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Straight Skeletons of Three-Dimensional Polyhedra

  • Gill Barequet
  • David Eppstein
  • Michael T. Goodrich
  • Amir Vaxman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5193)

Abstract

We study the straight skeleton of polyhedra in 3D. We first show that the skeleton of voxel-based polyhedra may be constructed by an algorithm taking constant time per voxel. We also describe a more complex algorithm for skeletons of voxel polyhedra, which takes time proportional to the surface-area of the skeleton rather than the volume of the polyhedron. We also show that any n-vertex axis-parallel polyhedron has a straight skeleton with O(n 2) features. We provide algorithms for constructing the skeleton, which run in O( min (n 2logn,klogO(1) n)) time, where k is the output complexity. Next, we show that the straight skeleton of a general nonconvex polyhedron has an ambiguity, suggesting a consistent method to resolve it. We prove that the skeleton of a general polyhedron has a superquadratic complexity in the worst case. Finally, we report on an implementation of an algorithm for the general case.

Keywords

Voronoi Diagram Medial Axis Full Version Binary Search Tree Event Queue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    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. 117–126. Springer, Heidelberg (1996)Google Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. J. of Universal Computer Science 1(12), 752–761 (1995)Google Scholar
  3. 3.
    Barequet, G., Goodrich, M.T., Levi-Steiner, A., Steiner, D.: Contour interpolation by straight skeletons. Graphical Models 66(4), 245–260 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bittar, E., Tsingos, N., Gascuel, M.-P.: Automatic reconstruction of unstructured 3D data: Combining a medial axis and implicit surfaces. Computer Graphics Forum 14(3), 457–468 (1995)CrossRefGoogle Scholar
  5. 5.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Wathen-Dunn, W. (ed.) Models for the Perception of Speech and Visual Form, pp. 362–380. MIT Press, Cambridge (1967)Google Scholar
  6. 6.
    Cheng, S.-W., Vigneron, A.: Motorcycle graphs and straight skeletons. In: Proc. 13th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 156–165 (January 2002)Google Scholar
  7. 7.
    Culver, T., Keyser, J., Manocha, D.: Accurate computation of the medial axis of a polyhedron. In: Proc. 5th ACM Symp. on Solid Modeling and Applications, New York, NY, pp. 179–190 (1999)Google 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)Google Scholar
  9. 9.
    Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and cutting paper. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 104–118. Springer, Heidelberg (2000)Google Scholar
  10. 10.
    Dey, T.K., Zhao, W.: Approximate medial axis as a Voronoi subcomplex. Computer-Aided Design 36, 195–202 (2004)CrossRefGoogle Scholar
  11. 11.
    Eppstein, D.: Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete & Computational Geometry 13, 111–122 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Eppstein, D.: Fast hierarchical clustering and other applications of dynamic closest pairs. ACM J. Experimental Algorithmics 5(1), 1–23 (2000)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete & Computational Geometry 22(4), 569–592 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Foskey, M., Lin, M.C., Manocha, D.: Efficient computation of a simplified medial axis. J. of Computing and Information Science in Engineering 3(4), 274–284 (2003)CrossRefGoogle Scholar
  15. 15.
    Haunert, J.-H., Sester, M.: Using the straight skeleton for generalisation in a multiple representation environment. In: ICA Workshop on Generalisation and Multiple Representation (2004)Google Scholar
  16. 16.
    Held, M.: On computing Voronoi diagrams of convex polyhedra by means of wavefront propagation. In: Proc. 6th Canadian Conf. on Computational Geometry, pp. 128–133 (August 1994)Google Scholar
  17. 17.
    Price, M.A., Armstrong, C.G., Sabin, M.A.: Hexahedral mesh generation by medial surface subdivision: Part I. Solids with convex edges. Int. J. for Numerical Methods in Engineering 38(19), 3335–3359 (1995)zbMATHCrossRefGoogle Scholar
  18. 18.
    Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Discrete & Computational Geometry 12, 327–345 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sheehy, D.J., Armstrong, C.G., Robinson, D.J.: Shape description by medial surface construction. IEEE Trans. on Visualization and Computer Graphics 2(1), 62–72 (1996)CrossRefGoogle Scholar
  20. 20.
    Sherbrooke, E.C., Patrikalakis, N.M., Brisson, E.: An algorithm for the medial axis transform of 3d polyhedral solids. IEEE Trans. on Visualization and Computer Graphics 2(1), 45–61 (1996)CrossRefGoogle Scholar
  21. 21.
    Tănase, M., Veltkamp, R.C.: Polygon decomposition based on the straight line skeleton. In: Proc. 19th Ann. ACM Symp. on Computational Geometry, pp. 58–67 (June 2003)Google Scholar
  22. 22.
    Wiernik, A., Sharir, M.: Planar realizations of nonlinear Davenport-Schinzel sequences by segments. Discrete & Computational Geometry 3, 15–47 (1988)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Gill Barequet
    • 1
  • David Eppstein
    • 2
  • Michael T. Goodrich
    • 2
  • Amir Vaxman
    • 1
  1. 1.Dept. of Computer ScienceTechnion—Israel Institute of TechnologyHaifaIsrael
  2. 2.Computer Science DepartmentUniv. of CaliforniaIrvine

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