There are two incompatible definitions of the squircle.
The first defines the squircle as the quartic plane curve which is special case of the superellipse with 
and 
,
namely
 |
(1)
|
illustrated above. This curve as arc length
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A186642), where 
is a Meijer
G-function (M. Trott, pers. comm., Oct. 21, 2011), encloses area
 |
(4)
|
and has area moment of inertia tensor
![I=a^4[pi/(2sqrt(2)) 0; 0 pi/(2sqrt(2))].](/images/equations/Squircle/NumberedEquation3.svg) |
(5)
|
The second definition of the squircle was given by Fernandez Guasti (1992), but apparently not dubbed with the name "squircle" until later (Fernández Guasti et al. 2005). This curve has quartic Cartesian equation
 |
(6)
|
with squareness parameter 
,
where 
corresponds to a circle with radius
and 
to a square of side length 
. This curve is actually semialgebraic, as it must be restricted
to 
to exclude other branches.
This squircle encloses area
 |
(7)
|
where 
is an elliptic
integral of the second kind, which can be verified reduces to 
for 
and 
for 
.
Both versions somewhat resemble the shape of the region swept out by a Reuleaux triangle.
The generalization of the squircle to the case with unequal 
- and 
-dimensions
might be dubbed the rectellipse.
