Behaviour of exponential three-point coordinates at the vertices of convex polygons

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle’s vertices and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications, it is desirable to extend the concept of barycentric coordinates from triangles to polygons, and several variants of such generalized barycentric coordinates have been proposed in recent years. In this paper we focus on exponential three-point coordinates, a particular one-parameter family for convex polygons, which contains Wachspress, mean value, and discrete harmonic coordinates as special cases. We analyse the behaviour of these coordinates and show that the whole family is C^0 at the vertices of the polygon and C^1 for a wide parameter range