Convex Hull of Grid Points below a Line or a Convex Curve

  • Authors:

  • Hélymar Balza-Gomez;Jean-Michel Moreau;Dominique Michelucci

  • Affiliations:

  • -;-;-

  • Venue:
  • DCGI '99 Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery
  • Year:
  • 1999

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Abstract

onsider a finite non-vertical, and non-degenerate straight-line segment s =[s0 s1] in the Euclidian plane E2. We give a method for constructing the boundary of the upper convex hull of all the points with integral coordinates below (or on) s, with abscissa in [x(s0),x(s1)]. The algorithm takes O(log n) time, if n is the length of the segment. We next show how to perform a similar construction in the case where s is a finite, non-degenerate, convex arc on a quadric curve. The associated method runs in O(klog n), where n is the arc's length and k the number of vertices on the boundary of the resulting hull. This method may also be used for a line segment; in this case, k = O(log n), and the second method takes O(k2) time, compared with O(k) for the first.