Abstract
This paper gives a brief survey of two kinds of generalized barycentric coordinates, Wachspress and mean value coordinates, and their applications. Applications include surface parameterization in geometric modeling, curve and surface deformation in computer graphics, and their use as nodal shape functions for polygonal and polyhedral finite element methods.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bruvoll, S., Floater, M.S.: Transfinite mean value interpolation in general dimension. J. Comp. Appl. Math. 233, 1631–1639 (2010)
Dyken, C., Floater, M.S.: Transfinite mean value interpolation. Comp. Aided Geom. Des. 26, 117–134 (2009)
Floater, M.S.: Parametrization and smooth approximation of surface triangulations. Comp. Aided Geom. Des. 14, 231–250 (1997)
Floater, M.S.: Mean value coordinates. Comp. Aided Geom. Des. 20, 19–27 (2003)
Floater, M.S.: One-to-one piecewise linear mappings over triangulations. Math. Comp. 72, 685–696 (2003)
Floater, M.S., Gillette, A., Sukumar, N.: Gradient bounds for Wachspress coordinates on polytopes. SIAM J. Numer. Anal. 52, 515–532 (2014)
Floater, M.S., Hormann, K., Kós, G.: A general construction of barycentric coordinates over convex polygons. Adv. Comp. Math. 24, 311–331 (2006)
Floater, M.S., Kos, G., Reimers, M.: Mean value coordinates in 3D. Comp. Aided Geom. Des. 22, 623–631 (2005)
Floater, M., Kosinka, J.: On the injectivity of Wachspress and mean value mappings between convex polygons. Adv. Comp. Math. 32, 163–174 (2010)
Floater, M., Schulz, C.: Pointwise radial minimization: Hermite interpolation on arbitrary domains. Comp. Graphics Forum (Proc. Symp. Geom. Process. 2008) 27, 1505–1512 (2008)
Gillette, A., Rand, A., Bajaj, C.: Error estimates for generalized barycentric interpolation. Adv. Comp. Math. 37, 417–439 (2012)
Gordon, W.J., Wixom, J.A.: Pseudo-harmonic interpolation on convex domains. SIAM J. Numer. Anal. 11, 909–933 (1974)
Hormann, K., Floater, M.S.: Mean value coordinates for arbitrary planar polygons. ACM Trans. Graph. 25, 1424–1441 (2006)
Joshi, P., Meyer, M., DeRose, T.., Green, B., Sanocki, T.: Harmonic coordinates for character articulation. ACM Trans. Graph. 26, 71 (2007)
Ju, T., Schaefer, S., Warren, J., Desbrun, M.: A geometric construction of coordinates for convex polyhedra using polar duals. In: Desbrun, M., Pottman H. (eds.) Geometry Processing 2005, Eurographics Association 2005, pp. 181–186 (2005)
Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. ACM TOG 24, 561–566 (2005)
Langer, T., Belyaev, A., Seidel, H.-P.: Spherical barycentric coordinates. In: Polthier, K., Sheffer A. (eds.) Eurographics Symposium on Geometry Processing, pp. 81–88 (2006)
Li, X.-Y., Hu, S.-M.: Poisson coordinates. IEEE Trans. Visual. Comput. Graphics 19, 344–352 (2013)
Li, X.-Y., Ju, T., Hu, S.-M.: Cubic mean value coordinates. ACM Trans. Graph. 32, 1–10 (2013)
Lipman, Y., Kopf, J., Cohen-Or, D., Levin, D.: GPU-assisted positive mean value coordinates for mesh deformation. In: Symposium on Geometry Processing, pp. 117–123 (2007)
Lipman, Y., Levin, D., Cohen-Or, D.: Green coordinates. ACM Trans. Graph. 27, 1–10 (2008)
Meyer, M., Barr, A., Lee, H., Desbrun, M.: Generalized barycentric coordinates on irregular polygons. J. Graph. Tools 7, 13–22 (2002)
Rand, A., Gillette, A., Bajaj, C.: Interpolation error estimates for mean value coordinates over convex polygons. Adv. Comp. Math. 39, 327–347 (2013)
Sibson, R.: A brief description of natural neighbour interpolation. In: Barnett, V. (ed.) Interpreting Multivariate Data, pp. 21–36. John Wiley, Chichester (1981)
Sukumar, N.: Construction of polygonal interpolants: a maximum entropy approach. Int. J. Num. Meth. Eng. 61, 2159–2181 (2004)
Sukumar, N., Tabarraei, A.: Conforming polygonal finite elements. Int. J. Num. Meth. Eng. 61, 2045–2066 (2004)
Talischi, C., Paulino, G.H., Le, C.H.: Honeycomb Wachspress finite elements for structural topology optimization. Struct. Multidisc. Optim. 37, 569–583 (2009)
Thiery, J.-M., Tierny, J., Boubekeur, T.: Jacobians and Hessians of mean value coordinates for closed triangular meshes. Vis. Comput. 29, 217–229 (2013)
Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13, 743–768 (1963)
Wachspress, E.: A rational finite element basis. Academic Press, New York (1975)
Wachspress, E.L.: Barycentric coordinates for polytopes. Comput. Math. Appl. 61, 3319–3321 (2011)
Warren, J.: Barycentric coordinates for convex polytopes. Adv. Comp. Math. 6, 97–108 (1996)
Warren, J., Schaefer, S., Hirani, A., Desbrun, M.: Barycentric coordinates for convex sets. Adv. Comp. Math. 27, 319–338 (2007)
Wicke, M., Botsch, M., Gross, M.: A finite element method on convex polyhedra. Proc. Eurograph. 07, 355–364 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Floater, M.S. (2014). Wachspress and Mean Value Coordinates. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-06404-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06403-1
Online ISBN: 978-3-319-06404-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)