Generating Realistic Roofs over a Rectilinear Polygon

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 $(n-4)/2 \choose \lfloor(n-4)/4\rfloor$ 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(n4) preprocessing time. We also present an O(n5)-time algorithm for computing a realistic roof with minimum height or volume.

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