A geometric characterization of parametric cubic curves

In this paper, we analyze planar parametric cubic curves to determine conditions for loops, cusps, or inflection points. By expressing the curve to be analyzed as a linear combination of control points, it can be transformed such that three of the control points are mapped to specific locations on the plane. We call this image curve the canonical curve. Affine maps do not affect inflection points, cusps, or loops, so the analysis can be applied to the canonical curve instead of the original one. Since the first three points are fixed, the canonical curve is completely characterized by the position of its fourth point. The analysis therefore reduces to observing which region of the canonical plane the fourth point occupies. We demonstrate that for all parametric cubes expressed in this form, the boundaries of these regions are tonics and straight lines. Special cases include Bézier curves, B-splines, and Beta-splines. Such a characterization forms the basis for an easy and efficient solution to this problem.