中文摘要：

几何不变量的使用是计算机视觉和图形学的一个重要手段.发现一个不变量后,如何找到它与其他不变量的关系,是实际应用中的一个重要问题,这种关系的探讨主要依靠在不变量层次上的代数运算.文中介绍了共形几何代数中的基本、高级和有理不变量如何在几何问题中自然出现,它们之间如何进行代数运算,以及如何通过不变量的化简,自然地得到几何条件的充分必要化和几何定理的完全化.几何定理的机器证明作为几何定理完全化的副产品,被发展成几何定理的关系定量化,这种量化的几何还原就是几何定理的自然推广.几何不变量之间的几何关系的计算是这些技术的一个具体应用.

英文摘要：

Geometric invariance is an important approach in computer vision and graphics. When a new invariant is found, an important problem is to find its “geometric” relationship （relationship with geometric meaning） with basic invariants. The discovery of this relationship is mainly based on algebraic manipulations of invariants. We show how the basic, advanced and rational invariants in conformal geometric algebra （CGA） appear naturally in geometric problems, how they are manipulated algebraically, and how to obtain the sufficient and necessary conditions and the complete form of geometric theorems by means of invariants manipulation. Automated geometric theorem proving, as a byproduct of the completion of geometric theorems, is further developed into automated quantitative description of geometric relations. The recovery of the geometric meaning of this quantitative description leads to a natural extension of the geometric theorem. Computing the geometric relationship among geometric invariants is a direct application of these techniques.