Talk:Evolute

I disagree with Mathworld's page on two points: I'm trying to raise Eric by email 142.177.124.178 14:29, 19 Jul 2004 (UTC)
 * involutes are not unique and thus "The original curve is then said to be the involute of its evolute" is wrong
 * what is the evolute of a sequence of circular arcs, if not disconnected points? And what would be an involute of that?

Continuing the "what to do without a natural parametrisation" stuff: it often happens that $$s'(t)^2$$ is simpler/easier than $$s'(t)$$. That may be useful.

Let $$q(t)=s'(t)^2$$. Differentiate once $$q'(t)=2s'(t)s''(t)$$. Fit these into the lower expression to get
 * $$r(s(t))={2q(t)x(t)-q'(t)x'(t)\over 2q(t)^2}$$

It so happens all the s's go away&mdash;that sqrt can be avoided. Of course we still need $$|r''|^2$$... 142.177.124.178 02:45, 22 Jul 2004 (UTC)
 * It can really help! Evolute of ellipse:
 * $$x=(aC,bS)$$ (with shorthand C=cos(t), S=sin(t))
 * $$x'=(-aS,bC)$$
 * $$x''=-x$$
 * $$q=|x'|^2=a^2S^2+b^2C^2$$
 * $$q'=2SC(a^2-b^2)$$
 * $$N=2qx''-q'x'=-2ab(bC,aS)$$
 * $$|N|^2=4a^2b^2(b^2C^2+a^2S^2)$$
 * $$r/|r|^2=2q^2N/|N|^2=-(b^2C^2+a^2S^2)(C/a,S/b)$$
 * evolute = $$(a^2-b^2)(C^3/a,-S^3/b)$$
 * the almost-astroid fell right out 142.177.124.178 15:20, 22 Jul 2004 (UTC)

Added animated gif
I've added an animated gif showing how the evolute is constructed as the locus of the centers of curvature. The tangent to the astroid is also shown, illustrating simultaneously that the evolute is the envelope of the normals to the curve.Juliusllb 23:32, 10 November 2013 (UTC) — Preceding unsigned comment added by Juliusllb (talk • contribs)

Deleted natural parameterization material
I deleted some of the material on natural parameterization, specifically the paragraph on dealing with the equations when s is not a closed form expression. First, the material was about simplifying the expression for derivatives in this case and not about evolutes. Second, the expression obtained should, when all the dust has settled, be the same as the expression in the general parameterization case (else there is an error somewhere). Third, I could not find similar material in references. --RDBury (talk) 14:32, 4 April 2008 (UTC)

Also deleted the line


 * Intrinsic equation of the evolute of a curve defined by an intrinsic equation r=f(s) is $$R[y]=\frac{rr'}{(r^{-1})'}$$.

I could not find a reference or a clear definition so I could verify it myself. If someone can clarify or site a source it could probably be re-added.--RDBury (talk) 20:09, 4 April 2008 (UTC)

Also moved statement about parallel curves to the article on parallel curves.--RDBury (talk) 20:58, 4 April 2008 (UTC)