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 pi/4 with the xy-plane. According to this definition, some roofs may have faces isolated from the boundary of P or even local minima, which are undesirable for several practical reasons. 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 that the straight skeleton induces a realistic roof with maximum height and volume. We also show that the maximum possible number of distinct realistic roofs over P is ((n-4)/2 left perpendicular(n-4)/4right perpendicular) 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. (C) 2013 Elsevier B.V. All rights reserved.