Talk:Cartesian oval

Untitled

I've put two dubious tags on parts of this page:

1. It says that the quartic equation forms two nested ovals, one for the underlying equation with m, a and the other for the underlying equation with -m, -a. But for example if m=1, a>0 the equation with m, a is an ellipse while the equation with -m, -a is a branch of a hyperbola that is imaginary if the ellipse is real; so they are not nested. So I think that at a minimum, there needs to be some parameter restriction to make them nested.

2. It says that the quartic equation gives the loci for both of the two stated underlying equations. This looks plausible to me, but I can't find a source to back it up. The only given reference shows nested ovals without saying they are from two different values of the parameter set, and the same is true for A Catalog of Special Plane Curves, p. 156. Duoduoduo (talk) 20:27, 2 April 2011 (UTC)

Also, the assertion that the quartic gives the points satisfying the two given sum-of-weighted-distance equations is dubious because obtaining the quartic from those two equations requires a squaring to get rid of a radical, and the squaring can introduce spurious points that satisfy the resulting quartic but do not satisfy either of the distance equations. Also, some sources (e.g. A Catalog of Special Plane Curves, p. 156-7) obtain a quartic without appealing to the introduction of a companion distance equation; to do so they square, rearrange, and square the equation again, and the squaring processes can introduce spurious points that don't satisfy the distance equation. Duoduoduo (talk) 18:26, 7 April 2011 (UTC)

I don't have a lot of information to help clear this up, but:

• Here's a curious fact: if you intersect two circular cones with parallel axes, and project the intersection onto a plane perpendicular to their axes, then the intersection is a Cartesian oval. [1]
• Four more old papers on these shapes: [2] [3] [4] [5].
• Here's one that seems to say that the quartic equation covers both choices of sign in the weighted sum of distances: [6].
• And one more recent reference, in one of the books of Martin Gardner: [7]

I also have a lemma in an unpublished paper showing that these ovals have exactly four local extrema of curvature, but because it's unpublished we can't yet include it I think (unless the same result can be found elsewhere). —David Eppstein (talk) 19:36, 7 April 2011 (UTC)

Thanks for all the references. I'll see what I can do to incorporate them into the article (though it may take some time). Duoduoduo (talk) 14:05, 8 April 2011 (UTC)

I went ahead and added the one that seemed most relevant for the "dubious" tags. It's probably still worth going through the others and finding what can be added from them, and also tracking down what's actually in the "Catalog of plane curves" reference so that it can be used as more than just additional reading. —David Eppstein (talk) 20:18, 8 April 2011 (UTC)

By the way, there's still something here that still bothers me wrt the removed "dubious" tags. It's obvious from the form of the quartic (it depends on m² rather than on any odd powers of m) that changing the sign of m doesn't change the equation. But for the same reason changing the sign of a doesn't change the equation. When m is positive, a must also be positive, so that rules out one of the four sign combinations. It seems (and the source says something about) only two of the remaining three sign combinations can have real solutions. But it's not clear that the other combination with a solution is always the one with −m and +a; why can't it be the one with −m and −a? So at the least we should be more careful in the lede when we talk about the equations for the two related ovals. —David Eppstein (talk) 06:33, 9 April 2011 (UTC)

That's exactly what's been bothering me too. Still need to think about it some more. Duoduoduo (talk) 15:53, 9 April 2011 (UTC)

Anallagmatic

The MacTutor reference states that these are Anallagmatic curves. That is, there is a point (where?) such that inversion through that point produces the same curve. But it's not clear from that source whether it means that a single oval inverts to itself, or whether one of the two ovals inverts to the other one. It would be nice to get a better source that clarifies this before adding it to the article. —David Eppstein (talk) 20:18, 8 April 2011 (UTC)

Construction by stretched thread

The article currently says

If one stretches a thread from a pin at one focus to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci.

I couldn't figure out why this doesn't just result in a circle centred on the second pin. After a bit of head-scratching, I realised that what it should say is something like

If one stretches a thread from a pin at one focus to wrap around a pen, back around the first pin, then around a pin at a second focus, and finally ties the free end of the thread to the pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci.

Of course, this is original research on my part, and I don't have a reference. Maybe someone who has the Martin Gardner book can check what he actually wrote? (Or Descartes, come to that.) Also, a diagram to clarify this long-winded description would be really useful. 79.64.187.238 (talk) 02:51, 28 February 2018 (UTC)

Lead

Should the first sentence really have four commas in it? I'd condense it or split up the sentence, but I don't want to accidently make the definition of the article's subject wrong (I don't know enough about Cartesian ovals to confidently change it). User:Heyoostorm^_talk! 22:12, 10 March 2021 (UTC)