A discrete curve-shortening equation
The authors propose a new discrete version of the so-called curve-shortening equation \(d{\mathbf r}/dt=\kappa{\mathbf n}\) where \(\kappa\) denotes the curvature and \({\mathbf n}\) the inner unit normal vector. It is chosen so that it inherits the following geometric feature of the continuous equation: the area \(S\) enclosed by the curve satisfies \(dS/dt=-2\pi\) which implies that every closed curve shrinks to a point in finite time. In the continuous case one can show that the asymptotic shape of such a shrinking curve is a circle. The authors conjecture that for their discrete version a similar property holds, namely that its asymptotic shape is an equiangular \(n\)-polygon. They treat in detail the case of a triangle, a parallelogram and of a hexagon.
scientific article; zbMATH DE number 1129314