Shape-preserving curve interpolation

Abstract

This work is a contribution towards the graphical display of 2D data when they are convex, monotone and positive. A piecewise rational function in a cubic/cubic form is proposed, which, in each interval, involves four free parameters in its construction. These four free parameters have a direct geometric interpretation, making their use straightforward. Illustrations of their effect on the shape of the rational function are given. Two of these free parameters are constrained to preserve the shape of convex, monotone and positive data, while the other two parameters are utilized for the modification of positive, monotone and convex curves to obtain a visually pleasing curve. The problem of shape preservation of data lying above a line is also discussed. The method that is presented applies equally well to data or data with derivatives. The developed scheme is computationally economical and pleasing. The error of rational interpolating function is also derived when the arbitrary function being interpolated is C ³ in an interpolating interval. The order of approximation is .

Keywords:

• spline
• positivity
• monotony
• convexity
• visualization

2000 AMS Subject Classifications :

• 68U05
• 65D05
• 65D07
• 65D18

Acknowledgements

The authors are grateful to the anonymous referees for their healthy and useful comments towards the improvement of this paper.