TITLE: 'Eigenvectors' of evolute operation
QUESTION [13 upvotes]: The evolute of a curve is the locus of its centers of curvature.
The evolute of some plane curves is a scaled, or scaled and
reflected/rotated, version of that curve.
For example, the evolute of a cardiod is a reflected cardiod at one-third
the scale:
     (source: MathWorld)
Loosely speaking, we might say that a cardiod is an eigenvector
of the evolute operation with eigenvalue $\frac{1}{3}$.
(I am not sure if this eigenvector language can be made
technically accurate.)
There are many classical curves that are evolute eigenvectors
in this sense.  Here is a partial list
(based on this table), where I indicate the scale but not reflection: 

 Evolute( cardiod ) $\mapsto$ $\frac{1}{3}$ cardiod
 Evolute( nephroid ) $\mapsto$ $\frac{1}{2}$ nephroid
 Evolute( astroid ) $\mapsto$ 2 astroid
 Evolute( cycloid ) $\mapsto$ 1 cycloid
 Evolute( logarithm spiral ) $\mapsto$ 1 logarithm spiral
 Evolute( deltoid ) $\mapsto$ 3 deltoid
 Evolute( epicycloid ) $\mapsto$ $\lambda$ epicycloid
 Evolute( hypocycloid ) $\mapsto$ $\lambda$ hypocycloid

My question is: Has the complete class of plane
curves whose evolutes are scaled versions of themselves been
studied?
The same question can be asked for evolute surfaces, and in 
higher dimensions.
This question is (very tenuously!) related to some research into
cut loci, but at this point I'm primarily curious and interested
in learning.
Edit.  If I may extend my question slightly: Can indeed "this eigenvector language" be made
"technically accurate"?  Is there a formalization where the plane curves are vectors (perhaps in Hilbert space?) and
evolution is an operator whose eigenvectors correspond to the class of curves I seek?
I am woefully ignorant in this area and formalization is beyond my expertise.  Thanks for any hints!

REPLY [6 votes]: All these curves are "multihedgehogs" : for clear definitions and neat examples see
Y. Martinez-Maure, A Sturm-type comparison theorem by a geometric study of plane multihedgehogs, Illinois Journal of Mathematics 52 (2008), 981-993
or
Y. Martinez-Maure, Les multihérissons et le théorème de Sturm-Hurwitz, Archiv der Mathematik 80, 2003, p. 79-86.
Such a curve can be defined by  a 2Nπ-periodic function of class $C^2$ on $R$, (the number $N$ is merely the number of full rotations of the coorienting normal vector $u (t) = (cos t, sin t)$. Its evolute as support function $h'(t-π/2)$. When you say that the curve is an "eigenvector", that means in fact that its support function is a spherical harmonic that is, an eigenfunction of the circular Laplacian.
There are of course extensions of these notions in higher dimensions.