Symbolic computation in the homogeneous geometric model with clifford algebra

Clifford algebra provides nice algebraic representations for Euclidean geometry via the homogeneous model, and is suitable for doing geometric reasoning through symbolic computation. In this paper, we propose various symbolic computation techniques in Clifford algebra. The content includes representation, elimination, expansion and simplification. Simplification includes contraction, combination and factorization. We apply the techniques to automated geometric deduction, and derive the conclusion in completely factored form in which every factor is a basic invariant. The efficiency of Clifford algebra in doing geometric reasoning is reflected in the short and readable procedure of deriving it sincere geometric factorization.

[1]  D. Hestenes.  The design of linear algebra and geometry , 1991 .

[2]  Hongbo Li,et al.  Automated short proof generation for projective geometric theorems with Cayley and bracket algebras: I. Incidence geometry , 2003, J. Symb. Comput..

[3]  David Hestenes,et al.  Generalized homogeneous coordinates for computational geometry , 2001 .

[4]  Henry Crapo,et al.  Automatic Proving of Geometric Theorems , 1995 .

[5]  Hongbo Li,et al.  Automated short proof generation for projective geometric theorems with Cayley and bracket algebras: II. Conic geometry , 2003, J. Symb. Comput..

[6]  S-C Chou,et al.  Machine Proofs in Geometry , 1994 .

[7]  A. Brini,et al.  Remarks on invariant geometric calculus. Cayley-Grassmann algebras and geometric Clifford algebras , 2001 .

[8]  Hongbo Li,et al.  Automated Geometric Theorem Proving, Clifford Bracket Algebra and Clifford Expansions , 2004 .

[9]  D. Hestenes,et al.  Clifford Algebra to Geometric Calculus , 1984 .

[10]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[11]  Xiao-Shan Gao,et al.  Automated geometry theorem proving by vector calculation , 1993, ISSAC '93.

[12]  S. Chou.  Mechanical Geometry Theorem Proving , 1987 .

[13]  Eduardo R. Caianiello,et al.  Combinatorics and renormalization in quantum field theory , 1973 .

[14]  Timothy F. Havel.  Some Examples of the Use of Distances as Coordinates for Euclidean Geometry , 1991, J. Symb. Comput..

[15]  Hongbo Li,et al.  Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra , 2002 .