The Cutting Pattern Problem for Tetrahedral Mesh Generation

Xiaotian Yin, Wei Han, Xianfeng Gu, Shing-Tung Yau

Abstract: In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a two-dimensional simplicial complex), assign each triangle with a triple of ±1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) and consistent (i.e. every edge shared by two triangles has opposite signs in these triangles). We show that this problem is the major challenge in converting a volumetric mesh consisting of prisms into a mesh consisting of tetrahedra, where each prism is cut into three tetrahedra. In this paper we provide a complete solution to this problem for topological disks under various boundary conditions ranging from very restricted one to the most flexible one. For each type of boundary conditions, we provide efficient algorithms to compute valid assignments if there is any, or report the obstructions otherwise. For all the proposed algorithms, the convergence is validated and the complexity is analyzed.