A double step algorithm for rendering parabolic curves

Abstract

Development of fast algorithms for rendering lines and curves has continued to be an active area of computer graphics research since Bresenham First published his line drawing algorithm in the mid sixties. Although there has been some work in developing a general curve rendering algorithm, most of the activity has concerned the development of algorithms for special cases of the quadratic curve described by the equation Ax^ + 2Bxy + 2Cx + Dy^ + 2Ey + F = 0. The most efficient of these algorithms are based on the midpoint method described by Akcn in which the implicit form, f(x,y) = 0, of the equation describing the curve is segmented based on its slope and convexity into cases where the choice of pixels to approximate the curve is reduced to two possibilities'. An integer value, the decision variable, can then be used to choose the closest pixel to the curve. Recently Wu and Rokne have developed a variation of this approach that calculates two pixel positions within each iteration of the algorithm when rendering lines and circles^. In this paper we incorporate elements of their double-step algorithm with the parabolic function rendering algorithm developed by Watson and Hodges to derive a fast algorithm for rendering parabolic functions of the form y = C jx^ + C2X + C3 3. The algorithm also provides a simple form of antialiasing. To achieve performance we have compromised slightly on accuracy, since the actual position of the curve in some cases may differ by as much as one unit from the plotted pixel position. Recent work by Wu and Rokne, however, argues that in the case when the choice of a more distant pixel agrees with the convexity of the curve, the overall shape of the curve may be even belter represented than by a