Apollonian de Casteljau-type algorithms for complex rational Bezier curves

We describe a new de Casteljau-type algorithm for complex rational Bezier curves. After proving that these curves exhibit the maximal possible circularity, we construct their points via a de Casteljau-type algorithm over complex numbers. Consequently, the line segments that correspond to convex linear combinations in affine spaces are replaced by circular arcs. In difference to the algorithm of Sanchez-Reyes (2009), the construction of all the points is governed by (generically complex) roots of the denominator, using one of them for each level. Moreover, one of the bi-polar coordinates is fixed at each level, independently of the parameter value. A rational curve of the complex degree n admits generically n! distinct de Casteljau-type algorithms, corresponding to the different orderings of the denominator's roots.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).