Description

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.

 

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 S^N 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
20.9 Comparison of Quaternion Frame Curve Lengths


21 3D SURFACES

21.1 Introduction to 3D Surfaces

21.1.1 Classical Gauss Map
21.1.2 Surface Frame Evolution
21.1.3 Examples of Surface Framings

21.2 Quaternion Weingarten Equations

21.2.1 Quaternion Frame Equations
21.2.2 Quaternion Surface Equations (Weingarten Equations)

21.3 Quaternion Gauss Map
21.4 Example: The Sphere

21.4.1 Quaternion Maps of Alternative Sphere Frames
21.4.2 Covering the Sphere and the Geodesic Reference Frame South Pole Singularity

21.5 Examples: Minimal Surface Quaternion Maps


22 OPTIMAL QUATERNION FRAMES

22.1 Background
22.2 Motivation
22.3 Methodology

22.3.1 The Space of Possible Frames
22.3.2 Parallel Transport and Minimal Measure

22.4 The Space of Frames

22.4.1 Full Space of Curve Frames
22.4.2 Full Space of Surface Maps

22.5 Choosing Paths in Quaternion Space

22.5.1 Optimal Path Choice Strategies
22.5.2 General Remarks on Optimization in Quaternion Space

22.6 Examples

22.6.1 Minimal Quaternion Frames for Space Curves
22.6.2 Minimal-quaternion-area Surface Patch Framings


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

23.3.1 Sequential Data
23.3.2 The Sequential Nearest-neighbor Algorithm
23.3.3 The Surface-based Nearest-neighbor Algorithm
23.3.4 The Volume-based Nearest-neighbor Algorithm

23.4 Application Data


24 QUATERNION MAPS OF STREAMLINES

24.1 Visualization Methods

24.1.1 Direct Plot of Quaternion Frame Fields
24.1.2 Similarity Measures for Quaternion Frames
24.1.3 Exploiting or Ignoring Double Points

24.2 3D Flow Data Visualizations

24.2.1 AVS Streamline Example
24.2.2 Deforming Solid Example

24.3 Brushing: Clusters and Inverse Clusters
24.4 Advanced Visualization Approaches

24.4.1 3D Rotations of Quaternion Displays
24.4.2 Probing Quaternion Frames with 4D Light


25 QUATERNION INTERPOLATION

25.1 Concepts of Euclidean Linear Interpolation

25.1.1 Constructing Higher-order Polynomial Splines
25.1.2 Matching
25.1.3 Schlag's Method
25.1.4 Control-point Method

25.2 The Double Quad
25.3 Direct Interpolation of 3D Rotations

25.3.1 Relation to Quaternions
25.3.2 Method for Arbitrary Origin
25.3.3 Exponential Version
25.3.4 Special Vector-Vector Case
25.3.5 Multiple-level Interpolation Matrices
25.3.6 Equivalence of Quaternion and Matrix Forms

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

25.9.1 Why a Cubic?
25.9.2 Extension to Quaternion Form


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

27.1.1 Complex Versus Real
27.1.2 What Is a Representation?

27.2 Basic Properties of Spherical Harmonics

27.2.1 Representations and Rotation-invariant Properties
27.2.2 Properties of Expansion Coefficients Under Rotations


28 SPHERICAL RIEMANNIAN GEOMETRY

28.1 Induced Metric on the Sphere
28.2 Induced Metrics of Spheres

28.2.1 S¹ Induced Metrics
28.2.2 S² Induced Metrics
28.2.3 S³ Induced Metrics
28.2.4 Toroidal Coordinates on S³
28.2.5 Axis-angle Coordinates on S³
28.2.6 General Form for the Square-root Induced Metric

28.3 Elements of Riemannian Geometry
28.4 Riemann Curvature of Spheres

28.4.1 S¹
28.4.2 S²
28.4.3 S³

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

30.1.1 The Number Systems with Dimensions 1, 2, 4, and 8
30.1.2 Parallelizable Spheres

30.2 Relation to Fiber Bundles
30.3 Constructing the Hopf Fibrations

30.3.1 Real: S⁰ fiber + S¹ base = S¹ bundle
30.3.2 Complex: S¹ fiber + S² base = S³ bundle
30.3.3 Quaternion: S³ fiber + S⁴ base = S⁷ bundle
30.3.4 Octonion: S⁷ fiber + S⁸ base = S¹⁵ bundle


31 CLIFFORD ALGEBRAS

31.1 Introduction to Clifford Algebras
31.2 Foundations

31.2.1 Clifford Algebras and Rotations
31.2.2 Higher-dimensional Clifford Algebra Rotations

31.3 Examples of Clifford Algebras

31.3.1 1D Clifford Algebra
31.3.2 2D Clifford Algebra
31.3.3 2D Rotations Done Right
31.3.4 3D Clifford Algebra
31.3.5 Clifford Implementation of 3D Rotations

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.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