Preserving Monotonicity of Ball Curve and it's Curvature Profile

Preserving the properties of real data is crucial especially to maintain the information and original shape of the data. This paper proposes a rational cubic Ball function that can interpolate real data as well as preserving the monotonicity of the data. Four shape parameters are proposed and sufficient conditions are imposed to the function. Some numerical results are presented together with comparisons of different values of shape parameters. The rational cubic Ball function proposed attained $C$¹ continuity and managed to preserve the monotonicity of the data. The curvature profile of the interpolants are used as the tool to measure the smoothness of curves. Real behaviour of the interpolated curve can be observed from the curvature profile. In conclusion, smooth interpolants are reflected by the low value of their curvature profile. Higher curvature profile indicates curve with high oscillations and do not produce a smooth interpolant.

I. Introduction

Cubic curves were found to be the most effective curve to represent a variety types of data [1]. Some researches used rational cubic Bézier curve [2], cubic B-spline curve [3], cubic Hermite polynomials [4] and a cubic Ball curve [5]. Cubic curves are favourable since cubic polynomials contain continuous first and second derivatives. Quartic or higher-order splines produced instabilities due to higher order polynomials.