Árpád Kurusa
mathematician, associate professor
 Department of Geometry
Bolyai Institute
Faculty of Science
University of Szeged
english
Árpád Kurusa

Maths-blog

How to form a closed curve to circle!

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Published on 2013. január 24. csütörtök, 14:20

The Schönflies Theorem states that if $C \subset \mathbb R^2$ is a simple closed curve in th eplane, i.e. Jordan curve, then there is a homeomorphism $f\colon\mathbb R^2 \to \mathbb R^2$ such that $f(C)$ is the unit circle in the plane.

If the curve $C$ is differentiable, then there is a continuous curve $f_{\varepsilon}$ in the set of diffeomorphisms that connects $f$ to the identity so that $f_{\varepsilon}(C)$ is a Jordan-curve for every $\varepsilon$.

When asked what kind of this curve in the set of diffeomorphisms can be, the following theorem is the answer.

Matt Grayson's theorem(1988): If a curve moves so that its velocity is always and everywhere equal to its curvature, and if it starts as a closed curve without self intersections, then it will never form self intersections, and it will become convex in finite time.

Here is an animation about how this works in practice.

Right clicking (or control-clicking) on the image (this was taken from [1]) allows you to start, stop and restart as well as zoom in&out of the animation.

More animations can be found in the subject of curve shortening:

References

[1] http://www.math.wisc.edu/~angenent/curveshortening/index.html

© 2024 Árpád Kurusa