共形几何代数——几何代数的新理论和计算框架

Conformal Geometric Algebra—A New Framework for Computational Geometry

共形几何代数是一个新的几何表示和计算工具.作为几何的高级不变量和协变量系统的结合,它为经典几何提供了统一和简洁的齐性代数框架,以及高效的展开、消元和化简算法,从而可以进行极其复杂的符号几何计算,在几何建模与计算方面表现出很大的优势.主要讲述共形几何代数的产生背景和意义,共形几何代数的数学理论和它最有特色的几个部分,包括Grassmann结构、统一几何表示和旋量作用、基本不变量系统和高级不变量系统、新的计算思想、展开和化简技术等.

Conformal Geometric Algebra （CGA） is a new tool for geometric representation and computation. As an integration of advanced geometric invariants and covariants systems, CGA provides a unified and compact homogeneous algebraic framework for classical geometries, together with a set of effective algorithms for expansion, elimination and simplification, which enables it to carry out extremely complicated symbolic geometric computations, and be used quite advantageously in geometric modeling and computing. This is the first of a series of papers dedicated to the introduction of CGA. The paper focuses on the background and importance of CGA, the mathematical theory and several most important features, including the Grassmann structure, the universal geometric representation and spin or action, the basic invariant system and advanced invariant system, the new idea in computing, the expansion and simplification techniques, etc.