Correspondence 437 С 5 Quaternion Exponential Form 437 D FRAME AND SURFACE EVOLUTION 439 D. 1 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 ä 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 R7 Useful Tensor Methods and Identities 457 F.7.1 Einstein Summation Convention 457

13 XXII CONTENTS.7.1 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 LI Definitions Metric Properties 468 REFERENCES 471 INDEX 487