Hodograph approach to geometric characterization of parametric cubic curves (Q1315883)

For a planar parametric cubic curve, the author discusses the possible zeros, loops, cusps and one or two inflection points. After reviewing previous approaches to this question the so-called hodograph is investigated. This quadratic curve is the derivative of the cubic Bézier curve realized with respect to the origin of the coordinate system. Now, the analysis reduces the curve-characterization problem into a point-localization problem among the regions in the characteristic diagram, where regions are defined by straight lines and/or a conic section. A simple characterization algorithm concludes this nice paper.