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

TL;DR

This paper introduces a method to reduce the spherical point-in-polygon problem to the planar case for a class of polygons that exclude antipodal boundary points, enabling efficient solutions.

Contribution

It proves that boundary antipode-excluding (BAE) spherical polygons can be reduced to planar point-in-polygon problems and provides two algorithms for this reduction.

Findings

Reduction methods enable solving SPiP for BAE polygons using planar algorithms.

All spherical polygons within an open hemisphere are BAE.

MATLAB code demonstrates the effectiveness of the reduction methods.

Abstract

Spherical polygons used in practice are nice, but the spherical point-in-polygon problem (SPiP) has long eluded solutions based on the winding number (wn). That a punctured sphere is simply connected is to blame. As a workaround, we prove that requiring the boundary of a spherical polygon to never intersect its antipode is sufficient to reduce its SPiP problem to the planar, point-in-polygon (PiP) problem, whose state-of-the-art solution uses wn and does not utilize known interior points (KIP). We refer to such spherical polygons as boundary antipode-excluding (BAE) and show that all spherical polygons fully contained within an open hemisphere is BAE. We document two successful reduction methods, one based on rotation and the other on shearing, and address a common concern. Both reduction algorithms, when combined with a wn-PiP algorithm, solve SPiP correctly and efficiently for BAE spherical polygons. The MATLAB code provided demonstrates scenarios that are problematic for previous work.