Computating the width of a set

Michael E. Houle, Godfried T. Toussaint

Abstract: Given a set of points P = {p1,p2,…,pn} in three dimensions, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in Ο(n log n + I) time and Ο(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I - Ο(n2). If P is a set of points in the plane, this complexity can be reduced to Ο(n log n). Finally, for simple polygons linear time suffices.