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Polyhedral Gauss Maps and Curvature Characterisation of Triangle Meshes

  • Lyuba Alboul
  • Gilberto Echeverria
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)

Abstract

We design a set of algorithms to construct and visualise unambiguous Gauss maps for a large class of triangulated polyhedral surfaces, including surfaces of non-convex objects and even non-manifold surfaces. The resulting Gauss map describes the surface by distinguishing its domains of positive and negative curvature, referred to as curvature domains. These domains are often only implicitly present in a polyhedral surface and cannot be revealed by means of the angle deficit. We call the collection of curvature domains of a surface the Gauss map signature. Using the concept of the Gauss map signature, we highlight why the angle deficit is sufficient neither to estimate the Gaussian curvature of the underlying smooth surface nor to capture the curvature information of a polyhedral surface. The Gauss map signature provides shape recognition and curvature characterisation of a triangle mesh and can be used further for optimising the mesh or for developing subdivision schemes.

Keywords

Negative Curvature Positive Curvature Triangle Mesh Polyhedral Surface Discrete Curvature 
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.
    Alboul, L., van Damme, R.: Polyhedral metrics in surface reconstruction. In: Mullineux, G. (ed.) The Mathematics of Surfaces VI, Oxford, pp. 171–200 (1996)Google Scholar
  2. 2.
    Alboul, L.: Optimising triangulated polyhedral surfaces with self-intersections. In: Wilson, M.J., Martin, R.R. (eds.) Mathematics of Surfaces. LNCS, vol. 2768, pp. 48–72. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Banchoff, T.F.: Critical points and curvature for embedded polyhedral surfaces. Amer. Math. Monthly 77, 475–485 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Banchoff, T.F., Kühnel, W.: Tight submanifolds, smooth and polyhedral. In: Cecil, T.E., Chern, S.-S. (eds.) Tight and Tight Submanifolds, vol. 32, pp. 52–117. MSRI Publications (1997)Google Scholar
  5. 5.
    Borelli, V., Cazals, F., Morvan, J.-M.: On the angular defect of triangulations and the pointwise approximation of curvature. Comp. Aided Geom. Design 20, 319–341 (2003)CrossRefGoogle Scholar
  6. 6.
    Brehm, U., Kühnel, W.: Smooth approximation of polyhedral surfaces with respect to curvature measures. Global differential geometry, 64–68 (1979)Google Scholar
  7. 7.
    Calladine, C.R.: Gaussian Curvature and Shell Structures. In: Gregory, J.A. (ed.) The Mathematics of Surfaces, pp. 179–196. University of Manchester (1984)Google Scholar
  8. 8.
    Dyn, N., et al.: Optimising 3D triangulations using discrete curvature analysis. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in CAGD, pp. 1–12. Vanderbilt University Press, Nashville (2001)Google Scholar
  9. 9.
    Kilian, M.: Differentialgeometrische Konzepte für Dreiecksnetze, Master thesis, University of Karlsruche, p. 135 (2004)Google Scholar
  10. 10.
    Kühnel, W.: Differential geometry. Curves-Surfaces-Manifolds. Amer. Math. Society (2002)Google Scholar
  11. 11.
    Little, J.J.: Extended Gaussian images, mixed volumes, shape reconstruction. In: Proc. of the first annual symposium on Comp. Geom., pp. 15–23 (1985)Google Scholar
  12. 12.
    Lowekamp, B., Rheingans, P., Yoo, T.S.: Exploring surface characteristics with interactive Gaussian images: a case study. In: Proc. of the conference on Visualization 2002, pp. 553–556 (2002)Google Scholar
  13. 13.
    Maltret, J.-L., Daniel, M.: Discrete curvatures and applications: a survey. Research report LSIS-02-004 (March 2002)Google Scholar
  14. 14.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.-Ch., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–59. Springer, Heidelberg (2003)Google Scholar
  15. 15.
    Peng, J., Li, Q., Ja Kuo, C.-C., Zhou, M.: Estimating Gaussian Curvatures from 3D Meshes. In: Rogowitz, B.E., Pappas, Th.N. (eds.)Google Scholar
  16. 16.
    Sullivan, J.: Curvature measures for discrete surfaces, http://torus.math.uiuc.edu/jms/Papers/dscrv.pdf

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Lyuba Alboul
    • 1
  • Gilberto Echeverria
    • 1
  1. 1.Sheffield Hallam UniversitySheffieldUK

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