Exponential parameterization to fit reduced data

Abstract

This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean space with Lagrange cubics γ^L and exponential parameterization which is governed by a single parameter λ ∈ [0, 1] and replaces a discrete set of unknown knots {ti}i=0m (ti ∈ I) with new values {t^i}i=0m (t^i∈I^). To compare γ with γ^L specific mapping ϕ:I→I^ must also be selected. For certain applications and theoretical considerations ϕ should be an injective mapping(e.g. in length estimation of γ which fits Qm). We justify two sufficient conditionsyielding ϕ as injective and analyze their asymptotic representatives for Qm getting dense. The algebraic constraints established here are also geometrically visualized with the aid of Mathematica 2D and 3D plots. Examples illustrating the latter and numerical investigation for the convergence rate in length estimation of γ are also supplemented. The reparameterization issue has its applications among all in computer graphics and robot navigation for trajectory planning once e.g. a new curve γ˜=γ∘ϕ controlled by the appropriate choice of interpolation knots and ϕ (and/or possibly Qm) needs to be constructed.