Inaccessibility-Inside Theorem for Point in Polygon

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.