Visualizing Quaternions

Andrew J. Hanson

  • 出版商: Morgan Kaufmann
  • 出版日期: 2006-01-12
  • 售價: $3,590
  • 貴賓價: 9.5$3,411
  • 語言: 英文
  • 頁數: 530
  • 裝訂: Hardcover
  • ISBN: 0120884003
  • ISBN-13: 9780120884001

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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.

 

Table Of Contents

    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
        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 ab 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