Throwing a Sofa Through the Window

TL;DR

This paper investigates the problem of moving convex polytopes through various types of windows in three dimensions, providing algorithms, thresholds, and characterizations for different motion constraints and window shapes.

Contribution

It introduces new algorithms and theoretical results for moving convex polytopes through windows, including translation-only motions, sliding equivalences, and thresholds for passage through circular windows.

Findings

Efficient algorithms for translation-only motions near O(n^{8/3})

Characterization of passage through unbounded gates via sliding

Thresholds for tetrahedron passing through circular windows

Abstract

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: $\bullet$ 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 $O(n^{8/3})$. $\bullet$ 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. $\bullet$ 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 $W$. $\bullet$ 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 $1 > \delta_1\approx 0.901388 > \delta_2\approx 0.895611$, such that (a) $K$ can slide through $W$ if the diameter $d$ of $W$ is $\ge 1$, (b) $K$ cannot slide through $W$ but can pass through it by a purely translational motion when $\delta_1\le d < 1$, (c) $K$ cannot pass through $W$ by a purely translational motion but can do it when rotations are allowed when $\delta_2 \le d < \delta_1$, and (d) $K$ cannot pass through $W$ at all when $d < \delta_2$. $\bullet$ 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 $O(n^4)$.