Abstract
In a recent paper by the first author, a simple proof was given of a result by Tutte on the validity of barycentric mappings, recast in terms of the injectivity of piecewise linear mappings over triangulations. In this note, we make a short extension to the proof to deal with arbitrary tilings. We also give a simple counterexample to show that convex combination mappings over tetrahedral meshes are not necessarily one-to-one.
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References
G. Choquet, Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. 69 (1945) 156–165.
E. Colin de Verdière, M. Pocchiola and G. Vegter, Tutte's barycenter method applied to isotopies, in: Electronic Proc. of the 13th Canadian Conf. on Computational Geometry, 2001, http://compgeo.math.uwaterloo.ca.
C.Ó. Dúnlaing, Personal communication (April 2002).
M.S. Floater, Parametrization and smooth approximation of surface triangulations, Comput. Aided Geom. Design 14 (1997) 231–250.
M.S. Floater, Parametric tilings and scattered data approximation, Internat. J. of Shape Modeling 4 (1998) 165–182.
M.S. Floater, One-to-one piecewise linear mappings over triangulations, Math. Comp. 72 (2003) 685–696.
M.S. Floater and C. Gotsman, How to morph tilings injectively, J. Comput. Appl. Math. 101 (1999) 117–129.
C. Gotsman and V. Surazhsky, Guaranteed intersection-free polygon morphing, Comput. Graphics 25(1) (2001) 67–75.
H. Kneser, Lösung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926) 123–124.
T. Radó, Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926) 49.
W.T. Tutte, How to draw a graph, Proc. London Math. Soc. 13 (1963) 743–768.
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Mathematics subject classifications (2000)
05C10, 05C85, 65D17, 58E20
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Floater, M.S., Pham-Trong, V. Convex combination maps over triangulations, tilings, and tetrahedral meshes. Adv Comput Math 25, 347–356 (2006). https://doi.org/10.1007/s10444-004-7620-5
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DOI: https://doi.org/10.1007/s10444-004-7620-5