Casting an Object with a Core

Abstract

This paper addresses geometric problems that concern manufacturing an object using a cast with a core. In casting, molten material is poured into the cavity of the cast and allowed to solidify. The cast has two main parts to be removed in opposite parting directions. To manufacture more complicated objects, the cast may also have a core to be removed in a direction skewed to the parting directions. In this paper, given an object and the parting and core directions, we give necessary and sufficient conditions to verify whether a cast can be constructed for these directions. In the case of polyhedral objects, we develop a discrete algorithm to perform the test in O(n 3 log n) time, where n is the object size. If the test result is positive, a cast with complexity O(n 3 ) can be constructed within the same time bound. We also present an example to show that a cast may have Θ(n 3 ) complexity in the worst case. Thus, the complexity of our cast is worst-case optimal.