  In this paper we classify convex compact ancient solutions to the affine
curve shortening flow: namely, any convex compact ancient solution to the
affine curve shortening flow must be a shrinking ellipse. The method combines a
rescaling argument inspired by \cite{Wang}, affine invariance of the equation
and monotonicity of the affine isoperimetric ratio. We will give two proofs.
The essential ideas are related, but the first one uses level set represenation
of the evolution. The second proof employs Schauder's estimates, and it also
provides a new simple proof for the corresponding classification result to the
higher dimensional affine normal flow.
