Paper Info
Title
There are 174 subdivisions of the hexahedron into tetrahedra.
Abstract
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.
Year
DOI
Venue
2018
10.1145/3272127.3275037
ACM Trans. Graph.
Keywords
Field
DocType
combination, isomorphism, meshing, oriented matroids, tetrahedrization, triangulation enumeration
Hexahedron,Mathematical optimization,Combinatorics,Tetrahedral meshes,Jacobian matrix and determinant,Vertex (geometry),Subdivision,Tetrahedron,Mathematics,Cube
Journal
Volume
Issue
ISSN
37
6
Transactions On Graphics, November 2018, Volume 37, Issue 6, SIGGRAPH Asia 2018 Technical Papers, Article 266
Citations 
PageRank 
References 
0
0.34
16
Authors
3
Name
Order
Citations
PageRank
Jeanne Pellerin1215.27
Kilian Verhetsel212.05
Jean-François Remacle324737.52