Inaccessibility-Inside Theorem for Point in Polygon

The paper explores the ideology behind the concept of inside a simple or self intersecting polygon by presenting new definitions on Inaccessibility and Inside for a point S related to a polygon P. It further goes on to give a theoretical proof to establish a relation as to when a point is inaccessible and inside a polygon. The proposed analytical solution depicts a novel way of tackling the well known 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 multi-ply inside a self intersecting polygon given by the winding number rule, the current solution gives unambiguous and singular result for both kinds of polygons. The solution also deals with the rare and subtle issues of ray crossing a odd or even number of vertices, an edge as well as analytically questions the idea of a point being multi-ply inside a polygon. Finally, the current solution proves to be mathematically correct for simple and self intersecting polygons.