Talk:Pedal curve

Epicycloids
Hm. Contrary to its pedal curve page, this mathworld page says the epicycloid does not give a rose. But hypo and epicycloids are such close cousins they should both or neither. I'll check carefully. 142.177.20.80 01:08, 2 Aug 2004 (UTC)
 * The page is mistaken. Working in the complex plane and starting with $$P=0$$ and $$z=k{\rm cis}(t)+{\rm cis}(kt)$$ (epicycloids have $$k>0$$, hypocycloids $$k<0$$) we get
 * $$z'=ik\left({\rm cis}(t)+{\rm cis}(kt)\right)$$
 * $$|z'|^2=z'\bar z'=2k^2\left(1+\cos(kt-t)\right)$$
 * $$\langle z',P-z\rangle=-\Re(z'\bar z)=k(k-1)\sin(kt-t)$$
 * $${\langle z',P-z\rangle\over|z'|^2}={k-1\over2k}\tan\left({k-1\over2}t\right)={k-1\over2ik}{{\rm cis}(kt)-{\rm cis}(t)\over{\rm cis}(kt)+{\rm cis}(t)}$$
 * and, finally, the pedal curve is
 * $$w=z+z'{\langle z',P-z\rangle\over|z'|^2}={k+1\over2}\left({\rm cis}(t) + {\rm cis}(kt)\right)$$
 * Noting $${\rm cis}A+{\rm cis}B=2\cos{A-B\over2}{\rm cis}{A+B\over2}$$ and letting $$t=2\theta/(k+1)$$
 * $$w=(1+k)\cos\left({1-k\over 1+k}\theta\right){\rm cis}(\theta)$$
 * which is obviously a rose. 142.177.126.230 16:25, 2 Aug 2004 (UTC)
 * Mathworld's been corrected =) 142.177.126.230 23:07, 5 Aug 2004 (UTC)

Contrapedals
More work needed on contrapedal curve; mainly, what's done in higher spaces? While one could sensibly use the curvature vector, one could also use the perpendicular subspace...and somehow the latter is more appealing. Kwantus 18:53, 2 Aug 2004 (UTC)

Merge with Orthotomic
The orthotomic is simply the pedal magnified by a factor of 2. The current orthotomic article is just a stub, so the merge should be easy and it seems silly (not to mention a content fork) to have two articles about essentially the same curve.--RDBury (talk) 23:09, 9 October 2009 (UTC)
 * The merge is done.--RDBury (talk) 13:57, 12 October 2009 (UTC)

A little history would be nice
It would be nice to know why on earth this construction is a thing, and why it's called a "pedal" curve. Something to do with feet or walking? 203.13.3.94 (talk) 23:56, 30 August 2022 (UTC)