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| About the Author | |
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| Foreword | |
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| Preface | |
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| Acknowledgments | |
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| Elements of Quaternions | |
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| The Discovery of Quaternions | |
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| Hamilton's Walk | |
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| Then Came Octonions | |
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| The Quaternion Revival | |
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| Folklore of Rotations | |
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| The Belt Trick | |
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| The Rolling Ball | |
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| The Apollo 10 Gimbal-lock Incident | |
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| 3D Game Developer's Nightmare | |
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| The Urban Legend of the Upside-down F16 | |
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| Quaternions to the Rescue | |
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| Basic Notation | |
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| Vectors | |
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| Length of a Vector | |
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| 3D Dot Product | |
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| 3D Cross Product | |
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| Unit Vectors | |
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| Spheres | |
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| Matrices | |
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| Complex Numbers | |
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| What Are Quaternions? | |
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| Road Map to Quaternion Visualization | |
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| The Complex Number Connection | |
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| The Cornerstones of Quaternion Visualization | |
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| Fundamentals of Rotations | |
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| 2D Rotations | |
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| Quaternions and 3D Rotations | |
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| Recovering [theta] and n | |
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| Euler Angles and Quaternions | |
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| Optional Remarks | |
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| Conclusion | |
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| Visualizing Algebraic Structure | |
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| Algebra of Complex Numbers | |
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| Quaternion Algebra | |
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| Visualizing Spheres | |
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| 2D: Visualizing an Edge-on Circle | |
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| The Square Root Method | |
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| 3D: Visualizing a Balloon | |
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| 4D: Visualizing Quaternion Geometry on S[superscript 3] | |
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| Visualizing Logarithms and Exponentials | |
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| Complex Numbers | |
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| Quaternions | |
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| Visualizing Interpolation Methods | |
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| Basics of Interpolation | |
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| Quaternion Interpolation | |
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| Equivalent 3 x 3 Matrix Method | |
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| Looking at Elementary Quaternion Frames | |
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| A Single Quaternion Frame | |
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| Several Isolated Frames | |
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| A Rotating Frame Sequence | |
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| Synopsis | |
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| Quaternions and the Belt Trick: Connecting to the Identity | |
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| Very Interesting, but Why? | |
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| The Details | |
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| Frame-sequence Visualization Methods | |
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| Quaternions and the Rolling Ball: Exploiting Order Dependence | |
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| Order Dependence | |
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| The Rolling Ball Controller | |
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| Rolling Ball Quaternions | |
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| Commutators | |
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| Three Degrees of Freedom From Two | |
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| Quaternions and Gimbal Lock: Limiting the Available Space | |
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| Guidance System Suspension | |
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| Mathematical Interpolation Singularities | |
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| Quaternion Viewpoint | |
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| Advanced Quaternion Topics | |
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| Alternative Ways of Writing Quaternions | |
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| Hamilton's Generalization of Complex Numbers | |
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| Pauli Matrices | |
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| Other Matrix Forms | |
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| Efficiency and Complexity Issues | |
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| Extracting a Quaternion | |
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| Efficiency of Vector Operations | |
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| Advanced Sphere Visualization | |
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| Projective Method | |
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| Distance-preserving Flattening Methods | |
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| More on Logarithms and Exponentials | |
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| 2D Rotations | |
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| 3D Rotations | |
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| Using Logarithms for Quaternion Calculus | |
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| Quaternion Interpolations Versus Log | |
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| Two-Dimensional Curves | |
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| Orientation Frames for 2D Space Curves | |
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| What Is a Map? | |
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| Tangent and Normal Maps | |
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| Square Root Form | |
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| Three-Dimensional Curves | |
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| Introduction to 3D Space Curves | |
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| General Curve Framings in 3D | |
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| Tubing | |
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| Classical Frames | |
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| Mapping the Curvature and Torsion | |
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| Theory of Quaternion Frames | |
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| Assigning Smooth Quaternion Frames | |
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| Examples: Torus Knot and Helix Quaternion Frames | |
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| Comparison of Quaternion Frame Curve Lengths | |
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| 3D Surfaces | |
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| Introduction to 3D Surfaces | |
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| Quaternion Weingarten Equations | |
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| Quaternion Gauss Map | |
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| Example: The Sphere | |
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| Examples: Minimal Surface Quaternion Maps | |
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| Optimal Quaternion Frames | |
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| Background | |
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| Motivation | |
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| Methodology | |
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| The Space of Frames | |
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| Choosing Paths in Quaternion Space | |
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| Examples | |
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| Quaternion Volumes | |
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| Three-degree-of-freedom Orientation Domains | |
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| Application to the Shoulder Joint | |
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| Data Acquisition and the Double-covering Problem | |
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| Application Data | |
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| Quaternion Maps of Streamlines | |
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| Visualization Methods | |
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| 3D Flow Data Visualizations | |
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| Brushing: Clusters and Inverse Clusters | |
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| Advanced Visualization Approaches | |
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| Quaternion Interpolation | |
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| Concepts of Euclidean Linear Interpolation | |
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| The Double Quad | |
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| Direct Interpolation of 3D Rotations | |
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| Quaternion Splines | |
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| Quaternion de Casteljau Splines | |
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| Equivalent Anchor Points | |
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| Angular Velocity Control | |
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| Exponential-map Quaternion Interpolation | |
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| Global Minimal Acceleration Method | |
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| Quaternion Rotator Dynamics | |
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| Static Frame | |
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| Torque | |
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| Quaternion Angular Momentum | |
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| Concepts of the Rotation Group | |
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| Brief Introduction to Group Representations | |
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| Basic Properties of Spherical Harmonics | |
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| Spherical Riemannian Geometry | |
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| Induced Metric on the Sphere | |
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| Induced Metrics of Spheres | |
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| Elements of Riemannian Geometry | |
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| Riemann Curvature of Spheres | |
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| Geodesics and Parallel Transport on the Sphere | |
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| Embedded-vector Viewpoint of the Geodesics | |
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| Beyond Quaternions | |
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| The Relationship of 4D Rotations to Quaternions | |
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| What Happened in Three Dimensions | |
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| Quaternions and Four Dimensions | |
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| Quaternions and the Four Division Algebras | |
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| Division Algebras | |
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| Relation to Fiber Bundles | |
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| Constructing the Hopf Fibrations | |
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| Clifford Algebras | |
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| Introduction to Clifford Algebras | |
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| Foundations | |
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| Examples of Clifford Algebras | |
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| Higher Dimensions | |
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| Pin(N), Spin(N), O(N), SO(N), and All That... | |
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| Conclusions | |
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| Appendices | |
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| Notation | |
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| Vectors | |
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| Length of a Vector | |
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| Unit Vectors | |
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| Polar Coordinates | |
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| Spheres | |
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| Matrix Transformations | |
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| Features of Square Matrices | |
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| Orthogonal Matrices | |
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| Vector Products | |
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| Complex Variables | |
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| 2D Complex Frames | |
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| 3D Quaternion Frames | |
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| Unit Norm | |
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| Multiplication Rule | |
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| Mapping to 3D rotations | |
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| Rotation Correspondence | |
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| Quaternion Exponential Form | |
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| Frame and Surface Evolution | |
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| Quaternion Frame Evolution | |
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| Quaternion Surface Evolution | |
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| Quaternion Survival Kit | |
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| Quaternion Methods | |
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| Quaternion Logarithms and Exponentials | |
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| The Quaternion Square Root Trick | |
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| The a to b formula simplified | |
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| Gram-Schmidt Spherical Interpolation | |
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| Direct Solution for Spherical Interpolation | |
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| Converting Linear Algebra to Quaternion Algebra | |
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| Useful Tensor Methods and Identities | |
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| Quaternion Path Optimization Using Surface Evolver | |
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| Quaternion Frame Integration | |
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| Hyperspherical Geometry | |
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| Definitions | |
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| Metric Properties | |
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| References | |
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| Index | |