Generic Smooth Connection Functions - A New Analytic Approach to Interpolation

Abstract

We present a generic solution to the fundamental problem of how to connect two points in a plane by a smooth curve that goes through these points with a given slope. The smoothness of any curve depends both on its curvature and its length. The smoothest curves correspond to a particular compromise between minimal curvature and minimal length. They can be described by a class of functions that satisfy certain boundary conditions and minimize a weight functional. The value of this functional is given essentially by the average of the curvature raised to some power νtimes the length of the curve. The parameter νdetermines the importance of minimal curvature with respect to minimal length. In order to find the functions that obtain the minimal weight, we use extensively notions that are well-known in classical mechanics. The minimization of the weight functional via the Euler-Lagrange formalism leads to a highly non-trivial differential equation. Using the symmetries of the problem it is possible to find conserved quantities, that help to simplify the problem to a level where the solution functions can be written in a closed form for any given ν. Applying the appropriate coordinate transformation to these solutions allows to adjust them to all possible boundary conditions.