Classification using normal curves

The analysis of parametric curves for the presence of shape characteristics such as loops, cusps, and inflections is considered. In contrast to previous methods that use heavily algebraic techniques to study polynomial cubic curves, our approach, based on the classical notion of normal curves, is predominately geometric. The approach not only provides new insight into the cubic polynomial case, it can also be applied to rational curves and to curves of degree greater than three. We demonstrate the use of the technique to analyze planar cubic curves, planar rational cubic curves, and polynomial curves of degree four in the plane and in three-space.