Rational cubic spline interpolation with shape control

A rational cubic spline, with shape control parameters, has been discussed here with the view to its application in computer graphics. It incorporates both conic sections and parametric cubic curves as special cases. An efficient scheme is presented which constructs a curve interpolating a set of given data points and allows subsequent interactive alteration of the shape of the curve by changing the shape control and shape preserving parameters associated with each curve segment. The parameters (weights), in the description of the spline curve can be used to modify the shape of the curve, locally and globally. The rational cubic spline retains parametric C2 smoothness. The stitching of the conic segments also preserves C2 continuity at the neighboring given points. An exact derivative as well as a very simple distance-based approximated derivative schemes are presented to calculate control points. The curve scheme is interpolatory and can plot parabolic, hyperbolic, elliptic, and circular splines independently as well as segments of a rational cubic spline. We discuss complex cases of elliptic arcs in space and introduce intermediate point interpolation scheme which can force the curve to pass through a given point between any segments.