In recent work by physicists, positive geometries are defined as certain semi-algebraic sets together with a meromorphic differential form called the canonical form: calculating scattering amplitudes reduces to determining the canonical form. Examples of planar positive geometries are polygons and, more generally, “curved” polygons, so-called polypols. The latter were proposed by Wachspress as generalized algebraic finite elements. In his quest to define barycentric coordinates for polypols, he introduced the adjoint curve of a rational polypol.
We show that a rational regular polypol gives a positive geometry and give an explicit expression for its canonical form in terms of the adjoint and boundary curves of the polypol. In the special case that the polypol is a convex polygon, we show that the adjoint curve is hyperbolic and describe its nested ovals.
This talk is based on joint work with K. Kohn, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn, M.-S. Sorea, and S. Telen.