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On the complexity of higher order abstract voronoi diagrams
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Casting is a manufacturing process in which molten material is poured into a cast (mould), which is opened after the material has solidified. As in all applications of robotics, we have to deal with imperfect control of the casting machinery. In this paper, we consider directional uncertainty: given a 3-dimensional polyhedral object, is there a polyhedral cast such that its two parts can be removed in opposite directions with uncertainty α without inflicting damage to the object or the cast parts? We give a necessary and sufficient condition for castability, and a randomized algorithm that verifies castability and produces two polyhedral cast parts for a polyhedral object of arbitrary genus. Its expected running time is O(n log n). The resulting cast parts have O(n) vertices in total. We also consider the case where the removal direction is not specified in advance, and give an algorithm that finds all feasible removal directions with uncertainty α in expected time O(n2 log n/α2).