Abstract
Wachspress and mean value coordinates are two generalizations of triangular barycentric coordinates to convex polygons and have recently been used to construct mappings between polygons, with application to curve deformation and image warping. We show that Wachspress mappings between convex polygons are always injective but that mean value mappings can fail to be so in extreme cases.
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Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
Belyaev, A.: On transfinite barycentric coordinates. In: Polthier, K., Sheffer, A. (eds.) Eurographics Symposium on Geometry Processing, pp. 89–99 (2006)
Carnicer, J.M., Garcia-Esnaola, M., Peña, J.M.: Convexity of rational curves and total positivity. J. Comput. Appl. Math. 71, 365–382 (1996)
Carnicer, J.M., Garcia-Esnaola, M.: Global convexity of curves and polygons. Comput. Aided Geom. Design 15, 893–907 (1998)
Dyken, C., Floater, M.S.: Transfinite mean value interpolation. Comput. Aided Geom. Design, (to appear, 2008)
Floater, M.S.: Total positivity and convexity preservation. J. Approx. Theory 96, 46–66 (1999)
Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Design 20, 19–27 (2003)
Floater, M.S., Hormann, K., Kós, G.: A general construction of barycentric coordinates over convex polygons. Adv. Comput. Math. 24, 311–331 (2006)
Floater, M.S., Kos, G., Reimers, M.: Mean value coordinates in 3D. Comput. Aided Geom. Design 22, 623–631 (2005)
Hormann, K., Floater, M.S.: Mean value coordinates for arbitrary planar polygons. ACM Trans. on Graphics 25, 1424–1441 (2006)
Gordon, W.J., Wixom, J.A.: Pseudo-harmonic interpolation on convex domains. SIAM J. Numer. Anal. 11, 909–933 (1974)
Kestelman, H.: Mappings with non-vanishing Jacobian. Amer. Math. Monthly 78, 662–663 (1971)
Meyer, M., Barr, A., Lee, H., Desbrun, M.: Generalized barycentric coordinates on irregular polygons. Journal of Graphics Tools 7, 13–22 (2002)
Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. ACM Trans. on Graphics 24, 561–566 (2005)
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, pp. 181–186 (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)
Lee, S.L.: Mean value representations and curvatures of compact convex hypersurfaces (preprint) (2007)
Lipman, Y., Kopf, J., Cohen-Or, D., Levin, D.: GPU-assisted positive mean value coordinates for mesh deformation. In: Symposium on Geometry Processing (2007)
Schaefer, S., Ju, T., Warren, J.: A unified integral construction for coordinates over closed curves. Comput. Aided Geom. Design 24, 481–493 (2007)
Wachspress, E.L.: A Rational Finite Element Basis. Academic Press, New York (1975)
Warren, J.: Barycentric coordinates for convex polytopes. Adv. Comput. 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)
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Communicated by Helmut Pottmann.
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Floater, M.S., Kosinka, J. On the injectivity of Wachspress and mean value mappings between convex polygons. Adv Comput Math 32, 163–174 (2010). https://doi.org/10.1007/s10444-008-9098-z
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DOI: https://doi.org/10.1007/s10444-008-9098-z