In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution. As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth. It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a singularity. If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point. The circle is the only simple closed curve that maintains its shape under the curve-shortening flow, but some curves that cross themselves or have infinite length keep their shape, including the grim reaper curve, an infinite curve that translates upwards, and spirals that rotate while remaining the same size and shape. An approximation to the curve-shortening flow can be computed numerically, by approximating the curve as a polygon and using the finite difference method to calculate the motion of each polygon vertex. Alternative methods include computing a convolution of polygon vertices and then resampling vertices on the resulting curve, or repeatedly applying a median filter to a digital image whose black and white pixels represent the inside and outside of the curve. The curve-shortening flow was originally studied as a model for annealing of metal sheets. Later, it was applied in image analysis to give a multi-scale representation of shapes. It can also model reaction–diffusion systems, and the behavior of cellular automata. The curve-shortening flow can be used to find closed geodesics on Riemannian manifolds, and as a model for the behavior of higher-dimensional flows.
Definitions A flow is a process in which the points of a space continuously change their locations or properties over time. More specifically, in a one-dimensional geometric flow such as the curve-shortening flow, the points undergoing the flow belong to a curve, and what changes is the shape of the curve, its embedding into the Euclidean plane determined by the locations of each of its points. In the curve-shortening flow, each point of a curve moves in the direction of a normal vector to the curve, at a rate proportional to the curvature. For an evolving curve represented by a two-parameter function C(s,t) where s parameterizes the arc length along the curve and t parameterizes a time in the evolution of the curve, the curve-shortening flow can be described by the parabolic partial differential equation
∂ C ∂ t = ∂ 2 C ∂ s 2 = κ n , {\displaystyle {\frac {\partial C}{\partial t}}={\frac {\partial ^{2}C}{\partial s^{2}}}=\kappa n,}
a form of the heat equation, where κ is the curvature and n is the unit normal vector. Because the ingredients of this equation, the arc length, curvature, and time, are all unaffected by translations and rotations of the Euclidean plane, it follows that the flow defined by this equation is invariant under translations and rotations (or more precisely, equivariant). If the plane is scaled by a constant dilation factor, the flow remains essentially unchanged, but is slowed down or sped up by the same factor.
Non-smooth curves In order for the flow to be well defined, the given curve must be sufficiently smooth that it has a continuous curvature. However, once the flow starts, the curve becomes analytic, and remains so until reaching a singularity at which the curvature blows up. For a smooth curve without crossings, the only possible singularity happens when the curve collapses to a point, but immersed curves can have other types of singularity. In such cases, with some care it is possible to continue the flow past these singularities until the whole curve shrinks to a single point. For a simple closed curve, using an extension of the flow to non-smooth curves based on the level-set method, there are only two possibilities. Curves with zero Lebesgue measure (including all polygons and piecewise-smooth curves) instantly evolve into smooth curves, after which they evolve as any smooth curve would. However, Osgood curves with nonzero measure instead immediately evolve into a topological annulus with nonzero area and smooth boundaries. The topologist's sine curve is an example that instantly becomes smooth, despite not even being locally connected; examples such as this show that the reverse evolution of the curve-shortening flow can take well-behaved curves to complicated singularities in a finite amount of time.
Non-Euclidean surfaces The curve-shortening flow, and many of the results about the curve-shortening flow, can be generalized from the Euclidean plane to any two-dimensional Riemannian manifold. In order to avoid additional types of singularity, it is important for the manifold to be convex at infinity; this is defined to mean that every compact set has a compact convex hull, as defined using geodesic convexity. The curve-shortening flow cannot cause a curve to depart from its convex hull, so this condition prevents parts of the curve from reaching the boundary of the manifold.
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