rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher.
Mathematical Basics of Motion and Deformation in Computer Graphics: Second Edition
Ken Anjyo and Hiroyuki Ochiai
www.morganclaypool.com
ISBN: 9781627056977 paperback
ISBN: 9781627059848 ebook
DOI 10.2200/S00766ED1V01Y201704VCP027
A Publication in the Morgan & Claypool Publishers series
SYNTHESIS LECTURES ON VISUAL COMPUTING: COMPUTER GRAPHICS, ANIMATION, COMPUTATIONAL PHOTOGRAPHY, AND IMAGING
Lecture #27
Series Editor: Brian A. Barsky, University of California, Berkeley
Series ISSN
Print 2469-4215 Electronic 2469-4223
Mathematical Basics of Motion and Deformation in Computer Graphics
Second Edition
Ken Anjyo
OLM Digital, Inc.
Hiroyuki Ochiai
Kyushu University
SYNTHESIS LECTURES ON VISUAL COMPUTING: COMPUTER GRAPHICS, ANIMATION, COMPUTATIONAL PHOTOGRAPHY, AND IMAGING #27
ABSTRACT
This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation.
This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.
KEYWORDS
motion, deformation, quaternion, Lie group, Lie algebra
Contents
2.3 2D Rigid Transformation
2.4 2D Reflection
2.5 3D Rotation: Axis-angle
2.6 3D Rotation: Euler Angle
2.7 3D Rotation: Quaternion
2.8 Dual Quaternion
2.9 Using Complex Numbers
2.10 Dual Complex Numbers
2.11 Homogeneous Expression of Rigid Transformations
3.1 Several Classes of Transformations
3.2 Semidirect Product
3.3 Decomposition of the Set of Matrices
3.3.1 Polar Decomposition
3.3.2 Diagonalization of Positive Definite Symmetric Matrix
3.3.3 Singular Value Decomposition (SVD)
4 Exponential and Logarithm of Matrices
4.1 Definitions and Basic Properties
4.2 Lie Algebra
4.3 Exponential Map from Lie Algebra
4.4 Another Definition of Lie Algebra
4.5 Lie Algebra and Decomposition
4.6 Loss of Continuity: Singularities of the Exponential Map
4.7 The Field of Blending
5 2D Affine Transformation between Two Triangles
5.1 Triangles and an Affine Transformation
5.2 Comparison of Three Interpolation Methods
6 Global 2D Shape Interpolation
6.2 Formulation
6.3 Error Function for Global Interpolation
6.4 Examples of Local Error Functions
6.5 Examples of Constraint
