Robust Geometric Modeling Free From Computational Errors

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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