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Visualizing Quaternions.pdf

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    Contents
    Foreword
    Preface
    Acknowledgments
    Elements of Quaternions
    The Discovery of Quaternions
    Hamilton's Walk
    Then Came Octonions
    The Quaternion Revival
    Folklore of Rotations
    The Belt Trick
    The Rolling Ball
    The Apollo 10 Gimbal-lock Incident
    3D Game Developer's Nightmare
    The Urban Legend of the Upside-down F16
    Quaternions to the Rescue
    Basic Notation
    Vectors
    Length of a Vector
    3D Dot Product
    3D 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
    2D Rotations
    Quaternions and 3D Rotations
    Recovering theta and n
    Euler Angles and Quaternions
    Optional Remarks
    Conclusion
    Visualizing Algebraic Structure
    Algebra of Complex Numbers
    Quaternion Algebra
    Visualizing Spheres
    2D: Visualizing an Edge-On Circle
    The Square Root Method
    3D: Visualizing a Balloon
    4D: Visualizing Quaternion Geometry on S3
    Visualizing Logarithms and Exponentials
    Complex Numbers
    Quaternions
    Visualizing Interpolation Methods
    Basics of Interpolation
    Quaternion Interpolation
    Equivalent 3 x 3 Matrix Method
    Looking at Elementary Quaternion Frames
    A Single Quaternion Frame
    Several Isolated Frames
    A Rotating Frame Sequence
    Synopsis
    Quaternions and the Belt Trick: Connecting to the Identity
    Very Interesting, but Why?
    The Details
    Frame-sequence Visualization Methods
    Quaternions and the Rolling Ball: Exploiting Order Dependence
    Order Dependence
    The Rolling Ball Controller
    Rolling Ball Quaternions
    Commutators
    Three degrees of freedom from two
    Quaternions and Gimbal Lock: Limiting the Available Space
    Guidance System Suspension
    Mathematical Interpolation Singularities
    Quaternion Viewpoint
    Advanced Quaternion Topics
    Alternative Ways of Writing Quaternions
    Hamilton's Generalization of Complex Numbers
    Pauli Matrices
    Other Matrix Forms
    Efficiency and Complexity Issues
    Extracting a Quaternion
    Efficiency of Vector Operations
    Advanced Sphere Visualization
    Projective Method
    Distance-preserving Flattening Methods
    More on Logarithms and Exponentials
    2D Rotations
    3D Rotations
    Using Logarithms for Quaternion Calculus
    Quaternion Interpolations Versus Log
    Two-Dimensional Curves
    Orientation Frames for 2D Space Curves
    What Is a Map?
    Tangent and Normal Maps
    Square Root Form
    Three-Dimensional Curves
    Introduction to 3D Space Curves
    General Curve Framings in 3D
    Tubing
    Classical Frames
    Mapping the Curvature and Torsion
    Theory of Quaternion Frames
    Assigning Smooth Quaternion Frames
    Examples: Torus Knot and Helix Quaternion Frames
    Comparison of Quaternion Frame Curve Lengths
    3D Surfaces
    Introduction to 3D Surfaces
    Quaternion Weingarten Equations
    Quaternion Gauss Map
    Example: The Sphere
    Examples: Minimal Surface Quaternion Maps
    Optimal Quaternion Frames
    Background
    Motivation
    Methodology
    The Space of Frames
    Choosing Paths in Quaternion Space
    Examples
    Quaternion Volumes
    Three-degree-of-freedom Orientation Domains
    Application to the Shoulder Joint
    Data Acquisition and the Double-covering Problem
    Application Data
    Quaternion Maps of Streamlines
    Visualization Methods
    3D Flow Data Visualizations
    Brushing: Clusters and Inverse Clusters
    Advanced Visualization Approaches
    Quaternion Interpolation
    Concepts of Euclidean Linear Interpolation
    The Double Quad
    Direct Interpolation of 3D Rotations
    Quaternion Splines
    Quaternion de Casteljau Splines
    Equivalent Anchor Points
    Angular Velocity Control
    Exponential-map Quaternion Interpolation
    Global Minimal Acceleration Method
    Quaternion Rotator Dynamics
    Static Frame
    Torque
    Quaternion Angular Momentum
    Concepts of the Rotation Group
    Brief Introduction to Group Representations
    Basic Properties of Spherical Harmonics
    Spherical Riemannian Geometry
    Induced Metric on the Sphere
    Induced Metrics of Spheres
    Elements of Riemannian Geometry
    Riemann Curvature of Spheres
    Geodesics and Parallel Transport on the Sphere
    Embedded-vector Viewpoint of the Geodesics
    Beyond Quaternions
    The Relationship of 4D Rotations to Quaternions
    What Happened in Three Dimensions
    Quaternions and Four Dimensions
    Quaternions and the Four Division Algebras
    Division Algebras
    Relation to Fiber Bundles
    Constructing the Hopf Fibrations
    Clifford Algebras
    Introduction to Clifford Algebras
    Foundations
    Examples of Clifford Algebras
    Higher Dimensions
    Pin(N), Spin(N), O(N), SO(N), and all that…
    Conclusions
    Part Appendices
    Notation
    Vectors
    Length of a Vector
    Unit Vectors
    Polar Coordinates
    Spheres
    Matrix Transformations
    Features of Square Matrices
    Orthogonal Matrices
    Vector Products
    Complex Variables
    2D Complex Frames
    3D Quaternion Frames
    Unit Norm
    Multiplication Rule
    Mapping to 3D rotations
    Rotation Correspondence
    Quaternion Exponential Form
    Frame and Surface Evolution
    Quaternion Frame Evolution
    Quaternion Surface Evolution
    Quaternion Survival Kit
    Quaternion Methods
    Quaternion Logarithms and Exponentials
    The Quaternion Square Root Trick
    The a->b formula simplified
    Gram-Schmidt Spherical Interpolation
    Direct Solution for Spherical Interpolation
    Converting Linear Algebra to Quaternion Algebra
    Useful Tensor Methods and Identities
    Quaternion Path Optimization Using Surface Evolver
    Quaternion Frame Integration
    Hyperspherical Geometry
    Definitions
    Metric Properties
    References
    Index
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    Almost all computer graphics practitioners have a good grasp of the 3D Cartesian space. However, in many graphics applications, orientations and rotations are equally important, and the concepts and tools related to rotations are less well-known. Quaternions are the key tool for understanding and manipulating orientations and rotations, and this book does a masterful job of making quaternions accessible. It excels not only in its scholarship, but also provides enough detailed figures and examples to expose the subtleties encountered when using quaternions. This is a book our field has needed for twenty years and I’m thrilled it is finally here. —Peter Shirley, Professor, University of Utah This book contains all that you would want to know about quaternions, including a great many things that you don’t yet realize that you want to know! —Alyn Rockwood, Vice President, ACM SIGGRAPH We need to use quaternions any time we have to interpolate orientations, for animating a camera move, simulating a rollercoaster ride, indicating fluid vorticity or displaying a folded protein, and it’s all too easy to do it wrong. This book presents gently but deeply the relationship between orientations in 3D and the differential geometry of the three-sphere in 4D that we all need to understand to be proficient in modern science and engineering, and especially computer graphics. — John C. Hart, Associate Professor, Department of Computer Science, Univer- sity of Illinois Urbana-Champaign, and Editor-in-Chief, ACM Transactions on Graphics Visualizing Quaternions is a comprehensive, yet superbly readable introduction to the concepts, mechanics, geometry, and graphical applications of Hamilton’s lasting contribution to the mathematical description of the real world. To write effectively on this subject, an author has to be a mathematician, physicist and computer scientist; Hanson is all three. Still, the reader can afford to be much less learned since the patient and detailed explanations makes this book an easy read. —George K. Francis, Professor, Mathematics Department, University of Illinois at Urbana-Champaign The new book, Visualizing Quaternions, will be welcomed by the many fans of Andy Hanson’s SIGGRAPH course. —Anselmo Lastra, University of North Carolina at Chapel Hill
    Andy Hanson’s expository yet scholarly book is a stunning tour de force; it is both long overdue, and a splendid surprise! Quaternions have been a perennial source of confusion for the computer graphics community, which sorely needs this book. His enthusiasm for and deep knowledge of the subject shines through his exceptionally clear prose, as he weaves together a story encompassing branches of mathematics from group theory to differential geometry to Fourier analysis. Hanson leads the reader through the thicket of interlocking mathematical frameworks using visualization as the path, providing geometric interpretations of quaternion properties. The first part of the book features a lucid explanation of how quaternions work that is suitable for a broad audience, covering such fundamental application areas as handling camera trajectories or the rolling ball interaction model. The middle section will inform even a mathematically sophisticated audience, with careful development of the more subtle implications of quaternions that have often been misunderstood, and presentation of less obvious quaternion applications such as visualizing vector field streamlines or the motion envelope of the human shoulder joint. The book concludes with a bridge to the mathematics of higher dimensional analogues to quaternions, namely octonions and Clifford algebra, that is designed to be accessible to computer scientists as well as mathematicians. —Tamara Munzner, University of British Columbia
    THE MORGAN KAUFMANN SERIES IN INTERACTIVE 3D TECHNOLOGY Series Editor: David H. Eberly, Geometric Tools, Inc. The game industry is a powerful and driving force in the evolution of computer technology. As the capabilities of personal computers, peripheral hardware, and game consoles have grown, so has the demand for quality information about the algorithms, tools, and descriptions needed to take advantage of this new technol- ogy. To satisfy this demand and establish a new level of professional reference for the game developer, we created the Morgan Kaufmann Series in Interactive 3D Technology. Books in the series are written for developers by leading industry professionals and academic researchers, and cover the state of the art in real-time 3D. The series emphasizes practical, working solutions and solid software-engineering principles. The goal is for the developer to be able to implement real systems from the funda- mental ideas, whether it be for games or for other applications. Visualizing Quaternions Andrew J. Hanson Better Game Characters by Design: A Psychological Approach Katherine Isbister Artificial Intelligence for Games Ian Millington 3D Game Engine Architecture: Engineering Real-Time Applications with Wild Magic David H. Eberly Real-Time Collision Detection Christer Ericson Physically Based Rendering: From Theory to Implementation Matt Pharr and Greg Humphreys Essential Mathematics for Games and Interactive Applications: A Programmer’s Guide James M. Van Verth and Lars M. Bishop Game Physics David H. Eberly Collision Detection in Interactive 3D Environments Gino van den Bergen 3D Game Engine Design: A Practical Approach to Real-Time Computer Graphics David H. Eberly Forthcoming Real-Time Cameras Mark Haigh-Hutchinson X3D: Extensible 3D Graphics for Web Authors Leonard Daly and Donald Brutzman Game Physics Engine Development Ian Millington
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    VISUALIZING QUATERNIONS
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