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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
Im H	the set of imaginary quaternions, 14
|q|	the absolute value of a quaternion, 14
S3	the set of unit quaternions, 15
exp	exponential map, 16, 29
slerp(q0, q1, t)	spherical linear interpolation, 16
ε	dual number, 16
M(2, H)	the set of square matrices of size 2 with entries in H, 16
	the set of anti-commutative dual complex numbers (DCN), 18
E(n)	rigid transformation group, 19
SE(n)	n-dimensional motion group, 20
GL(n)	general linear group, 23
Aff(n)	affine transformations group, 23
GL+(n)	general linear group with positive determinants, 23
Aff+(n)	the set of orientation-preserving affine transformations, 23
⋉	semi-direct product, 25
Sym+(n)	the set of positive definite symmetric matrices, 26
Diag+(n)	the set of diagonal matrices with positive diagonal entries, 27
SVD	Singular Value Decomposition, 28
exp(A)	exponential of a square matrix, 29
C×	the set of non-zero complex numbers, 31
gl(n)	Lie algebra of GL(n), 33
so(n)	Lie algebra of SO(n), 33
sl(n)	Lie algebra of SL(n), 33
aff(n)	Lie algebra of Aff(n), 34
se(n)	Lie algebra of SE(n), 34
[A, B]	Lie bracket, 34
Jx, Jy, Jz	basis of so(3), 37
log	logarithmic map (logarithm), 31, 42
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