Positive Gordon-Wixom Coordinates

Barycentric coordinates that are positive and have smoothness equal to the smoothness of the boundary. These coordinates are given by an integral rather than as the result of an iterative optimization.

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BibTeX

@article{Manson:2011:PGWC,
  author  = {Josiah Manson and
             Kuiyu Li and
             Scott Schaefer},
  title   = {Positive Gordon-Wixom Coordinates},
  journal = {Computer Aided Design (Proceedings of Geometric
             and Physical Modeling)},
  volume  = {43},
  number  = {11},
  year    = {2011},
  pages   = {1422--1426},
}

Abstract

We introduce a new construction of transfinite barycentric coordinates for arbitrary closed sets in two dimensions. Our method extends weighted Gordon-Wixom interpolation to non-convex shapes and produces coordinates that are positive everywhere in the interior of the domain and that are smooth for shapes with smooth boundaries. We achieve these properties by using the distance to lines tangent to the boundary curve to define a weight function that is positive and smooth. We derive closed-form expressions for arbitrary polygons in two dimensions and compare the basis functions of our coordinates with several other types of barycentric coordinates.

Supplemental

PGWC_SPM_2011.pptx

This is the set of slides that I used in my presentation of the paper at SPM 2011.

Bary2bezier.zip

Code for evaluating the different coordinate types shown in the paper. Sorry if this code is disorganized.

License

The images, executables, and code supplied are from the web page http://josiahmanson.com. These materials are free to use for non-commercial purposes. Any works that use materials from this web page should acknowledge Josiah Manson and the paper Positive Gordon-Wixom Coordinates. For commercial use, please contact Josiah Manson (josiahmanson@gmail.com).