Spin group and screw algebra, as extensions of quaternions and vector algebra, respectively, have important applications in geometry, physics and engineering. In threedimensional projective geometry, when acting on lines, each projective transformation can be decomposed into at most three harmonic projective reflections with respect to projective lines, or equivalently, each projective spinor can be decomposed into at most three orthogonal Minkowski bispinors, each inducing a harmonic projective line reflection. In this paper, we establish the corresponding result for three-dimensional affine geometry: with each affine transformation is found a minimal decomposition into general affine reflections, where the number of general affine reflections is at most three; equivalently, each affine spinor can be decomposed into at most three affine Minkowski bispinors, each inducing a general affine line reflection.
Chengran WU
,
Hongbo LI
. AFFINE SPINOR DECOMPOSITION IN THREE-DIMENSIONAL AFFINE GEOMETRY[J]. Acta mathematica scientia, Series B, 2022
, 42(6)
: 2301
-2335
.
DOI: 10.1007/s10473-022-0607-9
[1] Pottmann H, Wallner J. Computational Line Geometry. New York: Springer, 2001
[2] Merlet J P. Parallel Robots. 2nd ed. Dordrecht: Springer, 2005
[3] Bayro-Corrochano E, Lasenby J. Object modelling and motion analysis using Clifford algebra//Proceedings of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision. Xi’an, China, 1995: 143-149
[4] Daniilidis K. Hand-eye calibration using dual quaternions. The International Journal of Robotics Research, 1999, 18(3): 286-298
[5] Dai J. Geometrical Foundations and Screw Algebra for Mechanisms and Robotics. Beijing: Higher Education Press, 2014
[6] Selig J M. Geometric Fundamentals of Robotics. New York: Springer, 2005
[7] Iqbal H, Khan M U A, Yi B. Analysis of duality based interconnected kinematics of planar serial and parallel manipulators using screw theory. Intelligent Service Robotics, 2020, 13(1): 47-62
[8] Sariyildiz E, Temeltas H. A new formulation method for solving kinematic problems of multiarm robot systems using quaternion algebra in the screw theory framework. Turkish Journal of Electrical Engineering and Computer Sciences, 2012, 20(4): 9-30
[9] Li Z, Schicho J, Schrëcker H P. The rational motion of minimal dual quaternion degree with prescribed trajectory. Computer Aided Geometric Design, 2016, 41(3): 1-9
[10] Doran C, Lasenby A. Geometric Algebra for Physicists. Cambridge: Cambridge University Press, 2007
[11] Chasles M. Traité de Géométrie Supérieure. Paris: Gauthier-Villars, 1880
[12] Lounesto P. Clifford Algebras and Spinors. 2nd ed. Cambridge: Cambridge University Press, 2011
[13] Clifford W K. Elements of Dynamic: An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies. London: MacMillan and Company, 1878
[14] Meinrenken E. Clifford Algebras and Lie Theory. Heidelberg: Springer, 2013
[15] Li H. Three-dimensional projective geometry with geometric algebra. https://arxiv.org/pdf/1507.06634.pdf
[16] Hestenes D, Sobczyk G. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Dordrecht: D. Reidel Publishing Company, 1984
[17] Hestenes D, Ziegler R. Projective geometry with Clifford algebra. Acta Applicandae Mathematica, 1991, 23(1): 25-63
[18] Doran C, Hestenes D, Sommen F, et al. Lie groups as spin groups. Journal of Mathematical Physics, 1993, 34(8): 3642-3669
[19] Goldman R, Mann S, Jia X. Computing perspective projections in 3-dimensions using rotors in the homogeneous and conformal models of Clifford algebra. Journal of Mathematical Physics, 2014, 24(2): 465-491
[20] Li H, Zhang L. Line geometry in terms of the null geometric algebra over R3,3, and application to the inverse singularity analysis of generalized Stewart platforms//Dorst L, Lasenby J, eds. Guide to Geometric in Practice. London: Springer, 2011: 253-272
[21] Klawitter D. A Clifford algebraic approach to line geometry. Advances in Applied Clifford Algebra, 2013, 24(3): 737-761
