Casting an Object with a Core

This paper addresses geometric problems in manufacturing objects by casting. In casting, molten material is poured into the cavity of the cast and allowed to solidify, after which the cast is removed. The cast has two cast parts to be removed in opposite directions. To manufacture more complicated objects, the cast may also have a side core to be removed in a direction skewed to the removal directions for the cast parts. We address the following problem: Given an object and the removal directions for the cast parts and the side core, can a cast be constructed such that the cast parts and the side core can be removed in the directions specified without colliding with the object or each other? We give necessary and sufficient conditions for the problem, as well as a discrete algorithm to perform the test in O(n3log n) time for polyhedral objects, where n is the number of vertices, edges, and facets. If the test result is positive, a cast with complexity O(n3) can be constructed within the same time bound. We also present an example to show that a cast may have Ω(n3) complexity in the worst case.