Anti-commutative Dual Complex Numbers and 2D Rigid Transformation

  • Genki Matsuda
  • Shizuo Kaji
  • Hiroyuki Ochiai
Part of the Mathematics for Industry book series (MFI, volume 4)


We introduce a new presentation of the two dimensional rigid transformation which is more concise and efficient than the standard matrix presentation. By modifying the ordinary dual number construction for the complex numbers, we define the ring of anti-commutative dual complex numbers, which parametrizes two dimensional rotation and translation all together. With this presentation, one can easily interpolate or blend two or more rigid transformations at a low computational cost. We developed a library for C++ with the MIT-licensed source code [13].


2D deformation 2D animation Interpolation Skinning Rigid transformation Euclidian motion Dual number 



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 for S. Hirose at OLM Digital Inc., and Y. Mizoguchi, S. Yokoyama, H. Hamada, and K. Matsushita at Kyushu University for their valuable comments.


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityNishi-ku, FukuokaJapan
  2. 2.Yamaguchi University/JST CRESTYamaguchiJapan
  3. 3.Institute of Mathematics for IndustryKyushu University/JST CRESTNishi-ku, FukuokaJapan

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