G^2 Two-Point Hermite Rational Cubic Interpolation

We consider the problem of G^2 two-point Hermite interpolation by a rational cubic. Given two points with tangent vectors and curvatures, the necessary and sufficient conditions are placed on the weights of the rational cubic curve which ensures that (i) if the data suggest a C -shaped curve, the rational cubic interpolates a C -shaped curve without loops, cusps, or inflections, and (ii) if the data suggest an S -shaped curve, the rational cubic interpolates an S -shaped curve with a single inflection, no loops and no cusps.