A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics

In this paper we propose a non-stationary C¹-continuous interpolating 4-point scheme which provides users with a single tension parameter that can be either arbitrarily increased, to tighten the limit curve towards the piecewise linear interpolant between the data points, or appropriately chosen in order to represent elements of the linear spaces spanned respectively by the functions {1,x,x²,x³}, {1,x,e^{sx},e^{−sx}} and {1,x,e^{isx},e^{−isx}}. As a consequence, for special values of the tension parameter, such a scheme will be capable of reproducing all conic sections exactly. Exploiting the reproduction property of the scheme, we derive an algorithm that automatically provides the initial tension parameter required to exactly reproduce a curve belonging to one of the previously mentioned spaces, whenever the initial data are uniformly sampled on it. The performance of the scheme is illustrated by a number of examples that show the wide variety of effects we can achieve in correspondence of different tension values.