Searching for empty convex polygons

Authors:
D. P. Dobkin;H. Edelsbrunner;M. H. Overmars
Affiliations:
Department of Computer Science, Princeton University, Princeton, New Jersey;SDepaartment of Computer Science, University of ]llinois at Urbana-Champaign, Urbana, Illinois;aDepartment of Computer Science, University of Utrecht, P. O. Box 80012 NL-3508 TA Utrecht, the Netherlands
Venue:
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Year:
1988

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Abstract

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.