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Visualizing Quaternions
1st Edition - November 15, 2004
Authors: Steve Cunningham, Andrew J. Hanson
Language: English
Hardback ISBN:9780120884001
9 7 8 - 0 - 1 2 - 0 8 8 4 0 0 - 1
eBook ISBN:9780080474779
9 7 8 - 0 - 0 8 - 0 4 7 4 7 7 - 9
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…Read more
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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.
Programmers and developers in computer graphics and the game industry, scientists and engineers working in aerospace and scientific visualization, students of game development and computer graphics, and those interested in quaternions but who have limited math background.
ABOUT THE AUTHOR
FOREWORD by Steve Cunningham
PREFACE
ACKNOWLEDGMENTS
PART I ELEMENTS OF QUATERNIONS
01 THE DISCOVERY OF QUATERNIONS
1.1 Hamilton's Walk
1.2 Then Came Octonions
1.3 The Quaternion Revival
02 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
03 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
04 WHAT ARE QUATERNIONS?
05 ROAD MAP TO QUATERNION VISUALIZATION
5.1 The Complex Number Connection
5.2 The Cornerstones of Quaternion Visualization
06 FUNDAMENTALS OF ROTATIONS
6.1 2D Rotations
6.1.1 Relation to Complex Numbers
6.1.2 The Half-angle Form
6.1.3 Complex Exponential Version
6.2 Quaternions and 3D Rotations
6.2.1 Construction
6.2.2 Quaternions and Half Angles
6.2.3 Double Values
6.3 Recovering Θ and n
6.4 Euler Angles and Quaternions
6.5 † Optional Remarks
6.5.1 † Connections to Group Theory
6.5.2 † "Pure" Quaternion Derivation
6.5.3 † Quaternion Exponential Version
6.6 Conclusion
07 VISUALIZING ALGEBRAIC STRUCTURE
7.1 Algebra of Complex Numbers
7.1.1 Complex Numbers
7.1.2 Abstract View of Complex Multiplication
7.1.3 Restriction to Unit-length Case
7.2 Quaternion Algebra
7.2.1 The Multiplication Rule
7.2.2 Scalar Product
7.2.3 Modulus of the Quaternion Product
7.2.4 Preservation of the Unit Quaternions
08 VISUALIZING SPHERES
8.1 2D: Visualizing an Edge-on Circle
8.1.1 Trigonometric Function Method
8.1.2 Complex Variable Method
8.1.3 Square Root Method
8.2 The Square Root Method
8.3 3D: Visualizing a Balloon
8.3.1 Trigonometric Function Method
8.3.2 Square Root Method
8.4 4D: Visualizing Quaternion Geometry on S3
8.4.1 Seeing the Parameters of a Single Quaternion
8.4.2 Hemispheres in S3
09 VISUALIZING LOGARITHMS AND EXPONENTIALS
9.1 Complex Numbers
9.2 Quaternions
10 VISUALIZING INTERPOLATION METHODS
10.1 Basics of Interpolation
10.1.1 Interpolation Issues
10.1.2 Gram-Schmidt Derivation of the SLERP
10.1.3 † Alternative Derivation
10.2 Quaternion Interpolation
10.3 Equivalent 3×3 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?
12.1.1 The Intuitive Answer
12.1.2 † The Technical Answer
12.2 The Details
12.3 Frame-sequence Visualization Methods
12.3.1 One Rotation
12.3.2 Two Rotations
12.3.3 Synopsis
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
16.1.1 Positive Trace R
16.1.2 Nonpositive Trace R
16.2 Efficiency of Vector Operations
17 ADVANCED SPHERE VISUALIZATION
17.1 Projective Method
17.1.1 The Circle S1
17.1.2 General SN Polar Projection
17.2 Distance-preserving Flattening Methods
17.2.1 Unroll-and-Flatten S1
17.2.2 S2 Flattened Equal-area Method
17.2.3 S3 Flattened Equal-volume Method
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
19.1.1 2D Rotation Matrices
19.1.2 The Frame Matrix in 2D
19.1.3 Frame Evolution in 2D
19.2 What Is a Map?
19.3 Tangent and Normal Maps
19.4 Square Root Form
19.4.1 Frame Evolution in (a, b)
19.4.2 Simplifying the Frame Equations
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
20.4.1 Frenet-Serret Frame
20.4.2 Parallel Transport Frame
20.4.3 Geodesic Reference Frame
20.4.4 General Frames
20.5 Mapping the Curvature and Torsion
20.6 Theory of Quaternion Frames
20.6.1 Generic Quaternion Frame Equations
20.6.2 Quaternion Frenet Frames
20.6.3 Quaternion Parallel Transport Frames
20.7 Assigning Smooth Quaternion Frames
20.7.1 Assigning Quaternions to Frenet Frames
20.7.2 Assigning Quaternions to Parallel Transport Frames
20.8 Examples: Torus Knot and Helix Quaternion Frames
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.9.1 2D Dot Product
A.9.2 2D Cross Product
A.9.3 3D Dot Product
A.9.4 3D Cross Product
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
F.7.1 Einstein Summation Convention
F.7.2 Kronecker Delta
F.7.3 Levi-Civita Symbol
G QUATERNION PATH OPTIMIZATION USING SURFACE EVOLVER
H QUATERNION FRAME INTEGRATION
I HYPERSPHERICAL GEOMETRY
I.1 Definitions
I.2 Metric Properties
REFERENCES
INDEX
No. of pages: 536
Language: English
Edition: 1
Published: November 15, 2004
Imprint: Morgan Kaufmann
Hardback ISBN: 9780120884001
eBook ISBN: 9780080474779
SC
Steve Cunningham
Affiliations and expertise
California State University Stanislaus, U.S.A.
AH
Andrew J. Hanson
Andrew J. Hanson Ph.D. is an Emeritus Professor of Computer Science at Indiana University. He earned a bachelor’s degree in Chemistry and Physics from Harvard University in 1966 and a PhD in Theoretical Physics from MIT under Kerson Huang in 1971. His interests range from general relativity to computer graphics, artificial intelligence, and bioinformatics; he is particularly concerned with applications of quaternions and with exploitation of higher-dimensional graphics for the visualization of complex scientific contexts such as Calabi-Yau spaces. He is the co-discoverer of the Eguchi-Hanson “gravitational instanton” Einstein metric (1978), author of Visualizing Quaternions (Elsevier, 2006), and designer of the iPhone Apps “4Dice” and “4DRoom” (2012) for interacting with four-dimensional virtual reality.
Affiliations and expertise
Emeritus Professor of Computer Science, Indiana University, IN, USA