Efficiently approximating the minimum-volume bounding box of a point set in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Efficient Collision Detection Using Bounding Volume Hierarchies of k-DOPs
IEEE Transactions on Visualization and Computer Graphics
Transforming range queries to equivalent box queries to optimize page access
Proceedings of the VLDB Endowment
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In recent years there has been significant use of regular cross-polytopes (regular octahedrons or hyper-diamonds) as constructs to simplify problem solving in high-dimensional database queries, collision detection algorithms and graphic rendering techniques. Many of the algorithms for these applications use minimum volume bounding boxes as approximations of the polytopes to minimize computational complexity. The standard method [1] for finding these boxes in three dimensions uses the constraint of having two edges of a polyhedron coincident with two adjacent faces of the minimum bounding box. In this paper, we show that for a uniform cross-polytope in three dimensional space, a minimum volume bounding box would have a face flush with the convex hull of the polytope for all possible orientations of the polyhedron defined. We also show that the projections of the minimum bounding box of an n-dimensional regular cross-polytope are locally optimal with respect to the projections of the enclosed cross-polytope. We use this result to provide a necessary condition for the minimum bounding boxes of such polytopes. Finally, we show that if the two dimensional planar projections of a three dimensional uniform cross-polytope are simultaneously locally optimal then the polytope itself is optimally oriented.