On Geometric Theorem Proving with Null Geometric Algebra
The bottleneck in symbolic geometric computation is middle expression swell. Another embarrassing problem is geometric explanation of algebraic results, which is often impossible because the results are not invariant under coordinate transformations. In classical invariant-theoretical methods, the two difficulties are more or less alleviated but stay, while new difficulties arise.
In this chapter, we introduce a new framework for symbolic geometric computing based on conformal geometric algebra: the algebra for describing geometric configuration is null Grassmann–Cayley algebra, the algebra for advanced invariant manipulation is null bracket algebra, and the algebra underlying both algebras is null geometric algebra. When used in geometric computing, the new approach not only brings about amazing simplifications in algebraic manipulation, but can be used to extend and generalize existing theorems by removing some geometric constraints from the hypotheses.
Both authors are supported partially by NSFC 10871195, NSFC 60821002/F02 and NKBRSF 2011CB302404.
- 13.Li, H., Hestenes, D., Rockwood, A.: Generalized homogeneous coordinates for computational geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, pp. 27–60. Springer, Heidelberg (2001) Google Scholar
- 14.Muir, T.: A Treatise on the Theory of Determinants. Macmillan & Co., London (1882) Google Scholar
- 17.Sturmfels, B., White, N. (eds.): Invariant-Theoretic Algorithms in Geometry. Special Issue. J. Symb. Comput. 11 (5/6) (2002) Google Scholar
- 18.White, N. (ed.): Invariant Methods in Discrete and Computational Geometry. Kluwer, Dordrecht (1994) Google Scholar