IMR PROCEEDINGS

24th International Meshing Roundtable
Austin, TX
October 1214,2014
Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany
Email: Nadja.Goerigk@wiasberlin.de, Hang.Si@wiasberlin.de
Abstract
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 wellknown. However, the geometry and combinatorial structure of
such polyhedra are much less studied. In this article, we investigate the structure of some wellknown examples, the socalled
Schonhardt 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, socalled Steiner point, which can be used to decompose the Schonhardt and the Bagemihl's
polyhedra. We then provide a construction of a larger class of threedimensional indecomposable polyhedra which often appear in
grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schonhardt'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 (n5)/2 interior Steiner points. We also show that this number is optimal in the
worst case.
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