Generating Realistic Roofs over a Rectilinear Polygon

  • Hee-Kap Ahn
  • Sang Won Bae
  • Christian Knauer
  • Mira Lee
  • Chan-Su Shin
  • Antoine Vigneron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


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(n 4) preprocessing time. We also present an O(n 5)-time algorithm for computing a realistic roof with minimum height or volume.


Simple Polygon Candidate Pair Polygonal Surface Rectilinear Polygon Generalize Voronoi Diagram 
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  1. 1.
    Aichholzer, O., Albertsa, D., Aurenhammer, F., Gärtner, B.: A novel type of skeleton for polygons. J. Universal Comput. Sci. 1, 752–761 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F.: Straight Skeletons for General Polygonal Figures in the Plane. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 117–226. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  3. 3.
    Barequet, G., Goodrich, M.T., Levi-Steiner, A., Steiner, D.: Straight-skeleton based contour interpolation. In: Proc. 14th ACM-SIAM Symp. Discrete Alg. (SODA), pp. 119–127 (2003)Google Scholar
  4. 4.
    Brenner, C.: Interactive modelling tools for 3d building reconstruction. In: Fritsch, D., Spiller, R. (eds.) Photogrammetric Week 1999, pp. 23–34 (1999)Google Scholar
  5. 5.
    Brenner, C.: Towards fully automatic generation of city models. Int. Archives of Photogrammetry and Remote Sensing XXXIII(Part B3), 85–92 (2000)Google Scholar
  6. 6.
    Cheng, S.-W., Vigneron, A.: Motorcycle graphs and straight skeletons. Algorithmica 47, 159–182 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cloppet, F., Stamon, G., Olivia, J.-M.: Angular bisector network, a simplified generalized Voronoi diagram: Application to processing complex intersections in biomedical images. IEEE Trans. Pattern Anal. Mach. Intell. 22, 120–128 (2000)CrossRefGoogle Scholar
  8. 8.
    Demaine, E.D., Demaine, M.L., Lindy, J.F., Souvaine, D.L.: Hinged Dissection of Polypolyhedra. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 205–217. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete Comput. Geom. 22, 569–592 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Huber, S., Held, M.: Theoretical and practical results on straight skeletons of planar straight-line graphs. In: Symposium on Computational Geometry, pp. 171–178 (2011)Google Scholar
  11. 11.
    Huber, S., Held, M.: Theoretical and practical results on straight skeletons of planar straight-line graphs. In: Proc. 27th Annu. ACM Sympos. Comput. Geom., pp. 171–178 (2011)Google Scholar
  12. 12.
    Khoshelham, K., Li, Z.L.: A split-and-merge technique for automated reconstruction of roof planes. Photogrammetric Engineering and Remote Sensing 71(7), 855–863 (2005)CrossRefGoogle Scholar
  13. 13.
    Krauß, T., Lehner, M., Reinartz, P.: Generation of coarse 3D models of urban areas from high resolution stereo satellite images. Int. Archives of Photogrammetry and Remote Sensing XXXVII, 1091–1098 (2008)Google Scholar
  14. 14.
    Laycock, R.G., Day, A.M.: Automatically generating large urban environments based on the footprint data of buildings. In: Proc. 8th ACM Sympos. Solid Model. Appl., pp. 346–351 (2003)Google Scholar
  15. 15.
    Oliva, J.-M., Perrin, M., Coquillart, S.: 3D reconstruction of complex polyhedral shapes from contours using a simplified generalized Voronoi diagram. Comput. Graph. Forum 15(3), 397–408 (1996)CrossRefGoogle Scholar
  16. 16.
    Sohn, G., Huang, X.F., Tao, V.: Using a binary space partitioning tree for reconstructing polyhedral building models from airborne lidar data. Photogrammetric Engineering and Remote Sensing 74(11), 1425–1440 (2008)CrossRefGoogle Scholar
  17. 17.
    Tǎnase, M., Veltkamp, R.C.: Polygon decomposition based on the straight line skeleton. In: Proc. 19th ACM Sympos. Comput. Geom., pp. 58–67 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Sang Won Bae
    • 2
  • Christian Knauer
    • 3
  • Mira Lee
    • 4
  • Chan-Su Shin
    • 5
  • Antoine Vigneron
    • 6
  1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
  2. 2.Department of Computer ScienceKyonggi UniversitySuwonKorea
  3. 3.Institute of Computer ScienceUniversität BayreuthBayreuthGermany
  4. 4.Department of Computer ScienceKAISTDaejeonKorea
  5. 5.Department of Digital and Information EngineeringHankuk University of Foreign StudiesYonginKorea
  6. 6.Geometric Modeling and Scientific Visualization CenterKAUSTThuwalSaudi Arabia

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