Lima Bean Curve

The name lima bean curve is used here for the quartic curve with polar equation

(Wassenaar). It is also given by the Cartesian equation

Replacing the right-hand side by gives the distinct bean curve, a quartic curve that in its usual orientation is symmetric about the -axis rather than the line .

The left figure above shows the curve in its original orientation. The right figure shows the same curve after a counterclockwise rotation through . In this orientation, the curve lies in the half-plane and is symmetric about the -axis; its Cartesian equation is

The parametric equations for the original polar curve are

For , this curve has maximum values and minimum values , where is the real root of . The area enclosed by the curve is

The digits in the numerical value are those of OEIS A244978 shifted one place, since . The geometric centroid of the interior is given by

and the perimeter by

This perimeter has a closed form in terms of a Lauricella function. Set

Since and , the defining series for does not converge at these arguments. The required analytic continuation is specified by the following real integral. It is Euler-type because it is built from the beta integral kernel :

Therefore,

(OEIS A336501; E. Weisstein, Aug. 27, 2026).

The area moment of inertia tensor of the interior about the origin is given by