Placing the Largest Similar Copy of a Convex Polygon Among Polygonal Obstacles

Abstract

Given a convex polygon $P$ and an environment consisting of polygonal obstacles, we find the largest similar copy of $P$ that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem. Namely, if $P$ is a convex $k$-gon and if $Q$ has $n$ corners and edges then we can find the placement of the largest similar copy of $P$ in the environment $Q$ in time $O (k^{4}n \lambda_{4}(kn)$ log$n$), where \lambda_{4} is one of the almost-linear functions related to Davenport-Schinzel sequences. If the environment consists only of points then we can find the placement of the largest similar copy of $P$ in time $O (k^{2}n \lambda_{3}(kn)$ log $n$).