Curvature in Triangle Meshes

  • Jakob Andreas Bærentzen
  • Jens Gravesen
  • François Anton
  • Henrik Aanæs


In many cases notions from differential geometry can be usefully extended to piecewise planar surfaces, and this chapter covers curvature measures on triangle meshes. A frequently used principle is to obtain a smooth surface approximation and to estimate the curvature from this approximation. Alternatively, the integral of some curvature measures can be computed from a small region of the mesh and then normalized by dividing by the area of that region.

Following these principles, we first discuss how to extend the definition of a surface normal to the edges and vertices of a triangle mesh. Next, we cover the estimation of Gaußian and mean curvature on a triangle mesh. Finally, techniques for computing the shape operator are discussed—followed by a discussion on how to obtain the principal curvatures from the shape operator.


Smooth Surface Principal Curvature Principal Direction Curvature Measure Shape Operator 
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.


  1. 1.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99, pp. 317–324. ACM Press, New York (1999). doi: 10.1145/311535.311576 Google Scholar
  3. 3.
    Thürmer, G., Wüthrich, C.A.: Computing vertex normals from polygonal facets. J. Graph. Tools 3(1), 43–46 (1998) MATHCrossRefGoogle Scholar
  4. 4.
    Bærentzen, J., Aanæs, H.: Signed distance computation using the angle weighted pseudo-normal. IEEE Trans. Vis. Comput. Graph. 11(3), 243–253 (2005) CrossRefGoogle Scholar
  5. 5.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976) MATHGoogle Scholar
  6. 6.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.-C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  7. 7.
    Cohen-Steiner, D., Morvan, J.-M.: Restricted Delaunay triangulations and normal cycle. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, SCG ’03, pp. 312–321. ACM Press, New York (2003). doi: 10.1145/777792.777839 Google Scholar
  8. 8.
    Hildebrandt, K., Polthier, K.: Anisotropic filtering of non-linear surface features. Comput. Graph. Forum 23(3), 391–400 (2004) CrossRefGoogle Scholar
  9. 9.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins, Baltimore (1996) MATHGoogle Scholar
  10. 10.
    Hamann, B.: Curvature approximation for triangulated surfaces. In: Geometric Modelling, pp. 139–153. Springer, London (1993) CrossRefGoogle Scholar
  11. 11.
    Surazhsky, T., Magid, E., Soldea, O., Elber, G., Rivlin, E.: A comparison of Gaussian and mean curvatures triangular meshes. In: Proceedings of IEEE International Automation (ICRA2003), Taipei, Taiwan, 14–19 September, pp. 1021–1026 (2003) Google Scholar
  12. 12.
    Taubin, G.: Estimating the tensor of curvature of a surface from a polyhedral approximation. In: Proceedings of the Fifth International Conference on Computer Vision, ICCV ’95, p. 902. IEEE Comput. Soc., Washington (1995) Google Scholar
  13. 13.
    Hertzmann, A., Zorin, D.: Illustrating smooth surfaces. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’00, pp. 517–526. ACM, New York (2000). doi: 10.1145/344779.345074 Google Scholar
  14. 14.
    Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M.: Anisotropic polygonal remeshing. ACM Trans. Graph. 22(3), 485–493 (2003). doi: 10.1145/882262.882296 CrossRefGoogle Scholar
  15. 15.
    Kälberer, F., Nieser, M., Polthier, K.: QuadCover-Surface Parameterization using Branched Coverings. Comput. Graph. Forum 26(3), 375–384 (2007) MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Jakob Andreas Bærentzen
    • 1
  • Jens Gravesen
    • 2
  • François Anton
    • 1
  • Henrik Aanæs
    • 1
  1. 1.Department of Informatics and Mathematical ModellingTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark

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