A discrete curve-shortening equation

The usual "curve-shortening equation" describes the planar motion of a smooth curve that moves in a direction normal to itself with a speed proportional to its local curvature. We present here an analogous theory for the planar motion of a discrete (i.e., piecewise linear) curve. In the discrete case, an arbitrary nonintersecting, closed iV-sided curve shrinks in on itself, and its enclosed area vanishes in a finite time. We conjecture that the discrete curve tends to an equi-angle AT-polygon as it shrinks. Geometrical models which describe the motion of manifolds (curves and surfaces) in a higher dimensional space have been used successfully in various branches of science [12]. A simple example that has been studied extensively is the so-called curveshortening equation [4, 6, 7], which describes the planar motion of a smooth curve that moves normal to itself with a speed proportional to its local curvature: dr TT 17 = *, (1) Here r is the position vector of a point on the curve, n is the inward-facing unit normal vector of the curve, and K is its curvature. Two important features of such motion are: (1) that every smooth closed curve shrinks to a point in a finite time; and (2) that the asymptotic shape of such a shrinking curve is always a circle.