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One-to-one piecewise linear mappings over triangulations

Author: Michael S. Floater
Journal: Math. Comp. 72 (2003), 685-696
MSC (2000): Primary 65N30, 58E20; Secondary 05C85, 65M50
Published electronically: October 17, 2002
MathSciNet review: 1954962
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Abstract: We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are one-to-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can be viewed as a discrete version of the Radó-Kneser-Choquet theorem for harmonic mappings, but is also closely related to Tutte's theorem on barycentric mappings of planar graphs.

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Additional Information

Michael S. Floater
Affiliation: SINTEF Applied Mathematics, P.O. Box 124 Blindern, 0314 Oslo, Norway

Keywords: Triangulation, harmonic map, maximum principle, parameterization, numerical grid generation, planar embedding
Received by editor(s): July 9, 2001
Published electronically: October 17, 2002
Article copyright: © Copyright 2002 American Mathematical Society