Editing Superellipse (section)

== Specific cases ==

This formula defines a [[closed curve]] contained in the [[rectangle]] {{math|&minus;''a'' ≤ ''x'' ≤ +''a''}} and {{math|&minus;''b'' ≤ ''y'' ≤ +''b''}}.  The parameters <math>a</math> and <math>b</math> are the semi-diameters or semi-axes of the curve. The overall shape of the curve is determined by the value of the exponent <math>n</math>, as shown in the following table:

{| class="wikitable"
|-
| <math> 0 < n < 1</math>
| width="390px" | The superellipse looks like a four-armed star with [[wikt:concave|concave]] (inwards-curved) sides.<br>For <math>n=1/2</math>, in particular, each of the four arcs is a segment of a [[parabola]].<br>An [[astroid]] is the special case <math>a=b</math> , <math>n=2/3</math>
|[[File:Superellipse star.svg|thumb|200px|right|The superellipse with ''n''&nbsp;=&nbsp;{{fraction|1|2}}, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|-
| <math> n = 1</math>
| The curve is a [[rhombus]] with corners  (<math>\pm a,0 </math>) and (<math>0,\pm b</math>).  
|
|-
| <math> 1< n < 2</math>
| The curve looks like a rhombus with the same corners but with [[convex set|convex]] (outwards-curved) sides.<br>The [[curvature]] increases without [[Limit of a sequence|limit]] as one approaches its extreme points.
|[[File:Superellipse rounded diamond.svg|thumb|200px|right|The superellipse with ''n''&nbsp;=&nbsp;{{fraction|3|2}}, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|-
| <math>n=2</math>
| The curve is an ordinary [[ellipse]] (in particular, a [[circle]] if <math>a=b</math>).  
|
|-
| <math>n>2</math>
| The curve looks superficially like a [[rectangle]] with rounded corners.<br>The curvature is zero at the points  (<math>\pm a,0 </math>) and (<math>0,\pm b</math>).
| [[File:Superellipse chamfered square.svg|thumb|200px|right|[[Squircle]], the superellipse with ''n''&nbsp;=&nbsp;4, ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;1]]
|}

If <math>n<2</math>, the figure is also called a '''hypoellipse'''; if <math>n>2</math>, a '''hyperellipse'''. When <math>n\geq1</math> and <math>a=b</math>, the superellipse is the boundary of a [[ball (mathematics)|ball]] of <math>\R^2</math> in the <math>n</math>[[norm (mathematics)#p-norm|-norm]]. The extreme points of the superellipse are (<math>\pm a,0 </math>) and (<math>0,\pm b</math>), and its four "corners" are (<math>\pm s_{a}</math>,<math>\pm s_{b}</math>), where <math>s = 2^{-1/n}</math> (sometimes called the "superness"<ref>Donald Knuth: ''The METAFONTbook'', p. 126</ref>).