Talk:Lemniscate of Bernoulli

Equation, relation to Lissajous figures

1. How can I use the Lemniscate equation $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ to come up with a graph?
2. Is there a mathematical connection to the similar looking Lissajous figures? --Abdull 18:06, 27 May 2006 (UTC)

This article gives the rectangular formula as $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$. Meanwhile, Schaum's (ISBN 0070382034) says $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ -- note the factor of 2 is missing from the right-hand member. Either formula generates a lemniscate. Also, MathWorld says $(x^{2}+y^{2})^{2}=2c^{2}(x^{2}-y^{2})$ Later, MW gives two polar formulas: $r^{2}=2c^{2}\cos(2\theta )$ and $r^{2}=a^{2}\cos(2\theta )$, noting that $2c^{2}=a^{2}$. We can make this Wikipedia article more consistent with other mathematical literature by changing $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$ to $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$, and noting that some authorities write $(x^{2}+y^{2})^{2}=2c^{2}(x^{2}-y^{2})$. Xovimat 16:16, 6 October 2007 (UTC)

What would the other equations in the article be, if $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$ was changed to $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$? --Dogah (talk) 18:18, 29 November 2008 (UTC)

Derivative formulas

It's unclear what is meant by 'unbounded' in the formulas for the first and second derivates of y (respectively x) when thought of as a function of x (respectively y). Could someone please clarify what that is supposed to mean?Jrdodge (talk) 20:10, 17 November 2007 (UTC)

Other formulas: mistake?

I did the algebra and, assuming that "bipolar coordinates" refers to two-center bipolar coordinates, I'm pretty sure the formula there is wrong; it should be rr' = a², not rr' = a²/2 . This would be consistent with the definition of the lemniscate as given in the opening paragraph. I'm changing the formula; if I'm wrong, I'll be happy to learn why. --Jay (Histrion) (talk • contribs) 17:15, 14 October 2010 (UTC)

Removing OR

User:Xovimat added some derivatives and curvature formulas on 8 Sept. and 6 Oct. 2007, including the edit summary "Xovimat added the derivatives section, from his own calculations". This is self-admitted OR, and as far as I can see at least the first derivatives are wrong. (Both seem to be missing a minus sign, and one has extraneous 2's, etc.)

So I'm removing his calculations. 208.50.124.65 (talk) 20:10, 14 August 2014 (UTC)

These derivatives aren’t the most important feature to include in this article which is missing many relevant features of the Lemniscate, a discussion of its history, etc., and it seems okay to leave them out on the basis that they don’t fit in the current article’s narrative flow, or on the basis that they are incorrect. But mathematical calculations are not at all the same kind of “original research” as original work in other fields (they are either correct or they aren‘t, and anyone can easily verify them), and I don’t think it’s appropriate to remove them on that basis alone (not to mention the curvature of such a common curve is undoubtedly published in various other sources). –jacobolus (t) 19:28, 11 August 2021 (UTC)