Our current research project.

Curve Flattening

Begin with a simple, closed, planar curve, C.

Define

• X: the vector which parameterizes the curve
• T: the unit tangent to the curve, T = X' / |X'|
• N: the inward pointing unit normal
• 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) = ∫ k² 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.