An Accurate Vertex Normal Computation Scheme

  • Huanxi Zhao
  • Ping Xiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4035)


There are a number of applications in computer graphics and computer vision that require the accurate estimation of normal vectors at arbitrary vertices on a mesh surface. One common way to obtain a vertex normal over such models is to compute it as a weighted sum of the normals of facets sharing that vertex. But numerical tests and asymptotic analysis indicate that these proposed weighted average algorithms for vertex normal computation are all linear approximations. An open question proposed in [CAGD,17:521-543, 2000] is to find a linear combination scheme of the normals of the triangular faces, based on geometric considerations, that is quadratic convergence in the general mesh case. In this paper, we answer this question in general triangular mesh case. When tested on a few random mesh with valence 4, the scheme proposed by this paper is of second order accuracy, while the existing schemes only provide first order accuracy.


Normal Vector Order Accuracy Quadratic Convergence Vertex Angle Darboux Frame 
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.
    Langer, T., Belyaev, A.G., Seidel, H.P.: Asymptotic Analysis of Discrete Normals and Curvatures of Polylines. In: Juttler, B. (ed.) Spring Conference on Computer Graphics SCCG 2005, pp. 229–232 (2005)Google Scholar
  2. 2.
    Langer, T., Belyaev, A.G., Seidel, H.-P.: Analysis and design of discrete normals and curvatures. Research Report MPI-I-2005-4-003, Max-Planck Institut für Informatik (2005)Google Scholar
  3. 3.
    Chen, S.G., Wu, J.Y.: Estimating normal vectors and curvatures by centroid weights. Computer Aided Geometric Design 21(5), 447–458 (2004)MathSciNetGoogle Scholar
  4. 4.
    Cohen, D., Kaufman, A., Bakalash, R., Bergman, S.: Real-Time Discret hading. The Visual Computer 6(1), 16–27 (1990)CrossRefGoogle Scholar
  5. 5.
    Glassner, A.S.: Computing surface normals for 3D models. In: Glassner, A.S. (ed.) Graphics Gems, pp. 562–566. Academic Press, London (1990)Google Scholar
  6. 6.
    Gouraud, H.: Continuous shading of curved surfaces. IEEE Transactions on Computers C 20(6), 623–629 (1971)zbMATHCrossRefGoogle Scholar
  7. 7.
    Jin, S.S., Lewis, R.R., West, D.: A comparison of algorithms for vertex normal computation. The Visual Computer 21, 71–82 (2005)CrossRefGoogle Scholar
  8. 8.
    Max, N.: Weights for computing vertex normals from facet normals. J Graph Tools 4(2), 1–6 (1999)Google Scholar
  9. 9.
    Meek, D., Walton, D.: On surface normal and Gaussian curvature approximation given data sampled from a smooth surface. Computer-Aided Geometric Design 17, 521–543 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Thurmer, G., Wuthrich, C.: Computing vertex normals from polygonal facets. J Graph Tools 3(1), 43–46 (1998)Google Scholar
  11. 11.
    Webber, R.E.: Ray Tracing Voxel Based Data via Biquadratic Local Surfac nterpolation. The Visual Computer 6(1), 8–15 (1990)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Yagel, R., Cohen, D., Kaufman, A.: Discrete Ray Tracing. IEEE Computer Graphics & Applications 12(5), 19–28 (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Huanxi Zhao
    • 1
  • Ping Xiao
    • 2
  1. 1.School of Information Science and EngineeringCentral South UniversityChangshaChina
  2. 2.School of mathematical science and computing technologyCentral South UniversityChangshaChina

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