On de Casteljau-type algorithms for rational Bézier curves

We consider the space of rational functions of degree nn with a common denominator. It is shown that–in addition to the standard rational de Casteljau algorithm–the corresponding rational Bézier curves admit up to n!n! different de Casteljau-type algorithms, depending on the ordering of the elementary factors of the polynomial. Our observations generalize recent results of Han et al. (2014), which cover the case of denominators of the form ∏i=1n(1−t+qi−1t) where qq is a positive constant, to rational curves with general denominators.