loop quantum gravity
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So I was thinking about repeating an evolute of an evolute ad infinitum, it doesn't necessarily converge to some finite geometric shape, does it?
Suppose we have a curve ##\gamma: [a,b]\to \mathbb{R}^n##, and let ##\dot{\gamma}(t)## be the tangent to the curve, and ##n(t)## its normal to the tangent, ##k(t)## its curvature, then the evolue it defined as: ##E(t)=\gamma(t)+n(t)/k(t)##.
https://en.wikipedia.org/wiki/Evolute
So I was thinking why not repeat this process replace ##\gamma(t)## with ##E(t)##, and we calculate the evolute of the evolute... etc.
We can try it on an ellipsoid for starters; but for the above suggested n-th version, this may be interesting.
But how to prove that if I repeat this process, will it halt? or it's not possible, it's just a curiosity question.
Suppose we have a curve ##\gamma: [a,b]\to \mathbb{R}^n##, and let ##\dot{\gamma}(t)## be the tangent to the curve, and ##n(t)## its normal to the tangent, ##k(t)## its curvature, then the evolue it defined as: ##E(t)=\gamma(t)+n(t)/k(t)##.
https://en.wikipedia.org/wiki/Evolute
So I was thinking why not repeat this process replace ##\gamma(t)## with ##E(t)##, and we calculate the evolute of the evolute... etc.
We can try it on an ellipsoid for starters; but for the above suggested n-th version, this may be interesting.
But how to prove that if I repeat this process, will it halt? or it's not possible, it's just a curiosity question.