Evolute of an Evolute... ad infinitum?

[loop quantum gravity]

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.

 

[pbuk]

https://en.wikipedia.org/wiki/Evolute

Why not look at the examples on this page?

 

[FactChecker]

But how to prove that if I repeat this process, will it halt? or it's not possible, it's just a curiosity question.

By "halt", do you mean "converge to some finite geometric shape"?
I would bet anything that those are very special cases. It might be possible if you start with simple examples, but general cases are another thing completely.

 

[lavinia]

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.

The evolute of a parabola contains a cubic term so is not a parabola. I wonder if the degree of iterated evolutes of the parabola is increasing.

 

[loop quantum gravity]

By "halt", do you mean "converge to some finite geometric shape"?
I would bet anything that those are very special cases. It might be possible if you start with simple examples, but general cases are another thing completely.

So here I have an interesting classical DG research question. I also think that in the general case it doesn't converge to a finite geometric shape. But how to prove it?

 

[JimWhoKnew]

So here I have an interesting classical DG research question. I also think that in the general case it doesn't converge to a finite geometric shape. But how to prove it?

As @pbuk hinted in #2, the examples on the Wikipedia page answer your question (at least partially).

For circles, the evolute is a point, which is not a valid input for a further evolute. So the sequence halts right there.
For nephroids and cardioids the sequence converges (to a point as its limit).
For astroids and deltoids the sequence diverges (or "converges to infinity", as some may call it).
For logarithmic spirals the evolute is a rotation of the original curve. I haven't worked it out, but I'd expect that by an appropriate choice of the scale you can construct spirals for which some $~E^n~$ gets you back to the original curve. I'll leave it to you to check it out, and to prove that if it is possible then n>1.

 

[lavinia]

So here I have an interesting classical DG research question. I also think that in the general case it doesn't converge to a finite geometric shape. But how to prove it?

i tried to indicate a method of proof in reply #4

 

[lavinia]

For the parabola (x,x^2) plugging into the formula for the evolute gives the parametric equation (-4x^3,1/2+3x^2). This is not a parabola because of the cubic term. I thought perhaps an inductive proof would show that the parametric equation for the n'th evolute would be a pair of polynomials whose maximal degree would increase with each iteration. If so, then the sequence of evolutes would be infinite and at each stage the curves would be different.

There would also remain the question of whether the infinite sequence of iterated evolutes converges point wise.

 

[WWGD]

Why not work with Jet Bundles if we,generalize to Manifolds. Can think specifically how, but seems like a good fit.

 

[lavinia]

Why not work with Jet Bundles if we,generalize to Manifolds. Can think specifically how, but seems like a good fit.

I think for hypersurfaces of Euclidean n-space one gets n-1 evolute like hypersurfaces as the critical values of the normal mapping. This just generalizes the case of a curve in the plane. In the hypersurface case there are specific curves called principal curvatures along which the normal mapping has a critical point. I imagine the critical values of the normal mapping are the right generalization for surfaces of codimension greater than one although here the normal bundle is not 1 dimensional.

 

[lavinia]

@WWGD

I thought you'd like see see why evolutes are critical values of the normal map.

For a curve in the plane, the normal map is c(s) + tN(s) where N(s) is a unit normal to the curve c(s). If c(s) is parameterized by arc length and N(s) points in the direction of c'' , a classical equation says that

N'(s) = -k(s) c'(s).

Given this, the derivative with repect to s of the normal map is

c'(s) -tN'(s) =c(s) -tk(s)c'(s)

and this is zero when t=1/k. So the center of curvature is the critical value of the normal map.

In the case of a hypersurface, the classical equation doesn't hold in general but it does hold in cases where N'(s) is a multiple of c'(s). A classical theorem says that at each point there is an orthogonal basis of tangent vectors V_i such that V_i⋅N is a multiple of V_i. From this one gets n-1 critical values of the normal mapping.