Elastic splines III: Existence of stable nonlinear splines

Abstract

Given points P1,P2,…,Pn in the complex plane, a stable nonlinear spline is an interpolating curve, of arbitrary length, whose bending energy is minimal among all nearby interpolating curves. We show that if the chord angles of a restricted elastic spline f, at interior nodes, are less than π2 in magnitude, then f is a stable nonlinear spline. As a consequence, existence of stable nonlinear splines is now proved for the case when the stencil angles ψj≔argPj+1−PjPj−Pj−1 satisfy |ψj|<Ψ for j=2,3,…,n−1, where Ψ (≈37∘) is defined in our previous article. As in our previous articles, the optimal s-curves c1(α,β) play an important role and here we show that, when |α|,|β|<π2, they are also optimal among Hermite interpolating curves whose tangent directions are never orthogonal to the chord.