Approximating Polygonal Objects by Deformable Smooth Surfaces

  • Ho-lun Cheng
  • Tony Tan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


We propose a method to approximate a polygonal object by a deformable smooth surface, namely the t-skin defined by Edelsbrunner for all 0< t < 1. We guarantee that they are homeomorphic and their Hausdorff distance is at most ε >0. Such construction makes it possible for fully automatic, smooth and robust deformation between two polygonal objects with different topologies. En route to our results, we also give an approximation of a polygonal object with a union of balls.


Skin Surface Voronoi Cell Weighted Point Skin Body Voronoi Vertex 
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  1. 1.
    Amenta, N., Kolluri, R.K.: Accurate and Efficient Unions of Balls. In: Proc. 16th Sympos. Computational Geometry, pp. 119–128. ACM-SIAM (2000)Google Scholar
  2. 2.
    Cheng, H.-l., Edelsbrunner, H., Fu, P.: Shape Space from Deformation. Comput. Geometry: Theory and Applications 19, 191–204 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cheng, H.-l., Shi, X.-w.: Guaranteed Quality Triangulation of Molecular Skin Surfaces. In: Proc. IEEE Visualization, pp. 481–488 (2004)Google Scholar
  4. 4.
    Cheng, H.-l., Tan, T.: Subdividing alpha complex. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 179–190. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H.: Deformable Smooth Surface Design. Discrete and Computational Geometry 21, 87–115 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edelsbrunner, H., Mucke, E.P.: Simulation of Simplicity: a Technique to Cope with Degenerate Cases in Geometric Algorithms. ACM Trans. Graphics 9, 66–104 (1990)zbMATHCrossRefGoogle Scholar
  7. 7.
    Hubbard, P.M.: Approximating Polyhedra with Spheres for Time-critical Collision Detection. ACM Transactions on Graphics 15(3), 179–210 (1996)CrossRefGoogle Scholar
  8. 8.
    Kruithof, N., Vegter, G.: Approximation by Skin Surfaces. In: Proc. 8th Sympos. Solid Modeling and Applications, pp. 86–95. ACM-SIAM (2003)Google Scholar
  9. 9.
    Ranjan, V., Fournier, A.: Matching and Interpolation of Shapes Using Unions of Circles. Computer Graphics Forum 15(3), 129–142 (1996)CrossRefGoogle Scholar
  10. 10.
    Sharf, A., Shamir, A.: Feature-sensitive 3D Shape Matching. In: Proc. Computer Graphics International, pp. 596–599 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ho-lun Cheng
    • 1
  • Tony Tan
    • 1
  1. 1.School of ComputingNational University of Singapore 

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