Discrete Elastica

In this paper we shall address the problem of designing discretized, piecewise linear curves minimizing functionals of various types that could be regarded as discretized versions of the "curve energy" considered in the context of splines. We shall argue that solving the nonlinear equations that result from considering "Discrete elastica" is a feasible alternative to curve design. Furthermore we claim that almost noting is lost in the process of discretization since it is seldom the case that the differential equations arising from the continuous optimization problem can be explicitly solved. This being the case, those equations must be solved by numerical integration, hence we shall end up with a discretized solution anyhow. We therefore propose to start with a discretized problem and concentrate numerical procedures that cen effectively solve them.