The term "squircle" has two incompatible definitions.
The first defines the squircle as the quartic curve which is a special case of the superellipse with
and 
, namely
 |
(1)
|
illustrated above. This curve has 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)
|
For the superellipse form with 
, the greatest distance from
the origin is 
, the geometric mean
of the corresponding greatest distances 
and 
for a circle and a square of
the same half-width 
.
The second definition of the squircle was given by Fernandez Guasti (1992, pp. 898-899) and called a squircle by Fernández Guasti et al. (2005, p. 266). This curve has quartic Cartesian equation
 |
(6)
|
with 
and squareness parameter 
, where 
corresponds to a circle with
radius 
and 
to a square of side length 
. This curve is actually a semialgebraic
set, as it must be restricted to 
to exclude other branches. For 
, this squircle encloses area
 |
(7)
|
where 
is an elliptic
integral of the second kind, which can be verified to reduce 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.
The superellipse form of the squircle is among the imperfect-circle curves studied in squigonometry (Poodiack and Wood 2022).
The planar squircle is distinct from the squaricle, a creased D-form assembled from a disk and a square with equal perimeters (Sharp 2005, p. 123).
