A simple method for drawing a rational curve as two Bézier segments

Abstract

In this paper we give a simple method for drawing a closed rational curve specified in terms of control points as two Bézier segments. The main result is the following: For every affine frame (r, s) (where r < s), for every rational curve F(t) specified over [r, s] by some control polygon (β0, . . . , βm) (where the βi are weighted control points or control vectors), the control points (θ0, . . . , θm) (w.r.t. [r, s]) of the rational curve G(t) = F(φ(t)) are given by θi = (-1)iβi, where φ: RP1 → RP1 is the projectivity mapping [r, s] onto RP1 - ]r, s[. Thus, in order to draw the entire trace of the curve F over [-∞, +∞], we simply draw the curve segments F([r, s]) and G([r, s]). The correctness of the method is established using a simple geometric argument about ways of partitioning the real projective line into two disjoint segments. Other known methods for drawing rational curves can be justified using similar geometric arguments.