Abstract
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].
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References
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 ’00). ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, pp 157–164. doi:10.1145/344779.344859
Gleicher M, Witkin A (1992) Through-the-lens camera control. SIGGRAPH Comput. Graph 26(2):331–340. doi:10.1145/142920.134088
Humphreys JE (1978) Introduction to Lie algebras and representation theory. Second printing, revised. Graduate texts in mathematics, vol 9. Springer, New York
Igarashi T, Moscovich T, Hughes JF (2005) As-rigid-as-possible shape manipulation. In: Gross M (ed) ACM SIGGRAPH 2005 Papers (SIGGRAPH ’05), ACM, New York, NY, USA, pp 1134–1141. doi:10.1145/1186822.1073323
Jeffers J (2000) Lost theorems of geometry. Am Math Monthly 107(9):800–812
Kaji S, Hirose S, Ochiai H, Anjyo K (2013) A Lie theoretic parameterization of Affine transformation. In: Proceedings of MEIS2013 (Mathematical Progress in Expressive Image, Synthesis 2013), pp 134–140
Kavan L, Collins S, Zara J, O’Sullivan C (2008) Geometric skinning with approximate dual quaternion blending. ACM Trans Graph 27(4):105
Ochiai H, Anjyo K (2014) Mathematical formulation of motion and deformation and its applications. In: Mathematical Progress in Expressive Image Synthesis I, Springer-Verlag, Berlin, Germany
Ochiai H, Anjyo K (2013) Mathematical description of motion and deformation. In: SIGGRAPH Asia 2013 Course, http://portal.acm.org. Accessed 14 Mar 2014 16h30. ( Revised course notes are also available at http://mcg.imi.kyushu-u.ac.jp/english/project.php?record_id=95. Accessed 14 Mar 2014 16h30)
Pinheiro S, Gomes J, Velho L (1999) Interactive specification of 3D displacement vectors using arcball. In: Proceedings of the international conference on computer graphics (CGI ’99). IEEE Computer Society, Washington, DC, USA, p 70
Schaefer S, McPhail T, Warren J (2006) Image deformation using moving least squares. In: ACM SIGGRAPH 2006 Papers (SIGGRAPH ’06). ACM, New York, NY, USA, pp 533–540. doi:10.1145/1179352.1141920
Shoemake K (1985) Animating rotation with quaternion curves. In: ACM SIGGRAPH, pp 245–254
https://github.com/KyushuUniversityMathematics/iPad-ProbeDeformer. Accessed 14 Mar 2014 16h30
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 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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Matsuda, G., Kaji, S., Ochiai, H. (2014). Anti-commutative Dual Complex Numbers and 2D Rigid Transformation. In: Anjyo, K. (eds) Mathematical Progress in Expressive Image Synthesis I. Mathematics for Industry, vol 4. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55007-5_17
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DOI: https://doi.org/10.1007/978-4-431-55007-5_17
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