Visualizing quaternions / Andrew J. Hanson.

Includes bibliographical references (pages 471-486) and index.

Cover -- 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 --T.

"Andrew Hanson's new book is a fresh perspective on quaternions. Features include: illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing; covers both non-mathematical and mathematical approaches to quaternions; and a companion website with an assortment of quaternion utilities and sample code, data sets for the book's illustrations, and Mathematica notebooks with essential algebraic utilities."--Jacket.

Print version record.

English.

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