The ubiquitous ellipse

We discuss three different affine invariant evolution processes for smoothing planar curves. The first one is derived from a geometric heat-type flow, both the initial and the smoothed curves being differentiable. The second smoothing process is obtained from a discretization of this affine heat equation. In this case, the curves are represented by planar polygons. The third process is based on B-spline approximations. For this process, the initial curve is a planar polygon, and the smoothed curves are differentiable and even analytic. We show that, in the limit, all three affine invariant smoothing processes collapse any initial curve into an elliptic point.