Curve Shortening makes a plane curve convex
Why MathOverflow recommends it
MathOverflow users link it as a quick visual of what curve shortening flow does to a closed plane curve when asking or answering questions about the flow.
Recommended by 1 people across 1 different months, 2015 to 2020.
What it is
A 31-second animation by Sigurd Angenent showing curve shortening flow acting on a plane curve. It illustrates Grayson's 1987 theorem that the curve becomes convex, then (Gage-Hamilton) rounder and rounder until it shrinks to a point, with enclosed area dropping at 2 pi per second.
What people said
Recommends Gabe K · 2020-07 · MathOverflow · on “Surprising properties of closed planar curves”
One result that I initially found a bit surprising is Grayson's theorem. It's a little bit of a different flavor than the other examples but I think it's interesting and worth a mention. Given a closed planar curve $\gamma$ which is smooth enough ( $C^2$ is sufficient but it's possible to be deal with less regular curves), there is a process known as curve shortening flow, which deforms the curve using the flow $$ \frac{\partial \gamma}{\partial t} = \kappa N. $$ Here, $\kappa$ is the unsigned curvature and $N$ is the unit normal vector. This is called curve shortening flow because it tends to shrink curves. Grayson's theorem states that if you start with a closed curve that does not cross i
Cites Pablo Lessa · 2015-01 · MathOverflow · on “Continuity of the curve-shortening flow with respect to the curve”
The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in this video [this video] . The flow is supposed to formalize the idea of evolution by steepest-descent with respect to the length functional on a suitable space of curves. Letting $\alpha:S^1 \to M$ represent the closed curve (where $S^1$ is the circle) the evolution equation is $$\left\lbrace\begin{array}{l}u(0,x) = \alpha(x)\\ \partial_tu(t,x) = k(t,x)N(t,x)\end{array}\right.$$ where $k$ is the curvature of the curve $u(t,\cdot)$ with respect to the unit normal vector field $N$ (which is chosen to be
Cites Joseph O'Rourke · 2016-11 · MathOverflow · on “Gage-Grayson-Hamilton curve-shortening flow, at an angle”
The Gage-Grayson-Hamilton curve-shortening [LINK] flows along the normal to the curve: [LINK] (Image from Andrejs Treibergs presentation PDF [LINK] .) The result is remarkable avoidance of self-intersection, regardless of how convoluted is the original curve, and ultimate convergence to a "round point." See, e.g., this [this video] . My question is: Q . Has this flow been studied along vectors at some fixed angle to the normal? E.g., at $90^\circ$ counterclockwise of the normal, tangent to the curve? Presumably, for any angle $\alpha > 0$ counterclockwise of the normal, the curve might self-intersect during its evolution. But perhaps it still converges to a limit shape?