Realistic roofs over a rectilinear polygon

Authors:
Hee-Kap Ahn;Sang Won Bae;Christian Knauer;Mira Lee;Chan-Su Shin;Antoine Vigneron
Affiliations:
Department of Computer Science and Engineering, POSTECH, Pohang, Republic of Korea;Department of Computer Science, Kyonggi University, Suwon, Republic of Korea;Institute of Computer Science, Universität Bayreuth, 95440 Bayreuth, Germany;Department of Computer Science, KAIST, Daejeon, Republic of Korea;Department of Digital and Information Engineering, Hankuk University of Foreign Studies, Yongin, Republic of Korea;King Abdullah University of Science and Technology (KAUST), Geometric Modeling and Scientific Visualization Center, Thuwal 23955-6900, Saudi Arabia
Venue:
Computational Geometry: Theory and Applications
Year:
2013

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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 @p/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@?(n-4)/4@?) 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.