Characterization of Planar Cubic Alternative Curve

The investigation on a planar cubic Alternative curve is carried out to determine the conditions for the existence of convex, loops, cusps and inflection points. First, the cubic curve is represented by a linear combination of three control points with the basis functions formulated with two shape parameters. Upon algebraic manipulation, the constraint of shape parameters are determined and sufficient conditions which ensure that the curve is either strictly convex or with loops, cusps and inflection point are identified resulting in a shape diagram.