符号几何计算的一种方法：长几何积、BREEFS和Clifford分解
A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization
摘要：
在符号计算中,最困难的一个地方是中间计算过程的表达式快速膨胀.基于不变代数的符号几何计算可以缓解这一困难。例如，基于括号代数的射影几何计算的规模，往往可以通过使用终值多项式、面积法、Cayley展开式等将其约束为两项，这就是括号代数语言中射影几何计算的“二项式”特征。在这篇文章中，我们报道了欧几里得几何计算中的一个惊人发现：术语保持现象。用零括号代数（NBA）语言输入一个表达式，通过本文提出的方法，计算过程通常可以控制在与输入相同的项数之内，直到最后。特别是大多数欧氏几何定理的结论都可以用NBA中的单式来表示，证明过程中的表达式大小往往可以控制在一项之内！欧几里得几何计算现在可以被宣布为具有NBA语言中的“单性”特征。该方法由三部分组成：使用长几何积进行乘法表示和计算，使用BREEFS局部控制表达式大小，使用Clifford分解进行约简和代数到几何的转换。在撰写本文时，已经用\cite{chou}中的70多个例子测试了该配方，其中30多个例子有单项证明。其中著名的Miquel的五圆定理\cite{chou2}，其解析证明简单明了，但很难用符号计算，被发现有一个3项的优雅证明。
英文摘要：
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 \cite{chou}, among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose analytic proof is straightforward but very difficult symbolic computing, is discovered to have a 3-termed elegant proof with the recipe.
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