Geodesic
Point in Polygon Problem via Homotopy and Hopf's Degree Theorem
Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 283-3.

Voir la notice de l'article provenant de la source Heldermann Verlag

The current work revisits the point-in-polygon problem by providing a novel solution that explicitly employs the properties of epigraphs and hypographs. A new definition of inaccessibility and inside is provided in order to accurately specify the meaning of inclusion of a point within or without a polygon. Via Poincar�'s ideas on homotopy and Hopf's Degree Theorem from topology, a relationship between inaccessibility and inside is established, and it is shown that consistent results are obtained for peculiar cases of both non-intersecting and self-intersecting polygons while investigating the point inclusion test w.r.t. a polygon. Through illustrative examples, the novel method addresses the issues of ambiguous solutions given by the Cross Over for both non-intersecting and self-intersecting polygons and a point being labeled as multi-ply inside a self-intersecting polygon by the Winding Number Rule, by providing an unambiguous and singular result for both kinds of polygons. The proposed solution bridges the gap between Cross Over and Winding Number Rule for complex cases.
Classification : 65D18, 68U05
Mots-clés : Point in polygon, epigraph, hypograph, homotopy, Hopf's degree theorem 
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     author = {S. Sinha and L. Nanetti and R. J. Renken and G. J. ter Horst },
     title = {Point in {Polygon} {Problem} via {Homotopy} and {Hopf's} {Degree} {Theorem}},
     journal = {Journal for geometry and graphics},
     pages = {283--3},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2015},
     url = {http://geodesic.mathdoc.fr/item/JGG_2015_19_2_JGG_2015_19_2_a10/}
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S. Sinha; L. Nanetti; R. J. Renken; G. J. ter Horst . Point in Polygon Problem via Homotopy and Hopf's Degree Theorem. Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 283-3. http://geodesic.mathdoc.fr/item/JGG_2015_19_2_JGG_2015_19_2_a10/