Mathematical Formulation of Motion and Deformation and Its Applications

Part of the Mathematics for Industry book series (MFI, volume 4)


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.


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



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.


  1. 1.
    Alexa M, Cohen-Or D, Levin D (2000) As-rigid-as-possible shape interpolation. In: Proceedings of the 27th annual conference on computer graphics and interactive techniques. SIGGRAPH 2000, pp. 157–164Google Scholar
  2. 2.
    Kaji S, Hirose S, Ochiai H, Anjyo K (2013) A Lie theoretic parameterization of affine transformation. In: Mathematical progress in expressive image synthesis, MI Lecture Note, vol 50. Kyushu University, pp 134–140Google Scholar
  3. 3.
    Kaji S, Hirose S, Sakata S, Mizoguchi Y, Anjyo K (2012) Mathematical analysis on affine maps for 2D shape interpolation. In: Proceedings of SCA2012, pp 71–76Google Scholar
  4. 4.
    Matsuda G, Kaji S, Ochiai H (2013) Anti-commutative dual complex numbers and 2D rigid transformation. In: Mathematical progress in expressive image synthesis, MI Lecture Note, vol 50. Kyushu University, pp 128–133Google Scholar
  5. 5.
    Ochiai H, Anjyo K (2013) Mathematical description of motion and deformation—from basics to graphics applications. SIGGRAPH Asia 2013 course. (Revised course notes are also available at

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for Industry, Kyushu University/JST CRESTFukuokaJapan
  2. 2.OLM Digital, Inc./JST CRESTTokyoJapan

Personalised recommendations