4.3 a Data Structuring Approach

25] Seidel, R., Small-dimensional linear programming and convex hulls made easy. range searching and new zone theorems. A linear time algorithm for determining the separation of convex polyhedra. Two dimensional computation of the three dimensional reachable region for a welding head. 29 list some open problems related to this paper, and then list a few others in cast design that deserve attention. 1. What is the maximum number of distinct casting planes in case of arbitrary cast removal? This paper shows O(n 2), whereas the only lower bound we have is linear. 2. For a convex polyhedron P, what is the maximum summed complexity of the intersection of all distinct casting planes with P? This paper shows O(n log n) in case of opposite cast removal, but the trivial lower bound is linear. 3. Give simple algorithms for casting that improve our simple O(n log 2 n), O(n 2) and O(n 2 log n) time algorithms. Find algorithms that improve upon our O(n 3=2+) time algorithms. 4. Suppose that the casts may be removed with any motion. Give algorithms to determine whether a polyhedron is castable in this case. 5. Suppose that we wish to determine castability of an object with non-linear boundaries. Give (simple) algorithms that solve this problem. 6. Suppose that more cast parts are allowed. Determine for a polyhedron how many cast parts are necessary. 7. For some casting processes, is not necessary that the cast parts must be separated by a plane. In these cases, every convex polyhedron is castable. However, no algorithms are known for cast removal of simple polyhedra. The authors thank the organizer Godfried Toussaint and the other participants for creating a good research environment. Furthermore, we thank Pankaj Agarwal for answering several of our questions. Applications of a new space partitioning technique. 28 q d Figure 4: Left, rotating a plane about an edge through P. Right, the dual problem (in 2 dimensions), where the third plane intersecting the ray q + ~ d is found after wrap-around. past a vertical orientation. These adaptations to the query algorithm can easily be made within the same asymptotic time bounds. Hence, we conclude: Theorem 9 For any simple polyhedron P with n vertices and any constant > 0, one can determine in O(n 3=2+) time and space whether P is castable, when the cast parts can be removed in arbitrary directions. 6 Conclusions and …