Talk:Superellipse

"Characteristics"
I propose a new section called "Characteristics" or perhaps "Uses" that would mention some of the interesting and useful attributes of superellipses, and why they were used where they were. Comments? karlchwe

"Generalization" stub
I'm by no means a mathematics expert, but it looks like the article could be cleaned up a little, and the "Generalization" stub eliminated, if it were just incorporated into a ntoher section of the article. It doesn't look to me like there's much else to be said about the generalization. Zhankfor

Axell, here are the requested graphs for n=2 (it is really a circle as it is said in the article):



n=3/2:



and n=1/2 (really looks like an astroid):



Best regards. --XJamRastafire 16:59 Dec 19, 2002 (UTC)
 * In the n = 1/2 case, each of the four parts is really part of a parabola. Derivation:
 * $$\begin{align}

\sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} &= 1 \\ \sqrt{\frac{y}{b}} &= 1 - \sqrt{\frac{x}{a}} \\ \frac{y}{b} &= \left(1 - \sqrt{\frac{x}{a}}\right)^2 \\ \frac{y}{b} &= 1 - 2\sqrt{\frac{x}{a}} + \frac{x}{a} \\ 2\sqrt{\frac{x}{a}} &= 1 + \frac{x}{a} - \frac{y}{b} \\ 4\frac{x}{a} &= \left(1 + \frac{x}{a} - \frac{y}{b}\right)^2 \\ 4\frac{x}{a} &= 1 + 2\frac{x}{a} - 2\frac{y}{b} + \frac{x^2}{a^2} - 2\frac{xy}{ab} + \frac{y^2}{b^2} \\ 0 &= 1 - 2\frac{x}{a} - 2\frac{y}{b} + \frac{x^2}{a^2} - 2\frac{xy}{ab} + \frac{y^2}{b^2} \\ \end{align}$$
 * is a conic section with discriminant $$B^2 - 4AC = \left(\frac{2}{ab}\right)^2 - 4\frac{1}{a^2}\frac{1}{b^2} = 0$$, which means that it is a parabola. --Spoon! (talk) 23:21, 25 May 2008 (UTC)


 * - and in the case $$a=b=1$$, they have axes $$x=\pm y$$ and vertices $$(x,y)=(\pm\frac14,\pm\frac14)$$. I didn't know! Please find a nice way to include it in the article. If (part of) the derivation is included, there should be a link to something about conics and their discriminants.--Noe (talk) 12:37, 26 May 2008 (UTC)

Why is the ellipsis, a punctuation mark, listed in the "see also" section of this page?
Could this be moved to the more grammatically correct "superellipse"? It's not about an ellipse that is especially amazing, it's about a curve that goes beyond an ellipse. 84.70.169.233 10:57, 18 April 2006 (UTC)

Knuth's Metafont

 * Like Bezier curves, superellipses are easier to implement with integer arithmetic than are circular arcs, so Knuth used superellipses instead of circular arcs in his Metafont type-design software.

Say what?? The arithmetic of superellipses is if anything more difficult than that of ordinary ellipses. Knuth's definition of the Computer Modern type family contains a variable squarerootoftwo which may be set to 1.414 for classical ellipses, or to lower values for more square superellipses; but whatever the setting, Metafont approximates the curve with cubic splines. Can anyone cite something to contradict my understanding? &mdash;Tamfang 03:45, 3 August 2006 (UTC)

See also: Astroid
I removed the following two entries from the "See also" list: and replaced them by one:
 * Astroid, a specific superellipse
 * Astroid, a particular ellipsoid (n = $2⁄3$, a = b = 1)
 * Astroid, the superellipse with n = $2⁄3$ and a = b = 1

Intro image
is it a squircle? —The preceding unsigned comment was added by Circeus (talk • contribs) 18:46, 7 January 2007 (UTC).

Merge with Squircle
This article and Squircle are both short and about nearly the same thing; the squircle is a special case of the superellipse and the superellipse is a slight variation on a squircle. Perhaps it would be best to treat both in a single article.--RDBury (talk) 12:04, 27 February 2012 (UTC)


 * Hmm, I'm not convinced. The articles aren't long, but they're not so short for concern either. If your reasoning that one is a special case of the other is accepted, it would also be necessary to merge in astroid as well, since that's also a special case of superellipse. And following the idea to its logical conclusion, shouldn't superellipse in turn be merged into superformula? Or for that matter, ellipse into superellipse? I don't think there's any more need to merge squircle and superellipse than there is to merge circle and ellipse. Modest Genius talk 12:59, 27 February 2012 (UTC)
 * In the case of ellipse and astroid there is lots to say that's unique to those curves so it makes sense to have separate articles. But with those exceptions there isn't much to say about squircles that can't be generalized to the superellipse, in fact there is already a lot of overlap between the two articles. For example you may notice that all the images of superellipses are actually squircles. Actually after looking at Superformula I don't think it would be a bad idea to merge it with the rest, currently it's basically a haven for OR computer code.--RDBury (talk) 15:14, 27 February 2012 (UTC)
 * I think that squircle contains enough unique information (and is of itself a unique enough shape) to remain its own article. I mean, a square is just a special type of rectangle, and it has its own article. Why shouldn't the squircle have the same treatment? ~Sp K 17:50, 21 April 2012 (UTC)

Confusing for n = 1
The explanation for the n = 1 case is very confusing. The article states that the resulting figure is a rhombus, yet for the case a=b=1, you end up with the parametric equation x + y = 1, which is decidedly NOT a rhombus, and not even a polygon, but rather a straight line. In fact, any real value for a or b will provide a line. Where does the rhombus come from? It's not mentioned in the article at all. --JasonMacker (talk) 05:40, 13 February 2015 (UTC)


 * The norm is not $$x^n + y^n$$ but $$|x|^n + |y|^n$$. —Tamfang (talk) 05:56, 13 February 2015 (UTC)

Animation
The animation appears to be incorrect. The convex curves have n<2, and the concave curves have n>2. Paul Studier (talk) 04:41, 15 February 2015 (UTC)


 * You're right -- it should have the convex case with n > or = 1. I'll delete it. Loraof (talk) 15:53, 6 March 2015 (UTC)

Absolute value signs needed?
The section on mathematical properties currently says


 * ''The pedal curve is relatively straightforward to compute. Specifically, the pedal of
 * $$\left(\frac{x}{a}\right)^n\! + \left(\frac{y}{b}\right)^n\! = 1$$
 * ''is given in polar coordinates by[2]
 * $$(a \cos \theta)^{\tfrac{n}{n-1}}+(b \sin \theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1}}.$$

Should the first equation have absolute value signs? Loraof (talk) 16:44, 6 March 2015 (UTC)


 * I think so. A negative number taken to a non integer power is not meaningful. Paul Studier (talk) 03:55, 7 March 2015 (UTC)

Practical formulas to find perimeter and area of superellipse
L=a+b*(((2.5/(n+0.5))^(1/n))*b+a*(n-1)*0.566/n^2)/(b+a*(4.5/(0.5+n^2))) P=L*4 A=a*b*((0.5)^((n^(-1.52)))) At=A*4                          Maher ezzideen aldaher (talk) 07:48, 16 March 2017 (UTC)

These eqs. from my research "New Simpler Equations for Properties of Hypoellipse ,Ellipse and Superellipse Curves " which presented at ICMS2012 available at academia.edu,linkedin websites Maher ezzideen aldaher (talk) 18:40, 13 April 2017 (UTC)

superellipse in 3D
hi, i don't mind the removal of my 3D rendering. but i wonder why it's not a superellipse in three dimensions? the classical 3D example, of course, is the convex structure (like piet heins' super egg). pls check out this article by paul bourke and tell me if i was mistaken. btw: i got the inspiration from a 3D animation program which in its latest release has the superellipse as a highly modifyable 3D structure in its portofolio. Maximilian (talk) 08:07, 15 August 2017 (UTC)

ps (made today): Maximilian (talk) 11:56, 15 August 2017 (UTC)


 * As the website you linked says, in the 3D case it could be called a superellipsoid rather than a superellipse. If there is a reliable source for the superellipsoid concept, it could be put into the generalizations section; but the website you gave is not a WP:reliable source. Loraof (talk) 21:37, 29 September 2017 (UTC)

Case n = 0
One might be forgiven for briefly considering the curve degenerates to a cross or a single point in the case n = 0, but as far as I am aware the curve disappears entirely in that case.

Proof: In the case n = 0, the equation degenerates to 1 + 1 = 1, which is not satisfied with any values of x or y.

Is this correct? J I P &#124; Talk 17:01, 16 October 2021 (UTC)

That pesky parameter '$$t$$'..
the parameter t {\displaystyle t} t having no elementary geometric interpretation

Does $$t$$ have any interpretation, elementary or not? Jimw338 (talk) 16:56, 17 October 2021 (UTC)