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Discrete Harmonic Functions from Local Coordinates

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Mathematics of Surfaces XII (Mathematics of Surfaces 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4647))

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Abstract

In this work we focus on approximations of continuous harmonic functions by discrete harmonic functions based on the discrete Laplacian in a triangulation of a point set. We show how the choice of edge weights based on generalized barycentric coordinates influences the approximation quality of discrete harmonic functions. Furthermore, we consider a varying point set to demonstrate that generalized barycentric coordinates based on natural neighbors admit discrete harmonic functions that continuously depend on the point set.

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References

  1. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  2. Benjamini, I., Lovász, L.: Harmonic and analytic functions on graphs. J. of Geometry 76, 3–15 (2003)

    Article  MATH  Google Scholar 

  3. Hjelle, Ø., Dæhlen, M.: Triangulations and Applications. In: Mathematics and Visualization, Springer, Heidelberg (2006)

    Google Scholar 

  4. Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. Computer Graphics (Annual Conference Series) 29, 173–182 (1995)

    Google Scholar 

  5. Iserles, A.: A First Course in Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  6. Warren, J., Schaefer, S., Hirani, A.N., Desbrun, M.: Barycentric coordinates for convex sets. Advances in Computational Mathematics (to appear)

    Google Scholar 

  7. Hormann, K.: Barycentric coordinates for arbitrary polygons in the plane. Technical Report IfI-05-05, Department of Informatics, Clausthal University of Technology (February 2005)

    Google Scholar 

  8. Bobach, T., Umlauf, G.: Natural neighbor interpolation and order of continuity. In: GI Lecture Notes in Informatics: Visualization of Large and Unstructured Data Sets, pp. 69–86. Springer, Heidelberg (2006)

    Google Scholar 

  9. Christ, N.H., Friedberg, R., Lee, T.D.: Weights of links and plaquettes in a random lattice. Nuclear Physics B 210(3), 337–346 (1982)

    Article  MathSciNet  Google Scholar 

  10. Belikov, V., Ivanov, V., Kontorovich, V., Korytnik, S., Semenov, A.: The non-Sibsonian interpolation: A new method of interpolation of the values of a function on an arbitrary set of points. Comp. Mathematics and Mathematical Physics 37(1), 9–15 (1997)

    MathSciNet  Google Scholar 

  11. Sugihara, K.: Surface interpolation based on new local coordinates. Computer Aided Design 13(1), 51–58 (1999)

    Article  MathSciNet  Google Scholar 

  12. Sibson, R.: A vector identity for the Dirichlet tessellation. Math. Proc. of Cambridge Philosophical Society 87, 151–155 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hiyoshi, H., Sugihara, K.: Voronoi-based interpolation with higher continuity. In: Symposium on Comp. Geom., pp. 242–250 (2000)

    Google Scholar 

  14. Lovász, L.: Discrete analytic functions: a survey. Technical report, Microsoft Research (2000)

    Google Scholar 

  15. Varga, R.S.: Matrix Iterative Analysis. PrenticeHall, Englewood Cliffs, NJ, USA (1962)

    Google Scholar 

  16. Wachspress, E.: A rational finite element basis. Academic Press, San Diegoss (1975)

    MATH  Google Scholar 

  17. Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ju, T., Liepa, P., Warren, J.: A general geometric construction of coordinates in a convex simplicial polytope. Computer Aided Geometric Design 24, 161–178 (2007)

    Article  MathSciNet  Google Scholar 

  19. Flötotto, J.: A coordinate system associated to a point cloud issued from a manifold: definition, properties and applications. PhD thesis, Université de Nice-Sophia Antipolis (September 2003), http://www.inria.fr/rrrt/tu-0805.html

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Author information

Authors and Affiliations

  1. CS Dept., University of Kaiserslautern, Germany

    Tom Bobach & Georg Umlauf

  2. School of Computing and Informatics, Prism Lab., Arizona State University, USA

    Gerald Farin & Dianne Hansford

Authors
  1. Tom Bobach
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  2. Gerald Farin
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  3. Dianne Hansford
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  4. Georg Umlauf
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Editor information

Ralph Martin Malcolm Sabin Joab Winkler

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© 2007 Springer-Verlag Berlin Heidelberg

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Bobach, T., Farin, G., Hansford, D., Umlauf, G. (2007). Discrete Harmonic Functions from Local Coordinates. In: Martin, R., Sabin, M., Winkler, J. (eds) Mathematics of Surfaces XII. Mathematics of Surfaces 2007. Lecture Notes in Computer Science, vol 4647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73843-5_6

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  • DOI: https://doi.org/10.1007/978-3-540-73843-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73842-8

  • Online ISBN: 978-3-540-73843-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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