Geometric Hermite Interpolation Based on the Representation of Circular Arcs ⋆

A new heuristic method of geometric Hermite interpolation is presented to construct a planar cubic rational Bézier curve with two points and two unit tangent directions. The integral, which shows the change rate of the curvature, is taken as the energy function to measure the fairness of the parameter curve. Hence the curvature of the new curve is more stable. Since that the energy function of circular arc is minimum, the necessary and sufficient condition for the cubic rational Bézier curve representation of circular arc, instead of the sufficient condition mentioned in Farin’s scheme, is applied to construct the planar cubic rational Bézier curve. The energy function of new curve can be less then the one of Farin’s curve. The numerical example is presented to illustrate the validity of the algorithm.