A superellipse is a curve with Cartesian equation
 |
(1)
|
first discussed in 1818 by Lamé. A superellipse may be described parametrically by
 |  |  |
(2)
|
 |  |  |
(3)
|
The restriction to 
is sometimes made.
The generalization to a three-dimensional surface is known as a superellipsoid.
Superellipses with 
are also known as Lamé curves or Lamé ovals, and the case 
with 
is sometimes known as the squircle.
By analogy, the superellipse with 
and 
might be termed the rectellipse.
A range of superellipses are shown above, with special cases 
, 1, and 2 illustrated right above. The following table
summarizes a few special cases. Piet Hein used 
with a number of different 
ratios for various of his projects. For example, he used
for Sergels Torg (Sergel's Square)
in Stockholm, Sweden (Vestergaard), and 
for his table.
 | curve |
 | (squashed) astroid |
| 1 | (squashed) diamond |
| 2 | ellipse |
 | Piet Hein's "superellipse" |
| 4 | rectellipse |
If 
is a rational, then a superellipse is
algebraic. However, for irrational 
, it is transcendental. For even integers 
, the curve becomes closer to a rectangle as 
increases.
The area of a superellipse is given by
 |  | ![4bint_0^a[1-(x/a)^r]^(1/r)dx](/images/equations/Superellipse/Inline27.svg) |
(4)
|
 |  |  |
(5)
|
The above plots show generalization of the superellipse given by the function
 |
(6)
|
for 
,
..., 4 and 
,
..., 4.
Gielis (2003) has considered the further generalization of the superellipse given in polar coordinates by
![r(theta)=[|(cos(1/4mtheta))/a|^(n_2)+|(sin(1/4mtheta))/b|^(n_3)]^(-1/n_1).](/images/equations/Superellipse/NumberedEquation3.svg) |
(7)
|
Here, introduction of the parameter 
and use of polar coordinates gives rise to curves with 
-fold rotational symmetry. A number of
curves for different parameters are illustrated above, together with the names of
organisms that the curves resemble. While the above equation, dubbed the "superformula"
by Gielis (2003), is clearly capable of describing a number of diverse biological
shapes having a variety of symmetries, it seems unlikely that this formula has any
particularly fundamental biological significance (Peterson 2002, Whitfield 2003)
beyond as a possibly convenient parametrization. In fact, while the number of free
parameters in the "superformula" is six, Gielis (2003) also applies it
as a prefactor by which to multiply other polar curves (e.g., the logarithmic
spiral, rose curve curve, etc.), so the number
of parameters in the equation is effectively larger. Of course, any formula with
a large number of free parameters is capable of describing a very large parameter
space. (To emphasize this fact, it is sometimes humorously said that, given eight
or so free parameters, it is possible to describe an elephant.)
Families of curves generated by the "superformula" with 
and 
varying from 0 to 2 are illustrated above for values of 
varying from 1 to 8.
