A Simple Method For Drawing a Rational Curve as TwoB

In this paper, we give a simple method for drawing a closed rational curve speciied in terms of control points as two B ezier segments. The main result is the following: For every aane frame (r; s) (where r < s), for every rational curve F(t) speciied over r; s] by some control polygon (0 ; : : : ; m) (where the i are weighted control points or control vectors), the control points (0 ; : : : ; m) (w.r.t. r; s]) of the rational curve G(t) = F('(t)) are given by i = (?1) i i ; where ': RP 1 ! RP 1 is the projectivity mapping r; s] onto RP 1 ? ]r; s. Thus, in order to draw the entire trace of the curve F over ?1; +1], 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 justiied using similar geometric arguments.