k: the signed curvature of the curve (positive where C is convex)

T’ = k N |X'|

Let J(C) be the flatness energy of the curve,

J(C) = ∫ k^{2}ds.

Allow C to evolve by making local changes in order to minimize J.

We conjecture that C will evolve into a circle of infinite radius. We recall that k is the inverse of the radius, R, of the osculating circle, so k = 0 requires R = 8.

What We Do

Currently, we are studying the behavior of curves under a "curve flattening" evolution.