Mathematical Basics of Motion and Deformation in Computer Graphics. Ken Anjyo

       Preface

      In the computer graphics community, many technical terms, such as Euler angle, quaternion, and affine transformation, are fundamental and quite familiar words, and have a pure mathematical background. While we usually do not have to care about the deep mathematics, the graphical meaning of such basic terminology is sometimes slightly different from the original mathematical entities. This might cause misunderstanding or misuse of the mathematical techniques. Or, if we have just a bit more curiosity about pure mathematics relevant to computer graphics, it should be easier for us to explore a new possibility of mathematics in developing a new graphics technique or tool.

      This volume thus presents an intuitive introduction to several mathematical basics that are quite useful for various aspects of computer graphics, focusing on the fundamental procedures for deformation and animation of geometric objects, and curve/surface editing. The objective of this book, then, is to fill the gap between the original mathematical concepts and the practical meanings in computer graphics without assuming any prior knowledge of pure mathematics. We then restrict ourselves to the mathematics for matrices, while we know there are so many other mathematical approaches far beyond matrices in our graphics community. Though this book limits the topics to matrices, we hope you can easily understand and realize the power of mathematical approaches. In addition, this book demonstrates our ongoing work, which benefits from the mathematical formulation we develop in this book.

      This book is an extension of our early work that was given as SIGGRAPH Asia 2013 and SIGGRAPH 2014 courses. The exposition developed in this book has greatly benefited from the advice, discussions, and feedback of a lot of people. The authors are very much grateful to Shizuo Kaji at Yamaguchi University and J.P. Lewis at Victoria University of Wellington, who read a draft of this book and gave many invaluable ideas. The discussions and feedback from the audience at the SIGGRAPH courses are also very much appreciated. Many thanks also go to Gengdai Liu and Alexandre Derouet-Jourdan at OLM Digital for their help in making several animation examples included in this book.

      This work was partially supported by Core Research for Evolutional Science and Technology (CREST) program “Mathematics for Computer Graphics” of Japan Science and Technology Agency (JST). Many thanks especially to Yasumasa Nishiura at Tohoku University and Masato Wakayama at Institute of Mathematics for Industry, who gave long-term support to the authors.

      The authors wish to thank Ayumi Kimura for the constructive comments and suggestions made during the writing of this volume. Thanks also go to Yume Kurihara for the cute illustrations. Last, but not least, the authors are immensely grateful to Brian Barsky, the editor of the Synthesis

      Lectures on Compute Graphics and Animarion series, and Mike Morgan at Morgan & Claypool Publishers for giving the authors such an invaluable chance to publish this book in the series.

      Ken Anjyo and Hiroyuki Ochiai

      October 2014

       Preface to the Second Edition

      In this edition, we added an appendix where we derive several formulas for 3D rotation and deformation. We also incorporated a number of references, particularly relating to our SIGGRAPH 2016 course and its accompanying video. These additions will help readers to better understand the basic ideas developed in this book. We also resolved the mathematical notation inconsistency from the first edition, which makes this book more easily accessible.

      We would like to thank the graduate students at Kyushu University who carefully went through the book with the second author in his seminar. Finally, we are very grateful to Ayumi Kimura who worked with us for the SIGGRAPH 2016 course and helped significantly with the editing of the second edition.

      Ken Anjyo and Hiroyuki Ochiai

      April 2017

       Symbols and Notations

Tb	translation, 5
Rθ	2D rotation matrix, 6
SO(2)	2D rotation group (special orthogonal group), 7
M(n, R)	the set of square matrices of size n with real entries, 7
I, In	the identity matrix of size n, 7, 11, 30,
AT	transpose of a matrix A, 7
SE(2)	2D motion group (the set of non-flip rigid transformations), 8
O(2)	2D orthogonal group, 9
SO(n)	special orthogonal group, 10
O(n)	orthogonal group, 10
SO(3)	3D rotation group, 11
Rx (θ)	3D axis rotations, 11
H	the set of quaternions, 3, 13
q̄	conjugate of a quaternion, 13
Re(q)	real part of a quaternion, 14
Im(q)	imaginary part of a quaternion, 14
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