Parameterizations and Lagrange Cubics for Fitting Multidimensional Data

This paper discusses the issue of interpolating data points in arbitrary Euclidean space with the aid of Lagrange cubics [Formula: see text] and exponential parameterization. The latter is commonly used to either fit the so-called reduced data [Formula: see text] for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve [Formula: see text] passing through [Formula: see text]. The exponential parameterization governed by a single parameter [Formula: see text] replaces such discrete set of unavailable knots [Formula: see text] ([Formula: see text] - an internal clock) with some new values [Formula: see text] ([Formula: see text] - an external clock). In order to compare [Formula: see text] with [Formula: see text] the selection of some [Formula: see text] should be predetermined. For some applications and theoretical considerations the function [Formula: see text] needs to form an injective mapping (e.g. in length estimation of [Formula: see text] with any [Formula: see text] fitting [Formula: see text]). We formulate and prove two sufficient conditions yielding [Formula: see text] as injective for given [Formula: see text] and analyze their asymptotic character which forms an important question for [Formula: see text] getting sufficiently dense. The algebraic conditions established herein are also geometrically visualized in 3D plots with the aid of Mathematica. This work is supplemented with illustrative examples including numerical testing of the underpinning convergence rate in length estimation [Formula: see text] by [Formula: see text] (once [Formula: see text]). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curve [Formula: see text] controlled by the appropriate choice of interpolation knots and of mapping [Formula: see text] (and/or possibly [Formula: see text]).