Talk:Squircle/Archive 1

Archive 1

dinner plates?

The reference provided for dinner plates is insufficient. It doesn't say whether or not those plates are actually mathematical "squircles" or just random shapes somewhere in between a square and a circle.--345Kai 09:03, 22 November 2006 (UTC)

The "cupboards" bit needs changing. Squircles packing may be less than, equal to or better than circle packing depending on the parameters. Rich Farmbrough, 10:53 22 November 2006 (GMT).

Squircle or squirtle?

This article doesn't seem to be consistent about the terms "squircle" vs. "squirtle". -Chinju 11:55, 22 November 2006 (UTC)

I think someone was having a joke at the mathematicians' expense: Squirtle is a Pokémon! I've changed all instances of 'squirtle' to 'squircle'. DAllardyce 11:59, 22 November 2006 (UTC)

• This shape or something near it was a major element in 1960s design, and not just "modern" dinnerware and tumblers. A Scandinavian architect-designer made a brief splash with it as his "invention": his name is long gone from my fading brain. --Wetman 14:36, 22 November 2006 (UTC)

It's even in the article: Piet Hein ;) Modest Genius ^talk 21:10, 22 November 2006 (UTC)

Squircle vs. Rounded square

Why is the rounded square so difficult to generalize? What would its formula look like? Circeus 14:58, 22 November 2006 (UTC)

Because you would have to have at least 6 different defined curves (two of them being sets of parallel straight lines and 4 arcs of circles), each valid between different limits. I could scribble it down for you, but writing it out in math wikicode would take ages. As an example, one of the sections would be (x-r)^2+(y-r)^2=(r/4)^2, valid in the interval x>r AND y>r. there would be 4 of those, plus some straight lines. Modest Genius ^talk 21:16, 22 November 2006 (UTC)

Ford Interceptor

Uh. Why does this page link to the Ford Interceptor? And why does Ford Interceptor link back? Dextrose 02:43, 4 September 2007 (UTC)

Proaly due to http://media.ford.com/article_display.cfm?article_id=25109 . I've removed the link here, though not from the Ford page Modest Genius ^talk 15:53, 4 September 2007 (UTC)

"Professionally rounded circles"? Argyriou (talk) 16:12, 5 September 2007 (UTC)

Well, if Ford want to employ someone to round circles, I'm their man! Possibly the world's easiest job! Modest Genius ^talk 17:42, 5 September 2007 (UTC)

An Aside

I think this beats "tofurky" as the new "awesomest word in the English language".

That is all. Evilspoons (talk) 20:28, 26 November 2007 (UTC)

Squovals???

I have deleted the Squoval See also link because I thnk it has nothing to do with this article. Dirbaio (talk) 16:28, 29 June 2009 (UTC)

An odd coincidence

Thank you for writing an interesting article, Keith. By coincidence I signed up as a Wikipedia user on November 22, 2006, the same day this article was featured on the main page. I didn't run across this article until today, though.

Here's what makes that simple coincidence seem like an odd coincidence to me. I won second prize in the math division of the International Science Fair in 1969 with a paper entitled "The Hypercircle". The "hypercircle" I wrote about was the locus of points satisfying the equation x⁴ + y⁴ = r⁴ in the Cartesian plane, where r is an arbitrary positive constant (the "hyperradius"). This corresponds exactly with the article's definition of a "squircle" centered on the origin (0, 0). Besides calculating the circumference of the "hypercircle" (using Simpson's rule and an adding machine – I didn't have access to a digital computer, since those were pretty hard to find in 1969) and the length of the "minor diameters", I also worked out the properties of "hypercircular functions", analogues of the familiar trigonometric functions. The "hypersine" was just the altitude of a right triangle with central angle θ and inscribed inside the "hypercircle", the "hypercosine" was the base of the same triangle, and so forth. Oh, yeah – I also expanded the "hypercircular" functions as power series, and tied them to a class of definite integrals whose exact form I can't recall right now, except that the integrand was the inverse of the square root of a polynomial. That paper has to be lying around here somewhere ... if I run across it I may add some of that stuff to this article (or maybe link to it somehow).

I do remember the judges asking me if I had considered redefining angular measure in terms of the arc length of this particular curve – that notion seemed foreign to me at the time, because I was unfamiliar with contour integrals. I also remember some discussion about whether I might have stolen the idea from somebody else. I was entirely unaware of Piet Hein's work with a "squircle" back then, although I had already read some of his little "Grooks", which remain among my favorite poems.

Well, I've rambled on too long already. Thanks again for making me smile! DavidCBryant 17:45, 28 November 2006 (UTC)

Wow, that IS an odd coincidence ;). Please feel free to add any of that work to the article! The idea of redefined sine and cosines is especially interesting, any idea what possible use these would be? Modest Genius ^talk 19:06, 28 November 2006 (UTC)

Would the redefined trig functions be called the squine and cosquine? Argyriou (talk) 00:33, 29 November 2006 (UTC)

Maybe sound synthesis would have some use for them.Akilaa (talk) 19:02, 7 December 2009 (UTC)

See Also: Egg of Columbus

Why is the Egg of Columbus story linked after this article? I think that relation is a bit far-fetched. 74.66.134.47 (talk) 07:44, 16 February 2012 (UTC)

No idea. I've removed the link. Modest Genius ^talk 10:03, 16 February 2012 (UTC)

Squircle != RoundRect

Why does RoundRect redirect here? they're not the same shape at all. -- Ch'marr (talk) 18:36, 10 December 2010 (UTC)

But it's discussed in this article... AnonMoos (talk) 03:46, 11 December 2010 (UTC)

It may have been discussed once, but it isn't anymore. --Eva Ekeblad (talk) 06:48, 15 May 2012 (UTC)

The rounded square is, in the 'Similar shapes' section. A RoundRect is exactly the same shape as the rounded square just with different length sides. Maybe a sentence should be added to note this? Modest Genius ^talk 11:29, 15 May 2012 (UTC)

something must be wrong...

Just glancing at the picture, you can see that the points on the "squircle" aren't equidistant from the circle and the square (and I don't think it's an optical illusion). Either the text is wrong, or the picture. One or the other.--345Kai 08:58, 22 November 2006 (UTC)

As far as I can see, the squircle illustrated is a superellipse, and yes, I agree with you that the construction doesn't produce the given curve, so I've removed it from the article. -- The Anome 11:03, 22 November 2006 (UTC)

After annoying some people, I think I have independently found where the problem is:

The squircle is a specific case (found by setting $n=4$) of the class of shapes known as supercircles, which have the equation

$\left(x-a\right)^{n}+\left(y-b\right)^{n}=r^{n}$.

Unfortunately, the taxonomy is not consistent - some authors refer to the class as supercircles and the specific case as a squircle, while others adopt the opposite naming convention. Supercircles in turn are a subgroup of the even more general superellipses, which have the equation

$\left|{\frac {\left(x-a\right)}{r_{a}}}\right|^{n}\!+\left|{\frac {\left(y-b\right)}{r_{b}}}\right|^{n}\!=1$,

where $r_{a}$ and $r_{b}$ are the semimajor and semiminor axes. Superellipses were extensively studied and popularised by the Danish mathematician Piet Hein.

Please check the following link for a different definition where the square is inscribed inside the circle instead of the circle being inscribed inside the square. different squircle Is this the correct difinition or will both definitions work? It would be very nice to have matlab visually confirm either or both of these definitions with respect to the formula that purportedly defines a squircle. I don't have the technology to do this right now. I reformatted the comment immediately above so that it is indented the same amount throughout. I hope I did not cause irritation. Allan --68.73.227.43 15:15, 22 November 2006 (UTC)

You're correct - the description given there would appear to give the correct properties of the squircle, I must have got the square and circle the wrong way around. Modest Genius ^talk 21:09, 22 November 2006 (UTC)

I don't know about the formulae, but the picture (still? - this is my first visit to this page) does not look right. The sides are shorter than the top and bottom. That shouldn't be. A squircle should be an object that looks like a rounded SQUARE or a squarish CIRCLE rather than an ovalish rectangle or a rectangular oval (rectoval). 211.225.39.76 (talk) 07:14, 28 August 2013 (UTC)

Both images look square (not rectangular) to me. Have you checked that the images on square look correct? If not, it could be something odd with your browser or screen resolution. Modest Genius ^talk 11:49, 28 August 2013 (UTC)

I(Special:Contributions/211.225.39.76|211.225.39.76)'m on a different computer now and the image is all right. I'll have to go back to the other computer to check. 211.225.33.104 (talk) 11:04, 30 August 2013 (UTC)

Need more accurate image

If you click on the image of the squircle, the image's page says that the diagram is not completely accurate, and indeed it is not. By my measurements with a ruler, the ratio of the distance from the center to the northeast point to the distance from the center to the point directly above it is 1.24. But according to the correct formula on the superellipse page at the end of the lead, this ratio should be the fourth root of 2, or about 1.188.

Can someone with imaging skills draw it accurately using a graphing tool that can handle quartics and replace the present graph? Thanks. 208.50.124.65 (talk) 20:19, 13 July 2014 (UTC)

The mathematics of Bezier curves allowed in the SVG file format does not allow directly encoding 4th-order curves, so the two alternatives are to approximate the curve with cubic splines, or to use a large number of tiny straight-line segments. But if you mean "Squircle2.svg", then that's plenty accurate enough for display at ordinary computer screen resolutions. Run the PostScript source code on page File:Squircle2.svg, and the inaccuracy can be seen as the purple peeking out behind the black... AnonMoos (talk) 07:20, 15 July 2014 (UTC)

P.S. If you open the SVG file in a text editor, then you can see that the point (242.1782,242.1782) is √(242.1782²*2) distance away from the origin, or 342.4917, and 342.4917/288=1.1892... AnonMoos (talk) 07:45, 15 July 2014 (UTC)

Distance

If I take a square with sides of 2, and a circle inside the square with a radius of 1, and I draw the locations of the points with exactly the same distance to the square and the circle, would that be a Squircle? 81.205.204.122 (talk) 13:03, 8 March 2015 (UTC)

Strongly doubt it -- that would lead to a shape with a definite corner (tangent discontinuity). AnonMoos (talk) 17:12, 10 March 2015 (UTC)

What is the equation in polar coordinates?

The article gives the equation in rectangular coordinates. This is a suggestion to give the equation in polar coordinates too. --50.53.35.120 (talk) 19:33, 3 March 2016 (UTC)

Done. I simply inserted n=4 into the equivalent equation on Superellipse#Mathematical_properties, so there's no reference and I haven't verified the result. I don't think that counts as original research. Modest Genius ^talk 11:22, 4 March 2016 (UTC)

Thanks, but polar coordinates use r and θ (or ϕ). See the transformation here. Note that r is a variable, not a constant, so there is a notation conflict with the equation $\left(x-a\right)^{4}+\left(y-b\right)^{4}=r^{4}$. See, also, equation 1 in the cited article by Guasti et al: "LCD pixel shape and far-field diffraction patterns". If you cite Guasti et al, the equation won't be original research. --50.53.36.42 (talk) 16:27, 8 March 2016 (UTC)

Here is the Wolfram Alpha plot of

$r(\theta )={1 \over {\sqrt[{4}]{cos^{4}\,\theta +sin^{4}\,\theta }}}$.

I don't have a source, but the equation follows from a straight-forward substitution of

$x=r\cos \theta$

$y=r\sin \theta$

in

$x^{4}+y^{4}=1$. (plot)

Note that

$0<cos^{4}\,\theta +sin^{4}\,\theta$. (plot)

--50.53.38.206 (talk) 20:51, 8 March 2016 (UTC)

App icons

A squircle?

Squircles, rather than rounded squares, are used as the base shape for a lot of app icons nowadays. On recent Nokia Series 40, in iOS, in the new Instagram logo (right). Maybe the article should mention it. Here are two articles on the shape in iOS:

1. https://www.cocoanetics.com/2013/06/ios-7-icon-squircle/
2. http://www.designbygeometry.com/ios-icon-mask-corner-curve-study/

Nclm (talk) 14:22, 12 May 2016 (UTC)

Worth mentioning. One problem is that these icons aren't squircles, they're still just radiused corners of squares, as the icon sizes are still so low-res that there's no difference as yet. I suspect some bearded designers are looking at the high-res image on a large Mac screen, but not noticing what they're shipping out as the bitmap to the app coders. Andy Dingley (talk) 14:35, 12 May 2016 (UTC)

Arc length for 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

Area of superellipse

A=a*b*((0.5)^((n^(-1.52))))

At=A*4

Maher ezzideen aldaher 20:39, 12 March 2017 (UTC)

Do you have a reliable source for those equations? If not, they can't appear in the article. Modest Genius ^talk 10:59, 13 March 2017 (UTC)

Also, even if the formulas are correct, I would doubt extremely that the non-integer constants are exact as given ("0.566" etc). Finding the arc length of an ordinary ellipse leads to the intractable Elliptic integral problem, so a superellipse might be worse! ... AnonMoos (talk) 03:00, 20 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) 15:49, 13 April 2017 (UTC)

If you performed the research then you should avoid writing about it on Wikipedia. See WP:COI. Modest Genius ^talk 20:20, 13 April 2017 (UTC)