Extensions of Uniform Cubic B-Spline Curve with Local Shape Parameters

The construction of B-spline curves with shape parameters has become the hotspot in computer aided geometric design. However, the shape parameters of the curves in previous papers are all global parameters. In order to introduce B-spline curves with local shape parameters, two classes of polynomial blending functions with local shape parameters are presented in this paper. Both of them are extensions of cubic B-spline basic functions. The blending functions have similar properties of classical cubic B-spline basic functions. Based on the given blending functions, a method of generating piecewise polynomial curves with local shape parameters is proposed. By changing the values of the local shape parameters, the shape of the curves can be manipulated locally. The cubic curves can be manipulated to approximate the cubic B-spline curves from their sides away from the control polygons by changing the values of the shape parameter. Similarly, the quartic curves can also be manipulated to approximate the cubic B-spline curves from their both sides by changing the values of the shape parameters. The geometric meanings of the local shape parameters are also discussed. Their applications in curve design and interpolation are also presented. The modeling examples illustrate that these new curves are very valuable for computer aided geometric design.