Long Geom. Prod., BREEFS, Clifford Factorization

A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization
Hongbo Li
Mathematics Mechanization Research Center
AMSS, Chinese Academy of Sciences
Beijing 100080, China

In symbolic computing, a major bottleneck is middle expression swell. Symbolic geometric computing based on invariant algebras can alleviate this difficulty. For example, the size of projective geometric computing based on bracket algebra can often be restrained to two terms, using binomial polynomials, area method, Cayley expansion, etc. This is the “binomial” feature of projective geometric computing in he language of bracket algebra.
In this paper we report a stunning discovery in Euclidean geometric computing: the term preservation phenomenon. Input an expression in the language of Null Bracket Algebra (NBA), by the recipe we are to propose in this paper, the computing procedure can often be controlled to within he same number of terms as the input, through to the end. n particular, the conclusions of most Euclidean geometric
theorems can be expressed by monomials in NBA, and the expression size in the proving procedure can often be controlled to within one term! Euclidean geometric computing can now be announced as having a “monomial” feature in the language of NBA.
The recipe is composed of three parts: use long geometric product to represent and compute multiplicatively, use “BREEFS” to control the expression size locally, and use Clifford factorization for term reduction and transition from algebra to geometry.
By the time this paper is being written, the recipe has been tested by 70+ examples from [1], among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel’s five-circle theorem [2], whose analytic proofis straightforward but very difficult symbolic computing, is
discovered to have a 3-termed elegant proof with the recipe.
ACM Computing Classification: I.1.1 [Symbolic and Algebraic Manipulation]: Expressions and Their Representation; G.4 [Mathematical Software]: Efficiency.
General Terms: Theory; algorithm.
Keywords: Conformal geometric algebra, Null bracket algebra, Geometric invariance, Symbolic geometric computing, Geometric theorem proving.

Wed, 24 Jan 2007, by Hongo Li


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