Robust Geometric Modeling Free From Computational Errors

为了消除计算误差引起的几何建模系统的不稳定性，建立了基于面的数据结构的多面体处理算法，并将其推广到曲面对象；在多面体处理算法中，由于顶点、边、面等几何元素的退化会导致几何和拓扑的不一致，因此引入了关联准则。它是使用四个平面方程中的行列式的值计算的四面体的最小高度。结合关联度检验，得到了它们的发病规律。对于集合运算和特征运算，我们从干涉环中得到全局拓扑结构。在一个顶点的退化处，我们通过对那里的面段进行扇区测试来跟踪交点。为了防止顶点周围模糊形状的误差累积，并保持拓扑的一致性，我们在…中进一步介绍了一种确定局部拓扑的算法，该算法用树形结构表示，不能制作小的面，由于无法计算曲面的四面体与行列式的最小距离，因此必须计算从三个面的交点到一个面的垂线一英尺长，并考虑面、面和曲线以及曲线和曲线之间的切线。我们开发了三个面的交点、点到面的垂直线、两个面的最小距离位置以及边界和交点的最小距离位置的算法。然后，结合它们，我们确定了简并的模式。我们已经研究了检测和连接由于错误而没有形成环的交点的方法。我们已经给出了一个算法来确定曲线体顶点周围模糊形状的局部拓扑，就像确定多面体的算法一样。我们未来的研究是通过在系统上的实验来验证算法的有效性。较少

In order to make a robust geometric modeling system by eliminating instability caused by computational errors, we have established algorithms of processing polyhedra for an introduced face-based data structure, and then extended them to curved objects.In the algorithms for polyhedra, since inconsistency between geometry and topology occurs at the degeneracy of geometric elements such as vertices, edges and faces, we have introduced a criterion of incidence among them. It is the minimum height of a tetrahedron which is calculated using a determinant value from four plane equations. Combining incidence tests, we obtained their incidence patterns. We get a global topological structure from the interference loop for the set operation and feature operation. At the degeneracy for a vertex, we trace the intersection by the sector test for the face segments there. To prevent the accumulation of errors for the ambiguous shape around the vertex and to keep consistency of the topology, we have in … More troduced an algorithm to determine the local topology which is represented as a tree structure not to make small faces.Since we cannot calculate the minimum distance of the tetrahedron from the determinant for curved faces, we have to calculate a foot leng of perpendicular from an intersection point of three faces to a face and consider tangency between faces, a face and a curve and a curve and a curve. We have developed algorithms for an intersection point of three faces, a perpendicular from a point to a face, a minimum distance position of two faces, and a minimum distance position between a boundary and an intersection. Then, combining them, we determine the patterns of the degeneracy. We have examined methods for detecting and connecting the intersection points which do not make a loop because of errors. We have shown an algorithm to determine the local topology of the ambiguous shape around the vertex for a curved solid as similarly as for a polyhedron. Our future research is to confirm the algorithms by experiments on a system. Less