Talk:Curve-shortening flow

Normal and curvature not well-defined
The definition of the differential equation does not specify what is meant by "normal" (there are of course two options). Assuming a consistent definition of normal is used, then I believe it should be the "signed curvature", not the "curvature", to allow concave deformations to push "out".

Geometric heat flow for vector fields
There seems also to be the concept of a geometric heat flow on vector fields, as described in the arxiv paper A GEOMETRIC HEAT FLOW FOR VECTOR FIELDS. It implies that geometric heat flow is not a 1-1 synonym for curve-shortening flow, as there can be geometric heat flows on more than just curves or manifolds. Geometric heat flow on vector fields might also be worth mentioning in the Beyond curves section, but I'm not confident enough with the scope of the topic to say if it belongs. --Mark viking (talk) 04:33, 4 December 2015 (UTC)
 * I added a footnote remarking that the geometric heat flow name has other uses. —David Eppstein (talk) 22:37, 19 April 2016 (UTC)
 * Looks good, thanks. --Mark viking (talk) 23:25, 19 April 2016 (UTC)

Definition correct?
In the differential equation, the curve C is parameterized by arc length. Is that correct? I think it should be parameterized by the angle of the tangent vector; otherwise, the derivative with respect to t is not perpendicular to the tangent vector of C. I've tried to solve that differential equation for a circle, it seems to be wrong. — Preceding unsigned comment added by 92.218.115.27 (talk) 12:26, 2 February 2020 (UTC)

Inaccuracy?
This is a great article but I'm confused by this paragraph: "The avoidance principle implies that any smooth curve eventually either reaches a singularity (such as a point of infinite curvature) or collapses to a point. For, if a given smooth curve C is surrounded by a circle, both will remain disjoint until one or the other collapses or reaches a singularity. But the enclosing circle shrinks under the curvature flow, remaining circular, until it collapses, and by the avoidance principle C must remain contained within it. By the same reasoning, the radius of the smallest circle that encloses C must decrease at a rate that is at least as fast as the decrease in radius of a circle undergoing the same flow."

For one thing, I'm not sure exactly what is being concluded, as every smooth curve reaches a singularity, and the given argument says nothing about the specific subcase of collapsing to a point. It seems that what is being proved is that it is impossible for the flow to exist for all time. If so, the last sentence of the paragraph seems to be irrelevant, as it is sufficient to consider any surrounding circle. Gumshoe2 (talk) 20:05, 20 September 2020 (UTC)
 * The last sentence gives an explicit bound on how quickly the singularity/collapse happens. —David Eppstein (talk) 20:22, 20 September 2020 (UTC)
 * I see. The way I had read it, I'd thought that the previous sentence was making a simple observation which was setting up the last sentence, which was proving the claim. My mistake. I'm still confused by what I said about the first sentence though. Is the point just that infinite-time existence is impossible? Gumshoe2 (talk) 20:28, 20 September 2020 (UTC)
 * More or less. Either it collapses to a point (which is what always actually happens) or it becomes non-smooth (not ruled out by this argument but also not possible). —David Eppstein (talk) 21:22, 20 September 2020 (UTC)
 * I made a small edit to clarify. I think it was phrased a little confusingly. Gumshoe2 (talk) 05:19, 8 October 2020 (UTC)
 * Ok, thanks. I don't see anything to object to in your changes. —David Eppstein (talk) 05:27, 8 October 2020 (UTC)