  We consider curve shortening flow of arbitrary codimension in an Euclidean
background. We show that, close to a singularity, the flow is asymptotically
planar, paralleling Altschuler's work in the case of space curves, and analyse
the blow-up limits of the flow. Using these results, we then prove that the
curve shortening flow of initial curves with an entropy bound converges to a
round point in finite time.
