Throwing a Sofa Through the Window
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We study several variants of the problem of moving a convex polytope K, with n edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically: ∙ We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms for those variants, which run in time close to O(n^8/3). ∙ We consider the case of a `gate' (an unbounded window with two parallel infinite edges), and show that K can pass through such a window, by any collision-free rigid motion, if and only if it can slide through it. ∙ We consider arbitrary compact convex windows, and show that if K can pass through such a window W (by any motion) then K can slide through a gate of width equal to the diameter of W. ∙ We study the case of a circular window W, and show that, for the regular tetrahedron K of edge length 1, there are two thresholds 1 > δ_1≈ 0.901388 > δ_2≈ 0.895611, such that (a) K can slide through W if the diameter d of W is ≥ 1, (b) K cannot slide through W but can pass through it by a purely translational motion when δ_1≤ d < 1, (c) K cannot pass through W by a purely translational motion but can do it when rotations are allowed when δ_2 ≤ d < δ_1, and (d) K cannot pass through W at all when d < δ_2. ∙ Finally, we explore the general setup, where we want to plan a general motion (with all six degrees of freedom) for K through a rectangular window W, and present an efficient algorithm for this problem, with running time close to O(n^4).
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