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. Castable polyhedra can be manufactured easily using two cast parts, where each cast part can be removed from the object without breaking the cast part or the object. If we assume that the cast parts are each removed by a single translation, it is shown that for a simple polyhedron with n vertices, castability can be decided in $O(n^2\log n)$ time and linear space using a simple algorithm. A more complicated algorithm solves the problem in $O(n^{3/2+\epsilon})$ 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(n ² )-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 $O(n \log^2n)$ algorithm is presented.