Visualizing Quaternions 1st edition by Andrew Hanson – Ebook PDF Instant Download/Delivery: 0080474772, 9780080474779
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ISBN 10: 0080474772
ISBN 13: 9780080474779
Author: Andrew Hanson
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.
- Richly 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.
Visualizing Quaternions 1st Table of contents:
Part I Elements of Quaternions
1 The Discovery of Quaternions
1.1 Hamilton’s Walk
1.2 Then Came Octonions
1.3 The Quaternion Revival
2 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
3 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
4 What Are Quaternions?
5 Road Map to Quaternion Visualization
5.1 The Complex Number Connection
5.2 The Cornerstones of Quaternion Visualization
6 Fundamentals of Rotations
6.1 2D Rotations
6.2 Quaternions and 3D Rotations
6.3 Recovering theta and n
6.4 Euler Angles and Quaternions
6.5 † Optional Remarks
6.6 Conclusion
7 Visualizing Algebraic Structure
7.1 Algebra of Complex Numbers
7.2 Quaternion Algebra
8 Visualizing Spheres
8.1 2D: Visualizing an Edge-On Circle
8.2 The Square Root Method
8.3 3D: Visualizing a Balloon
8.4 4D: Visualizing Quaternion Geometry on S3
9 Visualizing Logarithms and Exponentials
9.1 Complex Numbers
9.2 Quaternions
10 Visualizing Interpolation Methods
10.1 Basics of Interpolation
10.2 Quaternion Interpolation
10.3 Equivalent 3 x 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?
12.2 The Details
12.3 Frame-sequence Visualization Methods
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
16.2 Efficiency of Vector Operations
17 Advanced Sphere Visualization
17.1 Projective Method
17.2 Distance-preserving Flattening Methods
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
19.2 What Is a Map?
19.3 Tangent and Normal Maps
19.4 Square Root Form
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
20.5 Mapping the Curvature and Torsion
20.6 Theory of Quaternion Frames
20.7 Assigning Smooth Quaternion 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
21.2 Quaternion Weingarten Equations
21.3 Quaternion Gauss Map
21.4 Example: The Sphere
21.5 Examples: Minimal Surface Quaternion Maps
22 Optimal Quaternion Frames
22.1 Background
22.2 Motivation
22.3 Methodology
22.4 The Space of Frames
22.5 Choosing Paths in Quaternion Space
22.6 Examples
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
23.4 Application Data
24 Quaternion Maps of Streamlines
24.1 Visualization Methods
24.2 3D Flow Data Visualizations
24.3 Brushing: Clusters and Inverse Clusters
24.4 Advanced Visualization Approaches
25 Quaternion Interpolation
25.1 Concepts of Euclidean Linear Interpolation
25.2 The Double Quad
25.3 Direct Interpolation of 3D Rotations
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
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
27.2 Basic Properties of Spherical Harmonics
28 Spherical Riemannian Geometry
28.1 Induced Metric on the Sphere
28.2 Induced Metrics of Spheres
28.3 Elements of Riemannian Geometry
28.4 Riemann Curvature of Spheres
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
30.2 Relation to Fiber Bundles
30.3 Constructing the Hopf Fibrations
31 Clifford Algebras
31.1 Introduction to Clifford Algebras
31.2 Foundations
31.3 Examples of Clifford Algebras
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.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
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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