A bean curve is one of several plane curves known by that name.
The bean curve identified by Cundy and Rowllet (1989, p. 72) is the quartic curve given by the implicit equation
 |
(1)
|
Replacing the right-hand side 
by 
gives the distinct quartic
curve referred to here as the lima bean curve,
which in its original orientation is symmetric about the line 
rather than the 
-axis.
It has horizontal tangents at 
and vertical
tangents at 
and 
. The area
enclosed by the curve is given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
(OEIS A193505).
The geometric centroid 
of the interior is given by
 |  |  |
(5)
|
 |  |  |
(6)
|
The area moment of inertia tensor of the interior is given by
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
(E. Weisstein, Feb. 3-5, 2018).
The perimeter admits an exact representation in terms of the seven-variable Lauricella function 
. Let 
, ..., 
be the roots of
 |
(10)
|
and set 
.
Then
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
(OEIS A193506; E. Weisstein, Aug. 28, 2026). The defining series does not converge at these
arguments, so 
is understood by analytic continuation through
its Euler integral representation. The Euler
integral independently reproduces the numerical value above.
