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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:
alt text
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!

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@Joseph: you may want to re-write your first two sentences. At present it isn't very clear what the evolute of a curve is defined as, since the first sentence, which describes what you want, reads too much like the definition. –  Willie Wong Jul 18 '10 at 21:15
    
    
@Willie: You were right about the ambiguity--thanks! –  Joseph O'Rourke Jul 18 '10 at 22:12

2 Answers 2

up vote 10 down vote accepted

Light studied the problem, but didn't solve it in his 1917 dissertation. A 1920 note of Light was followed by a 1921 note of Franklin showing the existence of infinitely many classes of examples not covered by Light, and pointing out that Puiseux had shown this in 1844. However, the question of determining all of them isn't addressed there, and I don't know what else is known.

Regarding your question about formalizing the notion of curves as eigenvectors, couldn't you just take the real vector space whose basis consists of representatives from each similarity class of curves? Scalar multiplication of a basis element by a real number corresponds to dilating or shrinking, and reflecting if negative. The evolute operator is formally extended to this vector space linearly, and the curves similar to their evolutes will be precisely (scalar multiples of) the basis vectors that are eigenvectors.

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Wow! Nearly a century ago! Cool!! "On curves whose evolutes are similar curves," by Philip Franklin, Princeton Univ., "who derives in a simple way the result, stated by Puiseux, that an infinite set of such curves [whose evolutes are similar curves] exist." @Jonas: Great historical sleuthing!! –  Joseph O'Rourke Jul 19 '10 at 1:52
    
And thanks re eigenvectors! –  Joseph O'Rourke Jul 19 '10 at 11:45

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.

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