Relative Convex Hull Determination from Convex Hulls in the Plane
A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.
KeywordsRelative convex hull Geodesic convex hull Shortest Jordan curve Shortest path Minimal length polygon Minimal perimeter polygon
The first author gratefully acknowledges support for this research from SEP and CONACYT Mexico, grant No. CB-2011-01-166223. The authors would like to thank very much to the reviewers for their careful study of the work, and for their constructive criticism and helpful comments which were important to improve the presentation of the paper.
- 3.Klette, G.: A recursive algorithm for calculating the relative convex hull. In: Proceedings of 25th International Conference on Image and Vision Computing, New Zealand, pp. 1-7. IEEE Computer Society (2010). doi: 10.1109/IVCNZ.2010.6148857, 978-1-4244-9631-0/10
- 6.Klette, R., Kovalevsky, V., Yip, B.: On the length estimation of digital curves. In: SPIE Proceedings of Vision Geometry VIII, vol. 3811, pp. 117–129. SPIE (1999)Google Scholar
- 16.Reyes Becerril, H.: Versión revisada de un algorítmo que determina la cubierta convexa relativa de polígonos simples en el plano, Master Thesis. Dept. of Automatic Control, CINVESTAV-IPN, Mexico City, September 2013Google Scholar
- 23.Sloboda, F., Zatco, B., Stoer, J.: On approximation of planar one-dimensional continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160. Springer, Singapore (1998)Google Scholar
- 24.Toussaint, G.T.: An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In: Proceedings of EURASIP, Signal Processing III: Theories and Applications, Part 2, pp. 853–856. North-Holland (1986)Google Scholar
- 25.Toussaint, G.T.: Computing geodesic properties inside a simple polygon. Invited paper, Special Issue on Geometric Reasoning, Revue D’Intelligence Artificielle 3(2), 9–42 (1989)Google Scholar
- 27.Wiederhold, P., Villafuerte, M.: Triangulation of cross-sectional digital straights segments and minimum length polygons for surface area estimation. In: Wiederhold, P., Barneva, R.P. (eds.) Progress in Combinatorial Image Analysis, pp. 79–92. Research Publishing Services, Singapore (2009)Google Scholar
- 28.Yu, L., Klette, R.: An approximative calculation of relative convex hulls for surface area estimation of 3D digital objects. ICPR 1, 131–134 (2002)Google Scholar