A constructive framework for minimal energy planar curves

Given points P1,P2,…,Pn in the plane, we are concerned with the problem of finding a fair curve which interpolates the points. We assume that we have a method in hand, called a basic curve method, for solving the geometric Hermite interpolation problem of fitting a regular C∞ curve between two points with prescribed tangent directions at the endpoints. We also assume that we have an energy functional which defines the energy of any basic curve. Using this basic curve method repeatedly, we can then construct G1 curves which interpolate the given points P1,P2,…,Pn. The tangent directions at the interpolation points are variable and the idea is to choose them so that the energy of the resulting curve (i.e., the sum of the energies of its pieces) is minimal. We give sufficient conditions on the basic curve method, the energy functional, and the interpolation points for (a) existence, (b) G2 regularity, and (c) uniqueness of minimal energy interpolating curves. We also identify a one-parameter family of basic curve methods, based on parametric cubics, whose minimal energy interpolating curves are unique and G2 under suitable conditions. One member of this family looks very promising and we suggest its use in place of conventional C2 parametric cubic splines.