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:

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!