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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Heidelberg (2001)
Benjamini, I., Lovász, L.: Harmonic and analytic functions on graphs. J. of Geometry 76, 3–15 (2003)
Hjelle, Ø., Dæhlen, M.: Triangulations and Applications. In: Mathematics and Visualization, Springer, Heidelberg (2006)
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)
Iserles, A.: A First Course in Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)
Warren, J., Schaefer, S., Hirani, A.N., Desbrun, M.: Barycentric coordinates for convex sets. Advances in Computational Mathematics (to appear)
Hormann, K.: Barycentric coordinates for arbitrary polygons in the plane. Technical Report IfI-05-05, Department of Informatics, Clausthal University of Technology (February 2005)
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)
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)
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)
Sugihara, K.: Surface interpolation based on new local coordinates. Computer Aided Design 13(1), 51–58 (1999)
Sibson, R.: A vector identity for the Dirichlet tessellation. Math. Proc. of Cambridge Philosophical Society 87, 151–155 (1980)
Hiyoshi, H., Sugihara, K.: Voronoi-based interpolation with higher continuity. In: Symposium on Comp. Geom., pp. 242–250 (2000)
Lovász, L.: Discrete analytic functions: a survey. Technical report, Microsoft Research (2000)
Varga, R.S.: Matrix Iterative Analysis. PrenticeHall, Englewood Cliffs, NJ, USA (1962)
Wachspress, E.: A rational finite element basis. Academic Press, San Diegoss (1975)
Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)
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)
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
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)