Talk:Cissoid of Diocles

Proof of roulette form
Proof the cissoid is the described roulette, in case someone's interested.
 * $$f(t)=-t^2+ibt$$  $$f'(t)=-2t+ib$$
 * $$r(t)=t^2+ibt$$  $$r'(t)=2t+ib$$
 * $$f(t)-r(t){f'(t)\over r'(t)}=2bt^2(b+2it)/(b^2+4t^2)$$

thus
 * $$y=4bt^3/(b^2+4t^2)$$
 * $$x=2b^2t^2/(b^2+4t^2)$$
 * $$y^2=16b^2t^6/(b^2+4t^2)^2$$
 * $$x^3/(2a-x)={8b^6t^6\over(2a(b^2+4t^2)-2b^2t^2)(b^2+4t^2)^2}$$

if $$b^2=4a$$ then
 * $$y^2={4at^6\over(a+t^2)^2}={x^3\over2a-x}$$

142.177.124.178 06:26, 21 Jul 2004 (UTC)


 * Well, it's nice to know that some anonymous user read an article which I started. I'll take your word that your proof is correct.  Maybe I will get around to adding something similar to it in the article. --AugPi 14:27, 27 Jul 2004 (UTC)
 * Holy...I thought I was putting too much math in articles like roulette and evolute. I knew it was so from Mathworld but rather than just copy (esp since Mathworld is occasionally wrong) I like to have done a proof first. 142.177.126.230 22:34, 2 Aug 2004 (UTC)
 * The roulette property can be shown to a special case of a similar property of pedal curves.--RDBury (talk) 02:37, 30 October 2009 (UTC)

Existence
"On the other hand, if one accepts that cissoids of Diocles do exist, then there must exist at least one example of such a cissoid."

This does not follow at all. We are talking about different types of existence here. Circles exist as a mathematical object, but there is no 'example of a circle - all the examples are out of round, and have edges that do not have zero width.

"This cissoid could then be translated, rotated, and expanded or contracted in size (without changing its proportional shape) at will to fit into any position."

Well, no. This ``example`` of a cissoid wouldn't actually be a perfect cissoid, and if it were it wouldn't be possible to transform it into another perfect cissoid.

"Then one would readily admit that such a cissoid can be used to correctly solve the Delian problem."

I am not at all persuaded that "one" would "readily admit" such a thing at all. This person is either terribly confused about mathematical objects, about what solving a mathematical problem is, or is making jokes about existentialism - which really has no place in an encyclopedia.

203.13.3.93 (talk) 00:18, 30 August 2022 (UTC)


 * I came to this talk page on reading that section. Going to remove it, it's WP:OR/entirely unsourced. 31.187.2.210 (talk) 21:23, 11 January 2024 (UTC)