Pancharatnam-Berry phase algorithm to calculate the area of arbitrary polygons on the Poincaré sphere.

We introduce a very efficient noniterative algorithm to calculate the signed area of a spherical polygon with arbitrary shape on the Poincaré sphere. The method is based on the concept of the geometric Berry phase. It can handle diverse scenarios like convex and concave angles, multiply connected domains, overlapped vertices, sides and areas, self-intersecting polygons, holes, islands, cogeodesic vertices, random polygons, and vertices connected with long segments of great circles. A set of MATLAB routines of the algorithm is included. The main benefits of the algorithm are the ability to handle all manner of degenerate shapes, the shortness of the program code, and the running time.