Visualizing Quaternions

Andrew J. Hanson - Collection Interactive 3D Technology

500 pages, parution le 15/02/2006

Ajouter à une liste

Indisponible

Résumé

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.

Sommaire

Elements of quaternions
- The discovery of quaternions
- Folklore of rotations
- Basic notation
- What are quaternions?
- Road map to quaternion visualization
- Fundamentals of rotations
- Visualizing algebraic structure
- Visualizing spheres
- Visualizing logarithms and exponentials
- Visualizing interpolation methods
- Looking at elementary quaternion frames
- Quaternions and the belt trick: connecting to the identity
- Quaternions and the rolling ball: exploiting order dependence
- Quaternions and gimbal lock: limiting the available space
Advanced quaternion topics
- Alternative ways of writing quaternions
- Efficiency and complexity issues
- Advanced sphere visualization
- More on logarithms and exponentials
- Two-dimensional curves
- Three-dimensional curves
- 3d surfaces
- Optimal quaternion frames
- Quaternion volumes
- Quaternion maps of streamlines
- Quaternion interpolation
- Quaternion rotator dynamics
- Concepts of the rotation group
- Spherical riemannian geometry
Beyond quaternions
- The relationship of 4d rotations to quaternions
- Quaternions and the four division algebras
- Clifford algebras
- Conclusions
- Appendices

Voir tout

Replier

Caractéristiques techniques

	PAPIER
Éditeur(s)	Morgan Kaufmann
Auteur(s)	Andrew J. Hanson
Collection	Interactive 3D Technology
Parution	15/02/2006
Nb. de pages	500
Format	20 x 24
Couverture	Relié
Poids	1240g
Intérieur	Noir et Blanc
EAN13	9780120884001
ISBN13	978-0-12-088400-1