Motivated by interpolation problems arising in image
analysis, computer vision, shape reconstruction and signal processing, we
develop an algorithm to simulate curve straightening flows under which
curves in $\rn$ of fixed length and prescribed boundary conditions to first
order evolve to {\em elasticae}, i.e., to (stable) critical points of the elastic
energy $E$
given by the integral of the square of the curvature function. 
We also consider variations in which the length $L$ is allowed to vary 
and the flows seek to minimize the scale-invariant elastic energy $E_{inv}$,
or the free elastic energy $E_\lambda$.   $E_{inv}$ is given by the product
of $L$ and the elastic energy $E$, and $E_\lambda$ is the energy functional
obtained by adding a term $\lambda$-proportional to the length
of the curve to $E$.  Details of the implementations, experimental results,
and applications to edge completion problems are also discussed.