Visualizing Quaternions

Visualizing Quaternions

Author: Andrew J. Hanson

Publisher: Elsevier

ISBN: 9780080474779

Category: Mathematics

Page: 530

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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. Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. Covers both non-mathematical and mathematical approaches to quaternions.
Visualizing Quaternions
Language: en
Pages: 530
Authors: Andrew J. Hanson
Categories: Mathematics
Type: BOOK - Published: 2006-02-06 - Publisher: Elsevier

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
Rethinking Quaternions
Language: en
Pages: 157
Authors: Ron Goldman
Categories: Computers
Type: BOOK - Published: 2010 - Publisher: Morgan & Claypool Publishers

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,
3D Math Primer for Graphics and Game Development, 2nd Edition
Language: en
Pages: 846
Authors: Fletcher Dunn, Ian Parberry
Categories: Computers
Type: BOOK - Published: 2011-11-02 - Publisher: CRC Press

This engaging book presents the essential mathematics needed to describe, simulate, and render a 3D world. Reflecting both academic and in-the-trenches practical experience, the authors teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. The text provides an introduction to mathematics for game
Quaternion Algebras
Language: en
Pages: 885
Authors: John Voight
Categories: Mathematics
Type: BOOK - Published: 2021-06-28 - Publisher: Springer Nature

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer
Quaternion Fourier Transforms for Signal and Image Processing
Language: en
Pages: 160
Authors: Todd A. Ell, Nicolas Le Bihan, Stephen J. Sangwine
Categories: Technology & Engineering
Type: BOOK - Published: 2014-06-02 - Publisher: John Wiley & Sons

Based on updates to signal and image processing technology made inthe last two decades, this text examines the most recent researchresults pertaining to Quaternion Fourier Transforms. QFT is acentral component of processing color images and complex valuedsignals. The book’s attention to mathematical concepts,imaging applications, and Matlab compatibility render it anirreplaceable