  Curve shortening in the $z$-plane in which, at a given point on the curve,
the normal velocity of the curve is equal to the curvature, is shown to satisfy
$S_tS_z=S_{zz}$, where $S(z,t)$ is the Schwarz function of the curve. This
equation is shown to have a parametric solution from which the known explicit
solutions for curve shortening flow; the circle, grim reaper, paperclip and
hairclip, can be recovered.
