Determining in Linear Time the Minimum Area Convex Hull of Two Polygons

Abstract

Abstract As preprocessing for the two-dimensional cutting stock problem or pallet loading problem, we compute the minimum area convex hull. Given two polygons P and Q, we find their relative positions such that the convex hull encasing them is minimum in area. Let N be the total number of vertices in P and Q. We determine the minimum area convex hull in O(N) time. Q is allowed to translate by any amount relative to P, while assuming a constant number of orientations. Instead of recomputing the convex hull after every translation, we update the computed area at certain critical points. Linearity follows by showing that there are O(N) such critical points.