Parameterizations and Lagrange Cubics for Fitting Multidimensional Data

Abstract

This paper discusses the issue of interpolating data points in arbitrary Euclidean space with the aid of Lagrange cubics hat{gamma }^L and exponential parameterization. The latter is commonly used to either fit the so-called reduced data Q_m={q_i}_{i=0}^m for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve gamma passing through Q_m. The exponential parameterization governed by a single parameter lambda in [0,1] replaces such discrete set of unavailable knots {t_i}_{i=0}^m (t_iin I - an internal clock) with some new values {hat{t}_i}_{i=0}^m (hat{t}_iin hat{I} - an external clock). In order to compare gamma with hat{gamma }^L the selection of some phi : Irightarrow hat{I} should be predetermined. For some applications and theoretical considerations the function phi :Irightarrow hat{I} needs to form an injective mapping (e.g. in length estimation of gamma with any hat{gamma } fitting Q_m). We formulate and prove two sufficient conditions yielding phi as injective for given Q_m and analyze their asymptotic character which forms an important question for Q_m 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 d(gamma ) by d(hat{gamma }) (once mrightarrow infty ). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curve tilde{gamma }=hat{gamma }circ phi controlled by the appropriate choice of interpolation knots and of mapping phi (and/or possibly Q_m).