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

Consider a nite non-vertical, and non-degenerate straight-line segment s = s0; s1] in the Euclidian plane E 2. 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 nite, non-degenerate, convex arc on a quadric curve. The associated method runs in O(k log 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(k 2) time, compared with O(k) for the rst. This paper will consider integral hulls, i.e. convex hulls of set of points with integral coordinates in the Euclidean plane (grid points). A paper on 3D lattice convex hulls is Reveill es and Yaacoub's RY95]. This section describes a method for computing the boundary of the upper convex hull of all the grid points located below a given non-degenerate, non-vertical, line segment s (including those possibly on it). Obviously, only the upper hull boundary requires some computation. Hence, we shall use the terms \hull" and \upper hull" indiierently in the sequel, unless where speciied otherwise. The supporting line of the segment has equation y = ax + b, and is unambiguously described by a and b, since s is supposed not to be vertical; typically, these coeecients are rational numbers, although the method also applies to any computable eld, i.e. a eld in which the usual exact arithmetic operations (sum, diierence, product, division, sign, oor) are available. Convention: We assume w.l.o.g. that the slope of s is non-negative (a 0): this allows a shorter presentation, and the other situation (a < 0) is deduced by symmetry. Since we want the convex hull of the integral points below s, we may restrict the problem to a segment the endpoints of which have integral abscisss, say x 0 = dx(s 0)e 2 Zand x 1 = bx(s 1)c 2 Z. Such a segment is the hypothenuse of the rectangle triangle with vertices (x 0 , ax 0 +b), (x 1 , ax 0 +b), and (x 1 , ax …