book

Visualizing Quaternions

Name: Visualizing Quaternions
Author: Andrew J. Hanson
ISBN: 9780080474779

by Andrew J. Hanson

February 2006

Intermediate to advanced

600 pages

8h 57m

English

Elsevier Science

Read now

Unlock full access

Copyright
Dedication
The Morgan Kaufmann Series in Interactive 3D Technology
About the Author
Foreword
Preface
Acknowledgments
I. Elements of Quaternions
01. The Discovery of Quaternions
1.1. Hamilton’s Walk1.2. Then Came Octonions1.3. The Quaternion Revival
02. Folklore of Rotations
2.1. The Belt Trick2.2. The Rolling Ball2.3. The Apollo 10 Gimbal-Lock Incident2.4. 3D Game Developer’s Nightmare2.5. The Urban Legend of the Upside-down F162.6. Quaternions to the Rescue

03. Basic Notation
3.1. Vectors3.2. Length of a Vector3.3. 3D Dot Product3.4. 3D Cross Product3.5. Unit Vectors3.6. Spheres3.7. Matrices3.8. Complex Numbers
04. What Are Quaternions?
05. Road Map to Quaternion Visualization
5.1. The Complex Number Connection5.2. The Cornerstones of Quaternion Visualization
06. Fundamentals of Rotations
6.1. 2D Rotations6.1.1. Relation to Complex Numbers6.1.2. The Half-angle Form6.1.3. Complex Exponential Version6.2. Quaternions and 3D Rotations6.2.1. Construction6.2.2. Quaternions and Half Angles6.2.3. Double Values6.3. Recovering θ and6.4. Euler Angles and Quaternions6.5. † Optional Remarks6.5.1. † Connections to Group Theory6.5.2. † “Pure” Quaternion Derivation6.5.3. † Quaternion Exponential Version6.6. Conclusion
07. Visualizing Algebraic Structure
7.1. Algebra of Complex Numbers7.1.1. Complex Numbers7.1.2. Abstract View of Complex Multiplication7.1.3. Restriction to Unit-length Case7.2. Quaternion Algebra7.2.1. The Multiplication Rule7.2.2. Scalar Product7.2.3. Modulus of the Quaternion Product7.2.4. Preservation of the Unit Quaternions
08. Visualizing Spheres
8.1. 2D: Visualizing an Edge-on Circle8.1.1. Trigonometric Function Method8.1.2. Complex Variable Method8.1.3. Square Root Method8.2. The Square Root Method8.3. 3D: Visualizing a Balloon8.3.1. Trigonometric Function Method8.3.2. Square Root Method8.4. 4D: Visualizing Quaternion Geometry on S38.4.1. Seeing the Parameters of a Single Quaternion8.4.2. Hemispheres in S3
09. Visualizing Logarithms and Exponentials
9.1. Complex Numbers9.2. Quaternions
10. Visualizing Interpolation Methods
10.1. Basics of Interpolation10.1.1. Interpolation Issues10.1.2. Gram-Schmidt Derivation of the SLERP10.1.3. † Alternative Derivation10.2. Quaternion Interpolation10.3. Equivalent 3×3 Matrix Method
11. Looking at Elementary Quaternion Frames
11.1. A Single Quaternion Frame11.2. Several Isolated Frames11.3. A Rotating Frame Sequence11.4. Synopsis
12. Quaternions and the Belt Trick: Connecting to the Identity
12.1. Very Interesting, but Why?12.1.1. The Intuitive Answer12.1.2. † The Technical Answer12.2. The Details: How Quaternion Visualization Explains the Belt Trick12.3. Frame-sequence Visualization Methods12.3.1. One Rotation12.3.2. Two Rotations12.3.3. Synopsis
13. Quaternions and the Rolling Ball: Exploiting Order Dependence
13.1. Order Dependence13.2. The Rolling Ball Controller13.3. Rolling Ball Quaternions13.4. † Commutators13.5. Three Degrees of Freedom From Two
14. Quaternions and Gimbal Lock: Limiting the Available Space
14.1. Guidance System Suspension14.2. Mathematical Interpolation Singularities14.3. Quaternion Viewpoint
II. Advanced Quaternion Topics
15. Alternative Ways of Writing Quaternions
15.1. Hamilton’s Generalization of Complex Numbers15.2. Pauli Matrices15.3. Other Matrix Forms
16. Efficiency and Complexity Issues
16.1. Extracting a Quaternion16.1.1. Positive Trace R16.1.2. Nonpositive Trace R16.2. Efficiency of Vector Operations
17. Advanced Sphere Visualization
17.1. Projective Method17.1.1. The Circle S117.1.2. General SN Polar Projection17.2. Distance-preserving Flattening Methods17.2.1. Unroll-and-Flatten S117.2.2. S2 Flattened Equal-Area Method17.2.3. S3 Flattened Equal-volume Method
18. More on Logarithms and Exponentials
18.1. 2D Rotations18.2. 3D Rotations18.3. Using Logarithms for Quaternion Calculus18.4. Quaternion Interpolations Versus Log
19. Two-Dimensional Curves
19.1. Orientation Frames for 2D Space Curves19.1.1. 2D Rotation Matrices19.1.2. The Frame Matrix in 2D19.1.3. Frame Evolution in 2D19.2. What Is a Map?19.3. Tangent and Normal Maps19.4. Square Root Form19.4.1. Frame Evolution in (a, b)19.4.2. Simplifying the Frame Equations
20. Three-Dimensional Curves
20.1. Introduction to 3D Space Curves20.2. General Curve Framings in 3D20.3. Tubing20.4. Classical Frames20.4.1. Frenet–Serret Frame20.4.2. Parallel Transport Frame20.4.3. Geodesic Reference Frame20.4.4. General Frames20.5. Mapping the Curvature and Torsion20.6. Theory of Quaternion Frames20.6.1. Generic Quaternion Frame Equations20.6.2. Quaternion Frenet Frames20.6.3. Quaternion Parallel Transport Frames20.7. Assigning Smooth Quaternion Frames20.7.1. Assigning Quaternions to Frenet Frames20.7.2. Assigning Quaternions to Parallel Transport Frames20.8. Examples: Torus Knot and Helix Quaternion Frames20.9. Comparison of Quaternion Frame Curve Lengths
21. 3D Surfaces
21.1. Introduction to 3D Surfaces21.1.1. Classical Gauss Map21.1.2. Surface Frame Evolution21.1.3. Examples of Surface Framings21.2. Quaternion Weingarten Equations21.2.1. Quaternion Frame Equations21.2.2. Quaternion Surface Equations (Weingarten Equations)21.3. Quaternion Gauss Map21.4. Example: The Sphere21.4.1. Quaternion Maps of Alternative Sphere Frames21.4.2. Covering the Sphere and the Geodesic Reference Frame South Pole Singularity21.5. Examples: Minimal Surface Quaternion Maps
22. Optimal Quaternion Frames
22.1. Background22.2. Motivation22.3. Methodology22.3.1. The Space of Possible Frames22.3.2. Parallel Transport and Minimal Measure22.4. The Space of Frames22.4.1. Full Space of Curve Frames22.4.2. Full Space of Surface Maps22.5. Choosing Paths in Quaternion Space22.5.1. Optimal Path Choice Strategies22.5.2. General Remarks on Optimization in Quaternion Space22.6. Examples22.6.1. Minimal Quaternion Frames for Space Curves22.6.2. Minimal-quaternion-area Surface Patch Framings
23. Quaternion Volumes
23.1. Three-degree-of-freedom Orientation Domains23.2. Application to the Shoulder Joint23.3. Data Acquisition and the Double-covering Problem23.3.1. Sequential Data23.3.2. The Sequential Nearest-neighbor Algorithm23.3.3. The Surface-based Nearest-neighbor Algorithm23.3.4. The Volume-based Nearest-neighbor Algorithm23.4. Application Data
24. Quaternion Maps of Streamlines
24.1. Visualization Methods24.1.1. Direct Plot of Quaternion Frame Fields24.1.2. Similarity Measures for Quaternion Frames24.1.3. Exploiting or Ignoring Double Points24.2. 3D Flow Data Visualizations24.2.1. AVS Streamline Example24.2.2. Deforming Solid Example24.3. Brushing: Clusters and Inverse Clusters24.4. Advanced Visualization Approaches24.4.1. 3D Rotations of Quaternion Displays24.4.2. Probing Quaternion Frames with 4D Light
25. Quaternion Interpolation
25.1. Concepts of Euclidean Linear Interpolation25.1.1. Constructing Higher-order Polynomial Splines25.1.2. Matching25.1.3. Schlag’s Method25.1.4. Control-point Method25.2. The Double Quad25.3. Direct Interpolation of 3D Rotations25.3.1. Relation to Quaternions25.3.2. Method for Arbitrary Origin25.3.3. Exponential Version25.3.4. Special Vector-Vector Case25.3.5. Multiple-level Interpolation Matrices25.3.6. Equivalence of Quaternion and Matrix Forms25.4. Quaternion Splines25.5. Quaternion de Casteljau Splines25.6. Equivalent Anchor Points25.7. Angular Velocity Control25.8. Exponential-map Quaternion Interpolation25.9. Global Minimal Acceleration Method25.9.1. Why a Cubic?25.9.2. Extension to Quaternion Form
26. Quaternion Rotator Dynamics
26.1. Static Frame26.2. Torque26.3. Quaternion Angular Momentum
27. Concepts of the Rotation Group
27.1. Brief Introduction to Group Representations27.1.1. Complex Versus Real27.1.2. What Is a Representation?27.2. Basic Properties of Spherical Harmonics27.2.1. Representations and Rotation-invariant Properties27.2.2. Properties of Expansion Coefficients Under Rotations
28. Spherical Riemannian Geometry
28.1. Induced Metric on the Sphere28.2. Induced Metrics of Spheres28.2.1. S1 Induced Metrics28.2.2. S2 Induced Metrics28.2.3. S3 Induced Metrics28.2.4. Toroidal Coordinates on S328.2.5. Axis-angle Coordinates on S328.2.6. General form for the Square-root Induced Metric28.3. Elements of Riemannian Geometry28.4. Riemann Curvature of Spheres28.4.1. S128.4.2. S228.4.3. S328.5. Geodesics and Parallel Transport on the Sphere28.6. Embedded-vector Viewpoint of the Geodesics
III. Beyond Quaternions
29. The Relationship of 4D Rotations to Quaternions
29.1. What Happened in Three Dimensions29.2. Quaternions and Four Dimensions
30. Quaternions and the Four Division Algebras
30.1. Division Algebras30.1.1. The Number Systems with Dimensions 1, 2, 4, and 830.1.2. Parallelizable Spheres30.2. Relation to Fiber Bundles30.3. Constructing the Hopf Fibrations30.3.1. Real: S0 fiber + S1 base = S1 bundle30.3.2. Complex: S1 fiber + S2 base = S3 bundle30.3.3. Quaternion: S3 fiber + S4 base = S7 bundle30.3.4. Octonion: S7 fiber + S8 base = S15 bundle
31. Clifford Algebras
31.1. Introduction to Clifford Algebras31.2. Foundations31.2.1. Clifford Algebras and Rotations31.2.2. Higher-dimensional Clifford Algebra Rotations31.3. Examples of Clifford Algebras31.3.1. 1D Clifford Algebra31.3.2. 2D Clifford Algebra31.3.3. 2D Rotations Done Right31.3.4. 3D Clifford Algebra31.3.5. Clifford Implementation of 3D Rotations31.4. Higher Dimensions31.5. Pin(N), Spin(N), O(N), SO(N), and All That...
32. Conclusions
Appendices
A. Notation
A.1. VectorsA.2. Length of a VectorA.3. Unit VectorsA.4. Polar CoordinatesA.5. SpheresA.6. Matrix TransformationsA.7. Features of Square MatricesA.8. Orthogonal MatricesA.9. Vector ProductsA.9.1. 2D Dot ProductA.9.2. 2D Cross ProductA.9.3. 3D Dot ProductA.9.4. 3D Cross ProductA.10. Complex Variables
B. 2D Complex Frames
C. 3D Quaternion Frames
C.1. Unit NormC.2. Multiplication RuleC.3. Mapping to 3D rotationsC.4. Rotation CorrespondenceC.5. Quaternion Exponential Form
D. Frame and Surface Evolution
D.1. Quaternion Frame EvolutionD.2. Quaternion Surface Evolution
E. Quaternion Survival Kit
F. Quaternion Methods
F.1. Quaternion Logarithms and ExponentialsF.2. The Quaternion Square Root TrickF.3. The â → Formula SimplifiedF.4. Gram–Schmidt Spherical InterpolationF.5. Direct Solution for Spherical InterpolationF.6. Converting Linear Algebra to Quaternion AlgebraF.7. Useful Tensor Methods and IdentitiesF.7.1. Einstein Summation ConventionF.7.2. Kronecker DeltaF.7.3. Levi-Civita Symbol
G. Quaternion Path Optimization Using Surface Evolver
H. Quaternion Frame Integration
I. Hyperspherical Geometry
I.1. DefinitionsI.2. Metric Properties
References

Content preview from Visualizing Quaternions

Chapter 14. Quaternions and Gimbal Lock: Limiting the Available Space

One of the classic problems with using ordinary 3 × 3 matrices to represent rotations is the possibility of gimbal lock, a phenomenon we explored qualitatively in our introduction. Now we have enough tools to look more carefully at the phenomenon, and to work toward exploring how to understand it, when to expect it, and how to avoid it.

There are basically two related, but quite distinct, phenomena that are referred to as gimbal lock. These are discussed in the sections that follow.

Guidance System Suspension

Figure 14.1 shows the basic configuration of a set of rotatable rings that typically house an inertial guidance system. There is a very specific mechanical problem that can ...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.

Read now

Unlock full access

More than 5,000 organizations count on O’Reilly

O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.

Julian F.

Head of Cybersecurity

I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.

Addison B.

Field Engineer

I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.

Amir M.

Data Platform Tech Lead

I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.

Mark W.

Embedded Software Engineer

Publisher Resources

ISBN: 9780120884001

Cloud Computing

Data Engineering

Data Science

AI & ML

Programming Languages

Software Architecture

IT/Ops

Security

Design

Business

Soft Skills

Visualizing Quaternions

by Andrew J. Hanson

Chapter 14. Quaternions and Gimbal Lock: Limiting the Available Space

Guidance System Suspension

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.