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.

中文翻译：

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Pancharatnam-Berry相位算法可计算Poincaré球面上任意多边形的面积。

我们引入了一种非常有效的非迭代算法来计算庞加莱球上任意形状的球形多边形的有符号面积。该方法基于几何贝里相的概念。它可以处理各种情况，例如凸角和凹角，多重连接的区域，重叠的顶点，侧面和区域，自相交多边形，孔，岛，共面顶点，随机多边形以及与长圆弧段相连的顶点。该算法包括一组MATLAB例程。该算法的主要优点是能够处理各种形式的简并形状，程序代码的简短性以及运行时间。