On the Convex Hull of the Integer Points in a Bi-circular Region

  • Authors:

  • Valentin E. Brimkov

  • Affiliations:

  • Mathematics Department, SUNY Buffalo State College, Buffalo, USA NY 14222

  • Venue:
  • IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
  • Year:
  • 2009

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Abstract

Given a set S ******2, denote $S_{\mathbb {Z}} = S \cap \mathbb {Z}^2$. We obtain bounds for the number of vertices of the convex hull of S *** , where S ******2 is a convex region bounded by two circular arcs. Two of the bounds are tight bounds--in terms of arc length and in terms of the width of the region and the radii of the circles, respectively. Moreover, an upper bound is given in terms of a new notion of "set oblongness." The results complement the well-known O (r 2/3) bound [2] which applies to a disc of radius r .