Visualizing quaternions

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.