Hodograph approach to geometric characterization of parametric cubic curves

Abstract

It is known that a planar parametric cubic curve may have zero, one or two inflection points, a loop or a cusp. These characteristics of a planar parametric cubic curve can be analysed with a hodograph, which is an instance of the first derivative of a given parametric curve. It is shown that a cubic curve can be characterized by the examination of the relative location of the origin of the coordinate system in a characteristic diagram. The analysis reduces the curve-characterization problem into a point-location problem among the regions in the characteristic diagram, where regions are defined by straight lines and/or a conic section.