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,The phrase "geometric heat flow" has also been used for flows on other kinds of object than curves, such as differential forms. 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., p.140: "a geometric flow [is] an evolution of the geometry of over time ." 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 where parameterizes the arc length along the curve and parameterizes a time in the evolution of the curve, the curve-shortening flow can be described by the parabolic partial differential equation
a form of the heat equation, where is the curvature and 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.
Space curves The curve-shortening flow has also been studied for curves in three-dimensional Euclidean space. The normal vector in this case can be defined (as in the plane) as the derivative of the tangent vector with respect to arc length, normalized to be a unit vector; it is one of the components of the Frenet–Serret frame. It is not well defined at points of zero curvature, but the product of the curvature and the normal vector remains well defined at those points, allowing the curve-shortening flow to be defined. Curves in space may cross each other or themselves according to this flow, and the flow may lead to singularities in the curves; every singularity is asymptotic to a plane. However, spherical curves and curves which can be orthogonally projected into a regular convex planar curve are known to remain simple. The curve shortening flow for space curves has been used as a way to define flow past singularities in plane curves.
Beyond curves It is possible to extend the definition of the flow to more general inputs than curves, for instance by using rectifiable varifolds or the level-set method. However, these extended definitions may allow parts of curves to vanish instantaneously or fatten into sets of nonzero area.; ; , "4.7.1 Brakke's varifold solution", p. 100. .
A commonly studied variation of the problem involves networks of interior-disjoint smooth curves, with junctions where three or more of the curves meet. When the junctions all have exactly three curves meeting at angles of 2/3 (the same conditions seen in an optimal Steiner tree or two-dimensional foam of soap bubbles) the flow is well-defined for the short term. However, it may eventually reach a singular state with four or more curves meeting at a junction, and there may be more than one way to continue the flow past such a singularity.
Behavior
Avoidance principle, radius, and stretch factor If two disjoint smooth simple closed curves undergo the curve-shortening flow simultaneously, they remain disjoint as the flow progresses. The reason is that, if two smooth curves move in a way that creates a crossing, then at the time of first crossing the curves would necessarily be tangent to each other, without crossing. But, in such a situation, the two curves' curvatures at the point of tangency would necessarily pull them apart rather than pushing them together into a crossing. For the same reason, a single simple closed curve can never evolve to cross itself. This phenomenon is known as the avoidance principle.
The avoidance principle implies that any smooth closed curve must eventually reach a singularity, such as a point of infinite curvature. For, if a given smooth curve is surrounded by a circle, both will remain disjoint for as long as they both exist. But the enclosing circle shrinks under the curvature flow, remaining circular, until it collapses, and by the avoidance principle must remain contained within it. So, if were to never reach a singularity, it would be trapped at a single point at the time when the circle collapses, which is impossible for a smooth curve. This can be quantified by observing that the radius of the smallest circle that encloses must decrease at a rate that is at least as fast as the decrease in radius of a circle undergoing the same flow.
quantifies the avoidance principle for a single curve in terms of the ratio between the arc length (of the shorter of two arcs) and Euclidean distance between pairs of points, sometimes called the stretch factor. He shows that the stretch factor is strictly decreasing at each of its local maxima, except for the case of the two ends of a diameter of a circle in which case the stretch factor is constant at . This monotonicity property implies the avoidance principle, for if the curve would ever touch itself the stretch factor would become infinite at the two touching points.
Length As a curve undergoes the curve-shortening flow, its length decreases at a rate given by the formula
where the integral is taken over the curve, is the curvature, and is arc length along the curve. The integrand is always non-negative, and for any smooth closed curve there exist arcs within which it is strictly positive, so the length decreases monotonically. More generally, for any evolution of curves whose normal speed is , the rate of change in length is
which can be interpreted as a negated inner product between the given evolution and the curve-shortening flow. Thus, the curve-shortening flow can be described as the gradient flow for length, the flow that (locally) decreases the length of the curve as quickly as possible relative to the norm of the flow. This property is the one that gives the curve-shortening flow its name., p. vii; , p. 526.
Area For a simple closed curve, the area enclosed by the curve shrinks, at the constant rate of 2 units of area per unit of time, independent of the curve. Therefore, the total time for a curve to shrink to a point is proportional to its area, regardless of its initial shape., Appendix B, Proposition 1, p. 230; , p. vii; , Theorem 1, p. 527. Because the area of a curve is reduced at a constant rate, and (by the isoperimetric inequality) a circle has the greatest possible area among simple closed curves of a given length, it follows that circles are the slowest curves to collapse to a point under the curve-shortening flow. All other curves take less time to collapse than a circle of the same length.
The constant rate of area reduction is the only conservation law satisfied by the curve-shortening flow. This implies that it is not possible to express the "vanishing point" where the curve eventually collapses as an integral over the curve of any function of its points and their derivatives, because such an expression would lead to a forbidden second conservation law. However, by combining the constant rate of area loss with the avoidance principle, it is possible to prove that the vanishing point always lies within a circle, concentric with the minimum enclosing circle, whose area is the difference in areas between the enclosing circle and the given curve.
Total absolute curvature The total absolute curvature of a smooth curve is the integral of the absolute value of the curvature along the arc length of the curve,