Piecewise polynomial curves with normalized derivatives

In this paper, a new method for constructing piecewise polynomial curves is presented. The first derivative and the second derivative of the given curve are non-negative normalized combination of the first order divided differences and non-negative combination of the second order divided differences of the control points respectively. Under the derivative constraint conditions, the given curves contain shape parameters that can locally be used to adjust a curve. The proper ranges for the adjustable shape parameters are given. The optimal value of the shape parameter is obtained for minimizing the non-negative combination coefficients of the second derivatives of the curves. Some geometric examples are provided to show the feasibility of such curve representations as well as the approximation effect to the control polygon.