Elastic splines III: Existence of stable nonlinear splines

Abstract Given points P 1 , P 2 , … , P n 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 π 2 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 ψ j ≔ arg P j + 1 − P j P j − P j − 1 satisfy | ψ j | Ψ for j = 2 , 3 , … , n − 1 , where Ψ ( ≈ 3 7 ∘ ) is defined in our previous article. As in our previous articles, the optimal s-curves c 1 ( α , β ) play an important role and here we show that, when | α | , | β | π 2 , they are also optimal among Hermite interpolating curves whose tangent directions are never orthogonal to the chord.