关键词: 插值曲线, 形状参数, 保正, 保单调, 保凸

Abstract: This article aims to construct a shape-preserving polynomial interpolation curve. Firstly, a
set of cubic polynomial functions with parameters in literature is proved to be a totally positive basis.
With this basis, we then define a piecewise interpolation polynomial curve with two local shape
parameters. The curve has G1 continuity at the join points. The sufficient conditions for the
interpolation curve to be positivity-preserving, monotonicity-preserving and convexity-preserving are
given. These conditions restrict the relationship between the two local shape parameters. By
transformation, no matter what kind of shape characteristic the interpolation curve of the data points
keeps, each segment still has two independent shape parameters. When the data points are both
positive and monotonous, just considering the monotonicity-preserving conditions, we can obtain the
interpolation curve not only monotonicity-preserving but also positivity-preserving. When the data
points are both monotonous and convex, just considering the convexity-preserving conditions, we can
obtain the interpolation curve not only convexity-preserving but also monotonicity-preserving. When
the data points are positive, monotonous and convex, just considering the convexity-preserving
conditions, we can get the interpolation curve with positivity-preserving, monotonicity-preserving,
and convexity-preserving simultaneously. The interpolation curve is proved to be bounded and its error is estimated.

Key words: interpolation curve, shape parameter, positivity-preserving, monotonicity-preserving;
convexity-preserving