A convex polygon among polygonal obstacles: placement and high-clearance motion

Given a convex polygon P and an environment consisting of polygonal obstacles, we find the placement for 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 W in time O(k⁴nλ₃(n)logn), where λ₃ is one of the almost-linear functions related to Davenport-Schinzel sequences. Based on our complexity analysis of the placement problem, we develop a high-clearance motion planning technique for a convex polygonal object moving among polygonal obstacles in the plane, allowing both rotation and translation (general motion). Given a k-sided convex polygonal object P, a set of polygonal obstacles with n corners and edges, and given initial and final positions for P, the time needed to determine a high-clearance, obstacle-avoiding path for P is O(k⁴nλ₃(n)log n).