Selecting Knots Locally for Curve Interpolation with Quadratic Precision

Abstract

There are several prevailing methods for selecting knots for curve interpolation. A desirable criterion for knot selection is whether the knots can assist an interpolation scheme to achieve the reproduction of polynomial curves of certain degree if the data points to be interpolated are taken from such a curve. For example, if the data points are sampled from an underlying quadratic polynomial curve, one would wish to have the knots selected such that the resulting interpolation curve reproduces the underlying quadratic curve; and in this case the knot selection scheme is said to have quadratic precision. In this paper we propose a local method for determining knots with quadratic precision. This method improves on upon our previous method that entails the solution of a global equation to produce a knot sequence with quadratic precision. We show that this new knot selection scheme results in better interpolation error than other existing methods, including the chord-length method, the centripetal method and Foley’s method, which do not possess quadratic precision.