On the singularity of a class of parametric curves

We consider parametric curves that are represented by combination of control points and basis functions. We let a control point vary while the rest is held fixed. We show that the locus of the moving control point that yields a zero curvature point on the curve is a developable surface, the regression curve of which is the locus that guarantees a cusp on the curve. We also specify the surface that is described by those positions of the moving control point that yield a loop on the curve. Then we apply this approach to detect cusps, inflection points and loops of C-Bézier curves. Finally, we compare cubic Bézier, cubic rational Bézier and C-Bézier curves from singularity point of view.