Clear curves

Curve-shortening flow is the simplest example of a curvature flow. It moves each point on a plane curve \(\gamma\) in the inwards normal direction \(-\nu\) with speed proportional to the signed curvature \(k\) at that point, as described by the equation

$$ \frac{\partial \gamma}{\partial t} = - k \nu.$$

The name "curve-shortening" comes from the fact that the curve is always moving so as to decrease its length as efficiently as possible.

For any initial curve that does not intersect itself, the flow will converge to a "round point" in finite time - this is the Gage-Hamilton-Grayson theorem.

The demonstration on this page scales time by a factor of \(1/\max |k|\) for each curve to avoid numerical errors.

Self-intersecting curves also jump over some cusps that are singularities of the smooth flow - this behaviour is qualitatively (and perhaps quantitatively) the same as that of the weak formulation of the flow. Red markers appear where the curvature is very large.

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