Exponential and polynomial methods for applying tension to an interpolating spline curve

Abstract

Two approaches are explained for the application of tension to a cubic interpolating spline curve. This was first accomplished by Schweikertusing an exponential-based formulation, the spline under tension, and later Nielson developed a polynomial alternative called the v-spline. A generalized form of the spline under tension and the v-spline are juxtaposed in a comparative, unified fashion, deriving each one from a variational principle. The v-spline is derived using Hermite basis functions, to emphasize its relation to the conventional cubic interpolatory spline. The derivation of each formulation is completed for two different choices of end conditions and the matrix corresponding to each of these conditions is analyzed. Various desirable properties are shown to be possessed by the matrices for the spline under tension for all tension values, and by the matrices for the v-spline for certain ranges of tension values, which are derived.