Generating realistic roofs over a rectilinear polygon

Abstract

Given a simple rectilinear polygon P in the xy-plane, a roof over P is a terrain over P whose faces are supported by planes through edges of P that make a dihedral angle π/4 with the xy-plane. In this paper, we introduce realistic roofs by imposing a few additional constraints. We investigate the geometric and combinatorial properties of realistic roofs, and show a connection with the straight skeleton of P. We show that the maximum possible number of distinct realistic roofs over P is ( ⌊(n-4)/4⌋ (n-4)/2) when P has n vertices. We present an algorithm that enumerates a combinatorial representation of each such roof in O(1) time per roof without repetition, after O(n 4) preprocessing time. We also present an O(n 5)-time algorithm for computing a realistic roof with minimum height or volume.