Determining the castability of simple polyhedra

Abstract

A polyhedron P is castable if its boundary can be partitioned by a plane into two polyhedral terrains. Such polyhedra can be manufactured easily using two cast parts. Assuming that the cast parts are removed by a single translation each, it is shown that for a simple polyhedron with n vertices, castability can be decided in O(n2 log n) time and linear space using a simple algorithm. Furthermore, a more complicated algorithm solves the problem in O(n3/2+ε) time and space, for any fixed ε > 0. In the case where the cast parts are to be removed in opposite directions, a simple O(n2) time algorithm is presented. Finally, if the object is a convex polyhedron and the cast parts are to be removed in opposite directions, a simple O(n log2 n) algorithm is presented.