Elastic Splines I: Existence

TL;DR

This paper proves the existence of minimal bending energy interpolating curves composed of s-curves between points in the plane, extending the understanding of elastic splines beyond the classical minimal energy solutions.

Contribution

It establishes the existence of minimal bending energy admissible interpolating curves made of s-curves, generalizing elastic spline theory to cases where classical solutions do not exist.

Findings

Existence of minimal bending energy admissible interpolating curves.

Constructive proof of minimal energy s-curve connecting two tangent vectors.

Extension of elastic spline theory to non-linearly aligned points.

Abstract

Given interpolation points $P_1,P_2,\ldots,P_n$ in the plane, it is known that there does not exist an interpolating curve with minimal bending energy, unless the given points lie sequentially along a line. We say than an interpolating curve is {\it admissable} if each piece, connecting two consecutive points $P_i$ and $P_{i+1}$, is an s-curve, where an {\it s-curve} is a planar curve which first turns at most $180^\circ$ in one direction and then turns at most $180^\circ$ in the opposite direction. Our main result is that among all admissable interpolating curves there exists a curve with minimal bending energy. We also prove, in a very constructive manner, the existence of an s-curve, with minimal bending energy, which connects two given unit tangent vectors.