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Spherical polygons used in practice are nice, but the sphericalpoint-in-polygon problem (SPiP) has long eluded solutions based on the windingnumber (wn). That a punctured sphere is simply connected is to blame. As aworkaround, we prove that requiring the boundary of a spherical polygon tonever intersect its antipode is sufficient to reduce its SPiP problem to theplanar, point-in-polygon (PiP) problem, whose state-of-the-art solution uses wnand does not utilize known interior points (KIP). We refer to such sphericalpolygons as boundary antipode-excluding (BAE) and show that all sphericalpolygons fully contained within an open hemisphere is BAE. We document twosuccessful reduction methods, one based on rotation and the other on shearing,and address a common concern. Both reduction algorithms, when combined with awn-PiP algorithm, solve SPiP correctly and efficiently for BAE sphericalpolygons. The MATLAB code provided demonstrates scenarios that are problematicfor previous work.

On the Reduction of the Spherical Point-in-Polygon Problem for Antipode-Excluding Spherical Polygons