Curve-shortening flow

It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution.

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.

[5] 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.

[7] 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.

[12] 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.

When the junctions all have exactly three curves meeting at angles of 2π/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.

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.

[15] The avoidance principle implies that any smooth closed curve must eventually reach a singularity, such as a point of infinite curvature.

So, if C 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.

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.

[27] Andrews & Bryan (2011) provide a simpler proof of Grayson's result, based on the monotonicity of the stretch factor.

[32] Immersed curves on Riemannian manifolds, with finitely many self-crossings, become self-tangent only at a discrete set of times, at each of which they lose a crossing.

[34] According to Huisken's monotonicity formula, the convolution of an evolving curve with a time-reversed heat kernel is non-increasing.

In particular,[36] An ancient solution to a flow problem is a curve whose evolution can be extrapolated backwards for all time, without singularities.

[42] In the Cartesian coordinate system, they may be given by the implicit curve equation[43] In the physics literature, the same shapes are known as the paperclip model.

Choosing a careful reparameterization can help redistribute the vertices more evenly along the curve in situations where perpendicular motion would cause them to bunch up.

For most such methods, Cao (2003) warns that "The conditions of stability cannot be determined easily and the time step must be chosen ad hoc.

"[50] Another finite differencing method by Crandall & Lions (1996) modifies the formula for the curvature at each vertex by adding to it a small term based on the Laplace operator.

This modification is called elliptic regularization, and it can be used to help prove the existence of generalized flows as well as in their numerical simulation.

Mokhtarian & Mackworth (1992) suggest a numerical method for computing an approximation to the curve-shortening flow that maintains a discrete approximation to the curve and alternates between two steps: As they show, this method converges to the curve-shortening distribution in the limit as the number of sample points grows and the normalized arc length of the convolution radius shrinks.

[53] Merriman, Bence & Osher (1992) describe a scheme operating on a two-dimensional square grid – effectively an array of pixels.

This representation is updated by alternating two steps: In order for this scheme to be accurate, the time step must be large enough to cause the curve to move by at least one pixel even at points of low curvature, but small enough to cause the radius of blurring to be less than the minimum radius of curvature.

Therefore, the size of a pixel must be O(min κ/max κ2), small enough to allow a suitable intermediate time step to be chosen.

[54] Instead of blurring and thresholding, this method can alternatively be described as applying a median filter with Gaussian weights to each pixel.

[56] An early reference to the curve-shortening flow by William W. Mullins (1956) motivates it as a model for the physical process of annealing, in which heat treatment causes the boundaries between grains of crystallized metal to shift.

The method of Mokhtarian and Mackworth involves computing the curve-shortening flow, tracking the inflection points of the curve as they progress through the flow, and drawing a graph that plots the positions of the inflection points around the curve against the time parameter.

They compare it experimentally against several related alternative definitions of a scale space for shapes, and find that the resampled curvature scale space is less computationally intensive, more robust against nonuniform noise, and less strongly influenced by small-scale shape differences.

[58] In reaction–diffusion systems modeled by the Allen–Cahn equation, the limiting behavior for fast reaction, slow diffusion, and two or more local minima of energy with the same energy level as each other is for the system to settle into regions of different local minima, with the fronts delimiting boundaries between these regions evolving according to the curve-shortening flow.

[61] The curve-shortening flow can be used to prove an isoperimetric inequality for surfaces whose Gaussian curvature is a non-increasing function of the distance from the origin, such as the paraboloid.