Clifford 代数,几何计算和几何推理

Clifford代数是一种深深根植于几何学之中的代数系统,被它的创始人称为几何代数．历史上,E．Cartan,R．Brauer,H．Weyl,C．Chevalley等数学大师都曾研究和应用过Clifford代数,对它的发展起了重要作用．近年来,Clifford代数在微分几何、理论物理、经典分析等方面取得了辉煌的成就,是现代理论数学和物理的一个核心工具,并在现代科技的各个领域,如机器人学、信号处理、计算机视觉、计算生物学、量子计算等方面有广泛的应用．本文主要介绍Clifford代数在几何计算和几何推理中的应用．作为一种优秀的描述和计算几何问题的代数语言,Clifford代数对于几何体,几何关系和几何变换有不依赖于坐标的、易于计算的多种表示,因而应用它进行几何自动推理,不仅使困难定理的证明往往变得极为简单,而且能够解决一些著名的公开问题,目前在国际上,几何自动推理已经成为Clifford代数的一个重要应用领域。

Clifford Algebra, Geometric Computing and Reasoning

Clifford algebra is an algebraic system deeply rooted in geometry. It was named "Geometric Algebra" by its discoverer W. K. Clifford. In history, many famous mathematicians, E. Cartan, R. Brauer, H. Weyl, C. Chevalley, to name a few, had contributed to its development. In recent years, Clifford algebra has made spectacular achievements in differential geometry, theoretical physics and classical analysis. It is a central tool in modern mathematics and physics, and has wide-ranged applications in robotics, signal processing, computer vision, computational biology, quantum computing, and other high technology fields. In this paper we introduce some applications of Clifford algebra in geometric computing and automated geometric theorem proving. As a very elegant algebraic language for describing and computing geometric problems, Clifford algebra has a variety of coordinate-free and computing-favorable representations for geometric entities, relations and transformations. Therefore, applying Clifford algebra in automated theorem proving can not only make the proof procedures often extremely simple, but also solve open mathematical problems. Nowadays in the world, automated theorem proving has become an important field for applying Clifford algebra.