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

In this paper we give a simple method for drawing a closed rational curve s pecified in terms of central points as two Bezier 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 (beta(o),...,beta(m)) (where the beta(i) are weighted control points or control vectors), the control po ints (theta(o),..., theta(m)) (w.r.t. [r, s]) of the rational curve G(t) = F(phi(t)) are given by theta(i) = (-1)(i)beta(i), where phi : 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 [-infinity, infinity], we simply draw the curve segments F([r, s]) and G([r, s]). The correctness of the method is established using a simple geometric argum ent about ways of partitioning the real projective line into two disjoint s egments. Other known methods for drawing rational curves can be justified u sing similar geometric arguments.