Inaccessibility-Inside Theorem for Point in Polygon

TL;DR

This paper introduces a new theoretical proof for determining whether a point lies inside a polygon, providing an unambiguous and mathematically sound solution applicable to both simple and self-intersecting polygons.

Contribution

It presents a novel analytical method based on properties of epigraphs and hypographs, improving upon ambiguous existing solutions like crossing number and winding number rules.

Findings

Provides a mathematically correct solution for simple polygons.

Offers an unambiguous method for self-intersecting polygons.

Enhances accuracy over traditional point-in-polygon algorithms.

Abstract

The manuscript presents a theoretical proof in conglomeration with new definitions on Inaccessibility and Inside for a point S related to a simple or self intersecting polygon P. The proposed analytical solution depicts a novel way of solving the point in polygon problem by employing the properties of epigraphs and hypographs, explicitly. Contrary to the ambiguous solutions given by the cross over for the simple and self intersecting polygons and the solution of a point being multiply inside a self intersecting polygon given by the winding number rule, the current solution gives unambiguous and singular result for both kinds of polygons. Finally, the current theoretical solution proves to be mathematically correct for simple and self intersecting polygons.