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

$z'=ik\left({\rm {cis}}(t)+{\rm {cis}}(kt)\right)$

$|z'|^{2}=z'{\bar {z}}'=2k^{2}\left(1+\cos(kt-t)\right)$

$\langle z',P-z\rangle =-\Re (z'{\bar {z}})=k(k-1)\sin(kt-t)$

${\langle z',P-z\rangle \over |z'|^{2}}={k-1 \over 2k}\tan \left({k-1 \over 2}t\right)={k-1 \over 2ik}{{\rm {cis}}(kt)-{\rm {cis}}(t) \over {\rm {cis}}(kt)+{\rm {cis}}(t)}$

and, finally, the pedal curve is

$w=z+z'{\langle z',P-z\rangle \over |z'|^{2}}={k+1 \over 2}\left({\rm {cis}}(t)+{\rm {cis}}(kt)\right)$

Noting ${\rm {cis}}A+{\rm {cis}}B=2\cos {A-B \over 2}{\rm {cis}}{A+B \over 2}$ 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)