Convex subdivision of a Bézier curve

Using the Walsh coincidence theorem, we show in this paper that the shape of the control polygon of a Bézier curve is closely related to the location of the complex roots of the corresponding polynomial. This explains why a convex polynomial over an interval does not necessarily produce a convex control polygon with respect to the same interval. Furthermore, our findings lead to an interesting algorithm of subdividing a Bézier curve into segments with convex control polygons.