Slope's choice in plane curve interpolation

The problem of plane curve construction, given by supporting points, is considered. The methods, based on the piece-wise Hermit interpolation by the polynomials of third degree (like the Fergusson and Bezier approaches), additionally require slope vectors in supporting points. Here two methods for slopes are suggested. In some sense they present extremal interpolation variants. Their convex combinations give multiparameter set of interpolating curves. There we select one-parameter set, whose parameter influences to the same extent on visual smoothness and curvature on all curve patches. On the base of numerical experiments the parameter limits have been found out and which provide the simplicity and sufficient smoothness of the interactively managed curve with the help of supporting points.