Elastic Splines I: Existence

Abstract

Given interpolation points $P_1,P_2,\ldots ,P_m$ 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 that an interpolating curve is admissible if each piece, connecting two consecutive points $P_i$ and $P_{i+1}$, is an s-curve, where an s-curve is a planar curve which first turns monotonically at most $180^\circ$ in one direction and then turns monotonically at most $180^\circ$ in the opposite direction. Our main result is that among all admissible 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, that connects two given unit tangent vectors.