Blended barycentric coordinates

Generalized barycentric coordinates are widely used to represent a point inside a polygon as an affine combination of the polygon's vertices, and it is desirable to have coordinates that are non-negative, smooth, and locally supported. Unfortunately, the existing coordinate functions that satisfy all these properties do not have a simple analytic expression, making them expensive to evaluate and difficult to differentiate. In this paper, we present a new closed-form construction of generalized barycentric coordinates, which are non-negative, smooth, and locally supported. Our construction is based on the idea of blending mean value coordinates over the triangles of the constrained Delaunay triangulation of the input polygon, which needs to be computed in a preprocessing step. We experimentally show that our construction compares favourably with other generalized barycentric coordinates, both in terms of quality and computational cost.