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  "body": "\n## Summary\n\nThe curve-shortening flow moves each point of a plane curve along its normal vector.\nThe speed is the [[curvature|7JQS6VKFEZ]] of the curve at that point.\nIt is the one-dimensional case of mean curvature flow and the steepest descent flow for the length of the curve.\nA heat-equation-like smoothing effect makes curves rounder over time, and embedded closed curves shrink to round points by the [[Gage–Hamilton–Grayson theorem|FVGW1381TZ]].\n\n## Explanations\n\n### L1 human\n\nImagine a closed loop of string in the plane in which every bit moves inward at a speed equal to how sharply the string bends there.\nSharp corners move fast and flat parts move slowly, so the loop smooths out.\nA circle of radius $r_0$ stays a circle, and its radius $r(t)$ satisfies $r' = -1/r$, so $r(t) = \\sqrt{r_0^2 - 2t}$ and the circle disappears at time $r_0^2 / 2$.\nThe flow decreases the length as fast as possible, which is the reason for its name.\n",
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