A new method for automatically constructing convexity-preserving interpolatory splines

Constructing a convexity-preserving interpolating curve according to the given planar data points is a problem to be solved in computer aided geometric design (CAGD). So far,almost all methods must solve a system of equations or recur to a complicated iterative process,and most of them can only generate some function-form convexity-preserving interpolating curves which are unaccommodated with the parametric curves,commonly used in CAGD systems. In order to overcome these drawbacks,this paper proposes a new method that can automatically generate some parametric convexity-preserving polynomial interpolating curves but dispensing with solving any system of equations or going at any iterative computation. The main idea is to construct a family of interpolating spline curves first with the shape parameter α as its family parameter;then,using the positive conditions of Bernstein polynomial to respectively find a range in which the shape parameter α takes its value for two cases of global convex data points and piecewise convex data points so as to make the corresponding interpolating curves convexity-preserving and C 2 (or G 1) continuous. The method is simple and convenient,and the resulting interpolating curves possess smooth distribution of curvature. Numerical examples illustrate the correctness and the validity of theoretical reasoning.