  Polypols are natural generalizations of polytopes, with boundaries given by
nonlinear algebraic hypersurfaces. We describe polypols in the plane and in
3-space that admit a unique adjoint hypersurface and study them from an
algebro-geometric perspective. We relate planar polypols to positive geometries
introduced originally in particle physics, and identify the adjoint curve of a
planar polypol with the numerator of the canonical differential form associated
with the positive geometry. We settle several cases of a conjecture by
Wachspress claiming that the adjoint curve of a regular planar polypol does not
intersect its interior. In particular, we provide a complete characterization
of the real topology of the adjoint curve for arbitrary convex polygons.
Finally, we determine all types of planar polypols such that the rational map
sending a polypol to its adjoint is finite, and explore connections of our
topic with algebraic statistics.
