Rethinking Quaternions

Book description


Quaternion multiplication can be used to rotate vectors in three-dimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is well-known in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this monograph are

to provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points;

to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane;

to derive the formula for quaternion multiplication from first principles;

to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection;

to show how to apply sandwiching to compute perspective projections.

In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolate between two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text.

Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography

Table of contents

  1. Rethinking Quaternions
  2. Synthesis Lectures on Computer Graphics and Animation
  3. ABSTRACT
  4. Keywords
  5. Contents
  6. Preface (1/2)
  7. Preface (2/2)
  8. part I Theory
  9. chapter 1 Complex Numbers (1/2)
  10. chapter 1 Complex Numbers (2/2)
  11. chapter 2 A Brief History of Number Systems and Multiplication
    1. 2.1 MULTIPLICATION IN DIMENSIONS GREATER THAN TWO
  12. chapter 3 Modeling Quaternions
    1. 3.1 MASS-POINTS: A CLASSICAL MODEL FOR CONTEMPORARY COMPUTER GRAPHICS
    2. 3.2 ARROWS IN FOUR DIMENSIONS
    3. 3.3 MUTUALLY ORTHOGONAL PLANES IN FOUR DIMENSIONS
  13. chapter 4 The Algebra of Quaternion Multiplication (1/2)
  14. chapter 4 The Algebra of Quaternion Multiplication (2/2)
  15. chapter 5 the Geometry of Quaternion Multiplication (1/2)
  16. chapter 5 the Geometry of Quaternion Multiplication (2/2)
  17. chapter 6 Affine, Semi-Affine, and Projective Transformations in Three Dimensions
    1. 6.1 ROTATION
    2. 6.2 MIRROR IMAGE
    3. 6.3 PERSPECTIVE PROJECTION
      1. 6.3.1 Perspective Projection and Singular 4 × 4 Matrices
      2. 6.3.2 Perspective Projection by Sandwiching with Quaternions (1/2)
      3. 6.3.2 Perspective Projection by Sandwiching with Quaternions (2/2)
    4. 6.4 ROTORPERSPECTIVES AND ROTOREFLECTIONS
  18. chapter 7 Recapitulation: Insights and Results
  19. part II Computation
  20. chapter 8 Matrix Representations for Rotations, Reflections, and Perspective Projections
    1. 8.1 MATRIX REPRESENTATIONS FOR QUATERNION MULTIPLICATION
    2. 8.2 MATRIX REPRESENTATIONS FOR ROTATIONS
    3. 8.3 MATRIX REPRESENTATIONS FOR MIRROR IMAGES
    4. 8.4 MATRIX REPRESENTATIONS FOR PERSPECTIVE PROJECTIONS
  21. chapter 9 Applications
    1. 9.1 EFFICIENCY: QUATERNIONS VERSUS MATRICES
    2. 9.2 AVOIDING DISTORTION BY RENORMALIZATION
    3. 9.3 KEY FRAME ANIMATION AND SPHERICAL LINEAR INTERPOLATION
  22. chapter 10 Summary—Formulas From Quaternion Algebra (1/2)
  23. chapter 10 Summary—Formulas From Quaternion Algebra (2/2)
  24. part III Rethinking Quaternions and Clifford Algebras
    1. chapter 11 Goals and Motivation
  25. Bookmark 3
  26. chapter 12 Clifford Algebras and Quaternions
  27. chapter 13 Clifford Algebra for the Plane
  28. chapter 14 The Standard Model of the Clifford Algebra for Three Dimensions
    1. 14.1 SCALARS, VECTORS, BIVECTORS, AND PSEUDOSCALARS
    2. 14.2 WEDGE PRODUCT AND CROSS PRODUCT
    3. 14.3 DUALITY
    4. 14.4 BIVECTORS
    5. 14.5 QUATERNIONS
  29. chapter 15 Operands and Operators-Mass---Points and Quaternions
    1. 15.1 ODD ORDER: MASS-POINTS
    2. 15.2 EVEN ORDER: QUATERNIONS
  30. chapter 16 Decomposing Mass-Points Into Two Mutually Orthogonal Planes
    1. 16.1 ACTION OF q (b, q ) ON b ⊥
    2. 16.2 ACTION OF q (b, q ) ON b||
    3. 16.3 SANDWICHING
  31. chapter 17 Rotation, Reflection, and Perspective Projection
    1. 17.1 ROTATION
    2. 17.2 MIRROR IMAGE
    3. 17.3 PERSPECTIVE PROJECTION
  32. chapter 18 Summary
  33. chapter 19 Some SImple Alternative Homogenous Models for Computer Graphics
  34. References
  35. Further Reading
  36. Author Biography

Product information

  • Title: Rethinking Quaternions
  • Author(s): Ron Goldman
  • Release date: May 2010
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781608454211