Description Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the
most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices
they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners
have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be
available. The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on
visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they
are important?a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second
part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the
full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras,
the all-encompassing framework for vectors and quaternions.
Audience
Programmers and developers in computer graphics and the game industry, scientists and engineers working in aerospace and scientific visualization,
students of game development and computer graphics, and those interested in quaternions but who have limited math background.
Contents ABOUT THE AUTHOR FOREWORD by Steve Cunningham PREFACE ACKNOWLEDGMENTS PART I ELEMENTS OF QUATERNIONS 01
THE DISCOVERY OF QUATERNIONS 1.1 Hamilton's Walk 1.2 Then Came Octonions 1.3 The Quaternion Revival 02
FOLKLORE OF ROTATIONS 2.1 The Belt Trick 2.2 The Rolling Ball 2.3 The Apollo 10 Gimbal-lock Incident 2.4
3D Game Developer's Nightmare 2.5 The Urban Legend of the Upside-down F16 2.6 Quaternions to the Rescue 03
BASIC NOTATION 3.1 Vectors 3.2 Length of a Vector 3.3 3D Dot Product 3.4 3D Cross Product 3.5
Unit Vectors 3.6 Spheres 3.7 Matrices 3.8 Complex Numbers 04 WHAT ARE QUATERNIONS? 05 ROAD
MAP TO QUATERNION VISUALIZATION 5.1 The Complex Number Connection 5.2 The Cornerstones of Quaternion Visualization 06
FUNDAMENTALS OF ROTATIONS 6.1 2D Rotations 6.1.1 Relation to Complex Numbers 6.1.2 The Half-angle Form 6.1.3
Complex Exponential Version 6.2 Quaternions and 3D Rotations 6.2.1 Construction 6.2.2 Quaternions and
Half Angles 6.2.3 Double Values 6.3 Recovering Θ and n 6.4 Euler Angles and Quaternions 6.5 † Optional Remarks 6.5.1 † Connections to Group Theory 6.5.2 † "Pure" Quaternion Derivation 6.5.3 † Quaternion Exponential Version 6.6 Conclusion 07 VISUALIZING ALGEBRAIC STRUCTURE 7.1 Algebra
of Complex Numbers 7.1.1 Complex Numbers 7.1.2 Abstract View of Complex Multiplication 7.1.3 Restriction
to Unit-length Case 7.2 Quaternion Algebra 7.2.1 The Multiplication Rule 7.2.2 Scalar Product 7.2.3
Modulus of the Quaternion Product 7.2.4 Preservation of the Unit Quaternions 08 VISUALIZING SPHERES 8.1
2D: Visualizing an Edge-on Circle 8.1.1 Trigonometric Function Method 8.1.2 Complex Variable Method 8.1.3
Square Root Method 8.2 The Square Root Method 8.3 3D: Visualizing a Balloon 8.3.1 Trigonometric Function
Method 8.3.2 Square Root Method 8.4 4D: Visualizing Quaternion Geometry on S 3 8.4.1
Seeing the Parameters of a Single Quaternion 8.4.2 Hemispheres in S 3 09 VISUALIZING LOGARITHMS
AND EXPONENTIALS 9.1 Complex Numbers 9.2 Quaternions 10 VISUALIZING INTERPOLATION METHODS 10.1
Basics of Interpolation 10.1.1 Interpolation Issues 10.1.2 Gram-Schmidt Derivation of the SLERP 10.1.3 †
Alternative Derivation 10.2 Quaternion Interpolation 10.3 Equivalent 3×3 Matrix Method 11 LOOKING
AT ELEMENTARY QUATERNION FRAMES 11.1 A Single Quaternion Frame 11.2 Several Isolated Frames 11.3 A Rotating
Frame Sequence 11.4 Synopsis 12 QUATERNIONS AND THE BELT TRICK: CONNECTING TO THE IDENTITY 12.1 Very
Interesting, but Why? 12.1.1 The Intuitive Answer 12.1.2 † The Technical Answer 12.2 The Details 12.3
Frame-sequence Visualization Methods 12.3.1 One Rotation 12.3.2 Two Rotations 12.3.3 Synopsis 13
QUATERNIONS AND THE ROLLING BALL: EXPLOITING ORDER DEPENDENCE 13.1 Order Dependence 13.2 The Rolling Ball Controller 13.3
Rolling Ball Quaternions 13.4 † Commutators 13.5 Three Degrees of Freedom From Two 14 QUATERNIONS AND
GIMBAL LOCK: LIMITING THE AVAILABLE SPACE 14.1 Guidance System Suspension 14.2 Mathematical Interpolation Singularities 14.3
Quaternion Viewpoint PART II ADVANCED QUATERNION TOPICS 15 ALTERNATIVE WAYS OF WRITING QUATERNIONS 15.1
Hamilton's Generalization of Complex Numbers 15.2 Pauli Matrices 15.3 Other Matrix Forms 16 EFFICIENCY AND
COMPLEXITY ISSUES 16.1 Extracting a Quaternion 16.1.1 Positive Trace R 16.1.2 Nonpositive Trace
R 16.2 Efficiency of Vector Operations 17 ADVANCED SPHERE VISUALIZATION 17.1 Projective
Method 17.1.1 The Circle S 1 17.1.2 General S N Polar Projection 17.2
Distance-preserving Flattening Methods 17.2.1 Unroll-and-Flatten S 1 17.2.2 S 2
Flattened Equal-area Method 17.2.3 S 3 Flattened Equal-volume Method 18 MORE ON LOGARITHMS
AND EXPONENTIALS 18.1 2D Rotations 18.2 3D Rotations 18.3 Using Logarithms for Quaternion Calculus 18.4
Quaternion Interpolations Versus Log 19 TWO-DIMENSIONAL CURVES 19.1 Orientation Frames for 2D Space Curves 19.1.1
2D Rotation Matrices 19.1.2 The Frame Matrix in 2D 19.1.3 Frame Evolution in 2D 19.2 What Is a Map? 19.3
Tangent and Normal Maps 19.4 Square Root Form 19.4.1 Frame Evolution in (a, b) 19.4.2 Simplifying
the Frame Equations 20 THREE-DIMENSIONAL CURVES 20.1 Introduction to 3D Space Curves 20.2 General
Curve Framings in 3D 20.3 Tubing 20.4 Classical Frames 20.4.1 Frenet-Serret Frame 20.4.2 Parallel
Transport Frame 20.4.3 Geodesic Reference Frame 20.4.4 General Frames 20.5 Mapping the Curvature and Torsion 20.6
Theory of Quaternion Frames 20.6.1 Generic Quaternion Frame Equations 20.6.2 Quaternion Frenet Frames 20.6.3
Quaternion Parallel Transport Frames 20.7 Assigning Smooth Quaternion Frames 20.7.1 Assigning Quaternions to
Frenet Frames 20.7.2 Assigning Quaternions to Parallel Transport Frames 20.8 Examples: Torus Knot and Helix Quaternion
Frames 20.9 Comparison of Quaternion Frame Curve Lengths 21 3D SURFACES 21.1 Introduction to 3D Surfaces 21.1.1
Classical Gauss Map 21.1.2 Surface Frame Evolution 21.1.3 Examples of Surface Framings 21.2 Quaternion Weingarten
Equations 21.2.1 Quaternion Frame Equations 21.2.2 Quaternion Surface Equations (Weingarten Equations) 21.3
Quaternion Gauss Map 21.4 Example: The Sphere 21.4.1 Quaternion Maps of Alternative Sphere Frames 21.4.2
Covering the Sphere and the Geodesic Reference Frame South Pole Singularity 21.5 Examples: Minimal Surface Quaternion Maps 22
OPTIMAL QUATERNION FRAMES 22.1 Background 22.2 Motivation 22.3 Methodology 22.3.1 The Space of
Possible Frames 22.3.2 Parallel Transport and Minimal Measure 22.4 The Space of Frames 22.4.1 Full Space
of Curve Frames 22.4.2 Full Space of Surface Maps 22.5 Choosing Paths in Quaternion Space 22.5.1 Optimal
Path Choice Strategies 22.5.2 General Remarks on Optimization in Quaternion Space 22.6 Examples 22.6.1
Minimal Quaternion Frames for Space Curves 22.6.2 Minimal-quaternion-area Surface Patch Framings 23 QUATERNION
VOLUMES 23.1 Three-degree-of-freedom Orientation Domains 23.2 Application to the Shoulder Joint 23.3 Data
Acquisition and the Double-covering Problem 23.3.1 Sequential Data 23.3.2 The Sequential Nearest-neighbor Algorithm 23.3.3
The Surface-based Nearest-neighbor Algorithm 23.3.4 The Volume-based Nearest-neighbor Algorithm 23.4 Application
Data 24 QUATERNION MAPS OF STREAMLINES 24.1 Visualization Methods 24.1.1 Direct Plot of Quaternion
Frame Fields 24.1.2 Similarity Measures for Quaternion Frames 24.1.3 Exploiting or Ignoring Double Points 24.2
3D Flow Data Visualizations 24.2.1 AVS Streamline Example 24.2.2 Deforming Solid Example 24.3 Brushing:
Clusters and Inverse Clusters 24.4 Advanced Visualization Approaches 24.4.1 3D Rotations of Quaternion Displays 24.4.2
Probing Quaternion Frames with 4D Light 25 QUATERNION INTERPOLATION 25.1 Concepts of Euclidean Linear Interpolation 25.1.1
Constructing Higher-order Polynomial Splines 25.1.2 Matching 25.1.3 Schlag's Method 25.1.4 Control-point Method 25.2
The Double Quad 25.3 Direct Interpolation of 3D Rotations 25.3.1 Relation to Quaternions 25.3.2 Method for
Arbitrary Origin 25.3.3 Exponential Version 25.3.4 Special Vector-Vector Case 25.3.5 Multiple-level Interpolation
Matrices 25.3.6 Equivalence of Quaternion and Matrix Forms 25.4 Quaternion Splines 25.5 Quaternion de Casteljau
Splines 25.6 Equivalent Anchor Points 25.7 Angular Velocity Control 25.8 Exponential-map Quaternion Interpolation 25.9
Global Minimal Acceleration Method 25.9.1 Why a Cubic? 25.9.2 Extension to Quaternion Form 26 QUATERNION
ROTATOR DYNAMICS 26.1 Static Frame 26.2 Torque 26.3 Quaternion Angular Momentum 27 CONCEPTS
OF THE ROTATION GROUP 27.1 Brief Introduction to Group Representations 27.1.1 Complex Versus Real 27.1.2
What Is a Representation? 27.2 Basic Properties of Spherical Harmonics 27.2.1 Representations and Rotation-invariant
Properties 27.2.2 Properties of Expansion Coefficients Under Rotations 28 SPHERICAL RIEMANNIAN GEOMETRY 28.1
Induced Metric on the Sphere 28.2 Induced Metrics of Spheres 28.2.1 S 1 Induced Metrics 28.2.2
S 2 Induced Metrics 28.2.3 S 3 Induced Metrics 28.2.4 Toroidal Coordinates on S 3 28.2.5
Axis-angle Coordinates on S 3 28.2.6 General Form for the Square-root Induced Metric 28.3 Elements
of Riemannian Geometry 28.4 Riemann Curvature of Spheres 28.4.1 S 1 28.4.2 S 2 28.4.3
S 3 28.5 Geodesics and Parallel Transport on the Sphere 28.6 Embedded-vector Viewpoint of the Geodesics PART
III BEYOND QUATERNIONS 29 THE RELATIONSHIP OF 4D ROTATIONS TO QUATERNIONS 29.1 What Happened in Three Dimensions 29.2
Quaternions and Four Dimensions 30 QUATERNIONS AND THE FOUR DIVISION ALGEBRAS 30.1 Division Algebras 30.1.1
The Number Systems with Dimensions 1, 2, 4, and 8 30.1.2 Parallelizable Spheres 30.2 Relation to Fiber Bundles 30.3
Constructing the Hopf Fibrations 30.3.1 Real: S 0 fiber + S 1 base = S 1
bundle 30.3.2 Complex: S 1 fiber + S 2 base = S 3 bundle 30.3.3 Quaternion:
S 3 fiber + S 4 base = S 7 bundle 30.3.4 Octonion: S 7 fiber
+ S 8 base = S 15 bundle 31 CLIFFORD ALGEBRAS 31.1 Introduction to Clifford
Algebras 31.2 Foundations 31.2.1 Clifford Algebras and Rotations 31.2.2 Higher-dimensional Clifford Algebra
Rotations 31.3 Examples of Clifford Algebras 31.3.1 1D Clifford Algebra 31.3.2 2D Clifford Algebra 31.3.3
2D Rotations Done Right 31.3.4 3D Clifford Algebra 31.3.5 Clifford Implementation of 3D Rotations 31.4 Higher
Dimensions 31.5 Pin( N ), Spin( N ), O( N ), SO( N ), and All That. . . 32 CONCLUSIONS APPENDICES A
NOTATION A.1 Vectors A.2 Length of a Vector A.3 Unit Vectors A.4 Polar Coordinates A.5 Spheres A.6
Matrix Transformations A.7 Features of Square Matrices A.8 Orthogonal Matrices A.9 Vector Products A.9.1
2D Dot Product A.9.2 2D Cross Product A.9.3 3D Dot Product A.9.4 3D Cross Product A.10 Complex Variables B
2D COMPLEX FRAMES C 3D QUATERNION FRAMES C.1 Unit Norm C.2 Multiplication Rule C.3 Mapping
to 3D rotations C.4 Rotation Correspondence C.5 Quaternion Exponential Form D FRAME AND SURFACE
EVOLUTION D.1 Quaternion Frame Evolution D.2 Quaternion Surface Evolution E QUATERNION SURVIVAL
KIT F QUATERNION METHODS F.1 Quaternion Logarithms and Exponentials F.2 The Quaternion Square Root
Trick F.3 The a → b formula simplified F.4 Gram-Schmidt Spherical Interpolation F.5 Direct
Solution for Spherical Interpolation F.6 Converting Linear Algebra to Quaternion Algebra F.7 Useful Tensor Methods and
Identities F.7.1 Einstein Summation Convention F.7.2 Kronecker Delta F.7.3 Levi-Civita Symbol G
QUATERNION PATH OPTIMIZATION USING SURFACE EVOLVER H QUATERNION FRAME INTEGRATION I HYPERSPHERICAL
GEOMETRY I.1 Definitions I.2 Metric Properties REFERENCES INDEX
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