数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (4): 991-1004.doi: 10.1016/S0252-9602(09)60083-7

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SYMBOLIC VERSOR COMPRESSION ALGORITHM

李洪波   

  1. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
  • 收稿日期:2009-01-16 出版日期:2009-07-20 发布日期:2009-07-20

SYMBOLIC VERSOR COMPRESSION ALGORITHM

Li Hongbo   

  1. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2009-01-16 Online:2009-07-20 Published:2009-07-20

摘要:

In an inner-product space, an invertible vector generates a reflection with re-spect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the
input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.

关键词: Clifford algebra, orthogonal transformation, symbolic computation, Grassmann-Cayley algebra, versor

Abstract:

In an inner-product space, an invertible vector generates a reflection with re-spect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the
input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.

Key words: Clifford algebra, orthogonal transformation, symbolic computation, Grassmann-Cayley algebra, versor compression

中图分类号: 

  • 68W30