Bézier curves: topological convergence of the control polygon

In terms of distance (e.g., Hausdorff distance), the control polygon of a Bzier curve converges to the curve via de Casteljau subdivision. This convergence is widely studied in the literature. In this paper, we adapt a different point of view for the convergence problem. We study whether the control polygon preserves the topology of the associated curve. For this purpose, we deal with two main topological features of the Bzier curve, the existence of multiple points and the convexity.