Finding an Optimal Bridge between Two Polygons

Abstract

Let π(a, b)denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a, b), denoted by gd(a, b), in contrast with the Euclidean distance between a and b, denoted by d(a, b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge)that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′, p), d(p, q)and gd(q, q′), with any p′ € P and any q′ € Q, is minimized. We present an O(n log³ n)time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n ²)time bound.