Geometric Accuracy Analysis for Discrete Surface Approximation

  • Junfei Dai
  • Wei Luo
  • Shing-Tung Yau
  • Xianfeng David Gu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surface. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density.

This theoretic work gives explicit formula to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formula can be directly applied to design sampling density for data acquisition and surface reconstructions.

Furthermore, we prove the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace-Beltrami operators on the meshes are also convergent. These theoretic results lay down the theoretic foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.


Voronoi Diagram Sampling Density Principal Curvature Surface Reconstruction Delaunay Triangulation 
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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  1. 1.
    Hoppe, H., DeRose, T., Duchamp, T., McDonald, J.A., Stuetzle, W.: Surface reconstruction from unorganized points. In: SIGGRAPH, pp. 71–78 (1992)Google Scholar
  2. 2.
    Eck, M., Hoppe, H.: Automatic reconstruction of b-spline surfaces of arbitrary topological type. In: SIGGRAPH, pp. 325–334 (1996)Google Scholar
  3. 3.
    Amenta, N., Bern, M.W., Kamvysselis, M.: A new voronoi-based surface reconstruction algorithm. In: SIGGRAPH, pp. 415–421 (1998)Google Scholar
  4. 4.
    Amenta, N., Bern, M.W.: Surface reconstruction by voronoi filtering. Discrete & Computational Geometry 22, 481–504 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geometry Appl. 12, 125–141 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bajaj, C.L., Bernardini, F., Xu, G.: Automatic reconstruction of surfaces and scalar fields from 3d scans. In: SIGGRAPH, pp. 109–118 (1995)Google Scholar
  7. 7.
    Bernardini, F., Bajaj, C.L.: Sampling and reconstructing manifolds using alpha-shapes. In: CCCG (1997)Google Scholar
  8. 8.
    Ju, T., Losasso, F., Schaefer, S., Warren, J.D.: Dual contouring of hermite data. In: SIGGRAPH, pp. 339–346 (2002)Google Scholar
  9. 9.
    Floater, M.S., Reimers, M.: Meshless parameterization and surface reconstruction. Computer Aided Geometric Design 18, 77–92 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Benkö, P., Martin, R.R., Várady, T.: Algorithms for reverse engineering boundary representation models. Computer-Aided Design 33, 839–851 (2001)CrossRefGoogle Scholar
  11. 11.
    He, Y., Qin, H.: Surface reconstruction with triangular b-splines. In: GMP, pp. 279–290 (2004)Google Scholar
  12. 12.
    Hoppe, H.: Progressive meshes. In: SIGGRAPH, pp. 99–108 (1996)Google Scholar
  13. 13.
    Leibon, G., Letscher, D.: Delaunay triangulations and voronoi diagrams for riemannian manifolds. In: Symposium on Computational Geometry, pp. 341–349 (2000)Google Scholar
  14. 14.
    Elber, G.: Error bounded piecewise linear approximation of freeform surfaces. Computer Aided Design 28, 51–57 (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    Morvan, J.M., Thibert, B.: On the approximation of a smooth surface with a triangulated mesh. Computational Geometry Theory and Application 23, 337–352 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Morvan, J.M., Thibert, B.: Approximation of the normal vector field and the area of a smooth surface. Discrete and Computational Geometr 32, 383–400 (2004)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S.J., Hoppe, H.: Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24, 553–560 (2005)CrossRefGoogle Scholar
  18. 18.
    Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in multiresolution for geometric modelling, pp. 157–186. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Ben-Chen, M., Gotsman, C.: On the optimality of spectral compression of mesh data. ACM Trans. Graph. 24, 60–80 (2005)CrossRefGoogle Scholar
  20. 20.
    Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces (submitted, 2005)Google Scholar
  21. 21.
    Federer, H.: Curvature measures. Transactions of the American Mathematical Society 93, 418–491 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Cambridge (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Junfei Dai
    • 1
  • Wei Luo
    • 1
  • Shing-Tung Yau
    • 2
  • Xianfeng David Gu
    • 3
  1. 1.Center of Mathematical SciencesZhejiang Univeristy 
  2. 2.Mathematics DepartmentHarvard University 
  3. 3.Center for Visual ComputingStony Brook University 

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