By analogy with the squircle, a name used by Fernández Guasti et al. (2005, p. 266), the term "rectellipse" (used here for the first time) is a natural generalization to the case of unequal vertical and horizontal dimensions.
The first definition of the rectellipse is the quartic curve which is a special case of the superellipse
with 
and 
, namely
 |
(1)
|
illustrated above. This curve encloses area
 |
(2)
|
and has area moment of inertia tensor
![I=[(ab^3pi)/(2sqrt(2)) 0; 0 (a^3bpi)/(2sqrt(2))].](/images/equations/Rectellipse/NumberedEquation3.svg) |
(3)
|
The second definition of the rectellipse was given, though not explicitly named, by Fernandez Guasti (1992, pp. 898-899). This curve has quartic Cartesian equation
 |
(4)
|
with 
and squareness parameter 
, where 
corresponds to an ellipse with
semiaxes 
and 
and 
to a rectangle with side lengths
and 
. This curve is actually a semialgebraic
set, as it must be restricted to 
and 
to exclude other branches. For 
, this rectellipse encloses area
 |
(5)
|
where 
is an elliptic integral of the second
kind, which can be verified reduces to 
for 
and 
for 
.
