Study on Representations and Processing of Geometric Objects in Terms of Mutual Constraints

从工程的角度考虑了用点、线、面等元素之间的相互约束而不是用坐标系来表示几何对象的可能性和用途，并得到了以下结果.阐明了顶点间的长度、面间的角度等相互约束之间的依赖关系的数学结构;对于几何元素处于一般位置的情形，依赖结构的特征在于几何对象的某些组合性质;而对于几何元素处于某些特殊位置从而发生退化的情形，依赖结构的特征在于几何对象的代数性质。2.上述结果被应用到的问题，发现不一致的定义中的几何结构，定位一个移动的机器人与一个单一的摄像机，并识别对象提取的范围图片。3.构造了一种避免几何退化的新方法。几何元素在某些特定位置处会发生几何退化，从而导致几何算法复杂化。一种称为“符号扰动”的技术被应用于避免这种退化的Voronoi图的建设和隐藏线消除。4.提出了一种使几何算法对数值误差稳定的新方法。该方法是基于这样的原则，即如果数值结果与拓扑结构相矛盾，则最优先考虑拓扑结构的一致性。将该方法应用于Voronoi图的构造，实验结果表明了该方法的有效性，所建立的新概念和新方法将在今后的工程应用中得到广泛的应用。

The possibility and the use of representing geometric objects, not in terms of coordinate systems, but in terms of mutual constraints among elements such as points, lines and faces of the objects was considered from an engineering point of view, and the following results have been obtained.1. The mathematical structure of dependence among mutual constraints such as lenghts between vertices and angles between faces was :clarified; the dependence structure was characterized by some combinatorial properties of the geometric objects for the case where geometric elements were in general positions, whereas the dependence structure was characterized by algebraic properties for the case where geometric elements were in some special position so that degeneracy took place. 2. The above result was applied to the problem of finding inconsistency in the definition of geometric structures, to that of locating a mobile robot with a single camera, and to that of recognizing objects extracted from range pictures. 3. A new method was constructed for avoiding geometric degeneracy. Geometric degeneracy takes place when geometric elements come to some special position, and it causes to make geometric algorithms complicated. A technique called "symbolic perturbation" was applied to avoidance of such degeneracy in Voronoi diagram construction and in hidden line elimination. 4. A new method was proposed for making geometric algorithms stable against numerical errors. The method is based on the principle that the highest priority is placed on the consistency of the topological structures if numerical results contradict topological structures. This method was applied to the construction of Voronoi diagrams, and experimental results showed the validity of the method.The new concepts and methods established in this projects will be applied to many other geometric problems in various fields of engineering in future.