# 3-dimensional shortest paths in the presence of polyhedral obstacles

## Abstract

We consider the problem of finding a minimum length path between two points in 3-dimensional Euclidean space which avoids a set of (not necessarily convex) polyhedral obstacles; we let *n* denote the number of the obstacle edges and *k* denote the number of "islands" in the obstacle space. An island is defined to be a maximal convex obstacle surface such that for any two points contained in the interior of the island, a minimal length path between these two points is strictly contained in the interior of the island; for example, a set of *i* disconnected convex polyhedra forms a set of *i* islands, however, a single non-convex polyhedron will constitute more that one island. Prior to this work, the best known algorithm required double-exponential time. We present an algorithm that runs in \(n^{k^{0(1)} }\) time and also one that runs in *O*(*n*^{log(k)}) space.

## Keywords

Mover's problem minimal movement problem shortest path Euclidean space robotics motion planning Voronoi diagram quadratic curves theory of real closed fields## Preview

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