Positive and smooth Gordon-Wixom coordinates

Generalized barycentric coordinates (GBC) are widely used in computer graphics and related areas and there are a few kinds of GBC in the literature. In this paper we propose positive and smooth Gordon-Wixom coordinates (PSGWC) for the interior points of planar polygons, which are smooth and inherit all nice properties of positive Gordon-Wixom coordinates (PGWC) that are only C⁰ for planar polygons. The basic idea of PSGWC is that we replace the polygon with an active curve for each inner point in the process of computing coordinates, and thus PSGWC can achieve the required continuity by adjusting the continuity order of the active curve. We also introduce a method for improving the quality of PSGWC by updating the polygon based on angle bisectors iteratively. Several numerical examples are included to show the effect of PSGWC.