# Mathematical Formulation of Motion and Deformation and Its Applications

## Abstract

This chapter is intended to give a summary of Lie groups and Lie algebras for computer graphics, including an example from interpolations and blending of motions and deformation. In animation and filmmaking procedure, we want to get a smooth transition of the drawing of given starting and ending point (the drawings at these points are called *key frame*). This problem can be understood as an *interpolation*, so that various approach have been proposed and used in Computer Graphics. After the seminal work so-called ARAP (as-rigid-as-possible), serious properties of matrix groups have been taken into account both in theoretical and in computational point of view. To understand and develop these properties, we here employ a Lie theoretic approach.

### Keywords

Blend shape Lie group Lie algebra Polar decomposition As-rigid-as-possible Deformation Exponential map Motion group## Notes

### Acknowledgments

This work was supported by Core Research for Evolutional Science and Technology (CREST) Program “Mathematics for Computer Graphics” of Japan Science and Technology Agency (JST). The authors are grateful to the collaborators including Shizuo Kaji at Yamaguchi University, Yoshihiro Mizoguchi, Shun’ichi Yokoyama, Hiroyasu Hamada, and Kohei Matsushita, Genki Matsuda at Kyushu University and Ayumi Kimura, Sampei Hirose, and Gengdai Liu at OLM Digital for their valuable discussions.

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