Mechanical geometry theorem proving
Mechanical geometry theorem proving
Multilinear cayley factorization
Journal of Symbolic Computation
Some examples of the use of distances as coordinates for Euclidean geometry
Journal of Symbolic Computation
Generalized homogeneous coordinates for computational geometry
Geometric computing with Clifford algebras
Journal of Symbolic Computation
Symbolic computation in the homogeneous geometric model with clifford algebra
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
On miquel's five-circle theorem
IWMM'04/GIAE'04 Proceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications
Complex brackets, balanced complex differences, and applications in symbolic geometric computing
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
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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 final polynomials, area method, Cayley expansion,etc. This is the "binomial "feature of projective geometric computing in the 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 the same number of terms as the input, through to the end. In 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 proof is straightforward but very difficult for symbolic computing, is discovered to have a 3-termed elegant proof with the recipe.