A univariate method for plane elastic curves

Abstract

The problem to interpolate Hermite-type data (i.e. two points with attached tangent vectors) with elastic curves of prescribed tension is known to have multiple solutions. A method is presented that finds all solutions of length not exceeding one period of its curvature function. The algorithm is based on algebraic relations between discrete curvature information which allow to transform the problem into a univariate one. The method operates with curves that by construction partially interpolat~ the given data. Hereby the objective function of the problem is drastically simplified. A bound on the maximum curvature value is established that provides an interval containing all solutions.

Second order spiral splines

“…6 On the other hand, the study of elastic splines does not reduce in this way. 7 The objection raised at the end of [16] does not really apply when T is given. 8 As noted on p.184 of [12], clothoidal splines are sometimes used to construct initial guesses for the computation of elastic splines.…”

“…The elastica are completely known in terms of elliptic functions [22], with simplifications for m = 2, which is the case of interest for the present paper. Although elastic splines (sometimes called nonlinear splines or true splines) are highly regarded 7 as interpolants, they are less widely used than cubic polynomial splines. This is because the interpolation conditions for elastic splines require the solution of a system of nonlinear equations that is even more complicated 8 than for clothoidal splines.…”

Low Energy Clamped Planar Elastica

“…This contrasts with the analogous problem where x is not necessarily unit-speed and the solution is a unique cubic polynomial 2. A different problem, where L is not considered in advance, is studied by Brunnett and Wendt[3].…”