有174个从六面体到四面体的细分
There are 174 Subdivisions of the Hexahedron into Tetrahedra
摘要：
这篇文章回答了一个重要的理论问题：有多少个不同的六面体细分成四面体？众所周知，立方体有五个细分成6个四面体和一个细分成5个四面体。然而，并非所有的六面体都是立方体，移动顶点位置会增加细分的数量。最近的六面体主要网格划分方法试图将这些配置考虑在内，以将四面体组合成六面体，但未能将它们全部枚举：它们仅使用我们在本文中发现的174个细分中的10个细分。将这174个六面体细分为四面体是我们的组合结果。174个细分中的每一个都有5到15个四面体，实际上是一个2到48个等价实例的类，这些实例直到顶点重新标记都是相同的。我们进一步表明，正是171这些细分有一个几何实现，即存在的八个六面体顶点在三维空间中的坐标，使几何四面体网格是有效的。我们展示了这些配置的四面体网格，并显示在特别细分的六面体与15个四面体，有一个严格的正雅可比矩阵。
英文摘要：
This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian.
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