Bean Curve -- from Wolfram MathWorld

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)

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). 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.

The left figure above shows a second bean curve that more closely resembles an actual bean (in particular, a lima bean), here called the "lima bean curve." In this original orientation, it is given by the simple polar equation

(14)

(Wassenaar). It is also a quartic curve, with Cartesian equation

(15)

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

(16)

The parametric equations for the left-hand original polar curve are

			(17)
			(18)

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

			(19)
			(20)

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

			(21)
			(22)

and the perimeter by

			(23)
			(24)
			(25)

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

			(26)
			(27)
			(28)
			(29)

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 :