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 γL and exponential parameterization. The latter is commonly used to either fit the so-called reduced data Qm={qi}i=0m for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve γ passing through Qm. The exponential parameterization governed by a single parameter (Formula Presented) replaces such discrete set of unavailable knots {ti}i=0m ((Formula Presented) - an internal clock) with some new values {ti}i=0m ((Formula Presented) - an external clock). In order to compare γ with γL the selection of some ɸI→ I should be predetermined. For some applications and theoretical considerations the function ɸI→ I needs to form an injective mapping (e.g. in length estimation of γ with any γ fitting Qm). We formulate and prove two sufficient conditions yielding ɸ as injective for given Qm and analyze their asymptotic character which forms an important question for Qm 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(γ) by d(γ) (once m→ ∞). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curve γ=γ O ɸ controlled by the appropriate choice of interpolation knots and of mapping ɸ (and/or possibly Qm).