Elastic Splines I: Existence

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.