边界表示的拓扑与几何一致性

The Consistency in Topology and Geometry of Boundary Representation

闭路拓扑原理和欧拉公式是检验实体模型边界表示的拓扑与几何一致性的重要工具 .有关文献对此问题的论述不妥 .文中在总结前人研究工作的基础上 ,给出曲面上简单闭路总旋转角的精确定义 ,并以球面为例分析曲面上简单闭路总旋转角的计算方法 ,从而导出角度超出量的概念 .对多面体进行严格的定义 ,给出欧拉公式及 Gauss- Bonnet定理对多面体的应用条件 .最后给出多面体的广义欧拉特征值、广义 Gauss- Bonnet定理及广义欧拉公式 .这些理论和方法 ,共同构成实体模型边界表示的拓扑与几何一致性检验的有效工具 .

For inspecting topological and geometrical unanimity of a solid model,principle of closed path topology and Euler′s formula are important tools.On which the exposition of related literatures was inappropriate.Starting from the summation of predecessors′ studies,the authors put forward their theory and method to make up effective tool for its inspection.Firstly,a precise definition is given to the angle of resultant rotation of a simple closed path on a curved surface,and an analysis is made on its calculation with spherical surface as example,and thus a concept of angle excess is derived.Secondly,a strict definition is given to polyhedra,and the conditions are given for applying Euler′s formula and Gauss Bonnet Theorem to polyhedra.Finally,the generalized Euler eigenvalue and the generalized Gauss Bonnet Theorem and the generalized Euler′s formula are given to polyhedra.