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 $z=k\mathrm{c}\mathrm{i}\mathrm{s}(t)+\mathrm{c}\mathrm{i}\mathrm{s}(kt)$ (epicycloids have $k>0$, hypocycloids $k<0$) we get

$z^{\prime }=ik(\mathrm{c}\mathrm{i}\mathrm{s}(t)+\mathrm{c}\mathrm{i}\mathrm{s}(kt))$

$|z^{\prime }{|}^2=z^{\prime }{\overline{z}}^{\prime }=2k^2(1+\cos (kt-t))$

$⟨z^{\prime },P-z⟩=-\mathrm{\Re }(z^{\prime }\overline{z})=k(k-1)\sin (kt-t)$

$\frac{⟨z^{\prime },P-z⟩}{|z^{\prime }{|}^2}=\frac{k-1}{2k}\tan \left(\frac{k-1}{2}t\right)=\frac{k-1}{2ik}\frac{\mathrm{c}\mathrm{i}\mathrm{s}(kt)-\mathrm{c}\mathrm{i}\mathrm{s}(t)}{\mathrm{c}\mathrm{i}\mathrm{s}(kt)+\mathrm{c}\mathrm{i}\mathrm{s}(t)}$

and, finally, the pedal curve is

$w=z+z^{\prime }\frac{⟨z^{\prime },P-z⟩}{|z^{\prime }{|}^2}=\frac{k+1}{2}(\mathrm{c}\mathrm{i}\mathrm{s}(t)+\mathrm{c}\mathrm{i}\mathrm{s}(kt))$

Noting $\mathrm{c}\mathrm{i}\mathrm{s}A+\mathrm{c}\mathrm{i}\mathrm{s}B=2\cos \frac{A-B}{2}\mathrm{c}\mathrm{i}\mathrm{s}\frac{A+B}{2}$ and letting $t=2\theta /(k+1)$

$w=(1+k)\cos \left(\frac{1-k}{1+k}\theta \right)\mathrm{c}\mathrm{i}\mathrm{s}(\theta )$

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)