The Characterization of Rational Alternative Cubic Curve

Parametric rational cubic curves to determine conditions for convex, inflections, loops, cusps and shear curves are analyzed. The rational alternative cubic curve is applied to construct the curve segment with fixed end points and end tangents. Expressing the end weights and shape parameters to be analyzed in terms of the barycentric coordinates of the shoulder point with respect to the control points, helps to determine the constraints of the curves. A method to control and preserve the segment shapes which is completely characterized by the position of its shoulder point is presented. Subsequently, it is demonstrated that for any specific shoulder point, a family of correspondent curves can be generated.