On Indecomposable Polyhedra and the Number of Steiner Points

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• 作者：Nadja Goerigk ^{Nadja.Goerigk@wias-berlin.de" class="auth_mail" title="E-mail the corresponding author} ; Hang Si ^{Hang.Si@wias-berlin.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Indecomposable polyhedron ; Steiner point ; Schö ; nhardt polyhedron ; Bagemihl's polyhedron ; tetrahedralization
• 刊名：Procedia Engineering
• 出版年：2015
• 出版时间：2015
• 年：2015
• 卷：124
• 期：Complete
• 页码：343-355
• 全文大小：760 K

文摘

The existence of indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. However, the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we investigate the structure of some well-known examples, the so-called Schönhardt polyhedron [10] and the Bagemihl's generalization of it [1], which will be called Bagemihl's polyhedra. We provide a construction of an additional point, so-called Steiner point, which can be used to decompose the Schönhardt and the Bagemihl's polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt's and Bagemihl's polyhedra, but they may need more than one Steiner point to be decomposed. Given such a polyhedron with n ≥ 6 vertices, we show that it can be decomposed by adding at most interior Steiner points. We also show that this number is optimal in theworst case.