Talk:Evolute
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I disagree with Mathworld's page (http://mathworld.wolfram.com/Evolute.html) on two points:
- 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?
I'm trying to raise Eric by email 142.177.124.178 14:29, 19 Jul 2004 (UTC)
Continuing the "what to do without a natural parametrisation" stuff: it often happens that <math>s'(t)^2<math> is simpler/easier than <math>s'(t)<math>. That may be useful.
Let <math>q(t)=s'(t)^2<math>. Differentiate once <math>q'(t)=2s'(t)s''(t)<math>. Fit these into the lower expression to get
- <math>r''(s(t))={2q(t)x''(t)-q'(t)x'(t)\over 2q(t)^2}<math>
It so happens all the s's go away—that sqrt can be avoided. Of course we still need <math>|r''|^2<math>... 142.177.124.178 02:45, 22 Jul 2004 (UTC)
- It can really help! Evolute of ellipse:
- <math>x=(aC,bS)<math> (with shorthand C=cos(t), S=sin(t))
- <math>x'=(-aS,bC)<math>
- <math>x''=-x<math>
- <math>q=|x'|^2=a^2S^2+b^2C^2<math>
- <math>q'=2SC(a^2-b^2)<math>
- <math>N=2qx''-q'x'=-2ab(bC,aS)<math>
- <math>|N|^2=4a^2b^2(b^2C^2+a^2S^2)<math>
- <math>r''/|r''|^2=2q^2N/|N|^2=-(b^2C^2+a^2S^2)(C/a,S/b)<math>
- evolute = <math>(a^2-b^2)(C^3/a,-S^3/b)<math>
- the almost-astroid fell right out 142.177.124.178 15:20, 22 Jul 2004 (UTC)