TOPICS
Search

Bean Curve


There are a few plane curves known as "bean curves."

BeanCurve

The bean curve identified by Cundy and Rowllet (1989, p. 72) is the quartic curve given by the implicit equation

 x^4+x^2y^2+y^4=ax(x^2+y^2).
(1)

It has horizontal tangents at (2/3a,+/-2/3a) and vertical tangents at (0,0) and (a,0). The area enclosed by the curve is given by

A=sqrt(2)a^2int_0^1sqrt(x(1-x+sqrt(1+(2-3x)x)))dx
(2)
=(7pia^2)/(12sqrt(3))
(3)
=1.058049...a^2
(4)

(OEIS A193505). The geometric centroid (x^_,y^_) of the interior by

x^_=(23)/(42)a
(5)
y^_=0
(6)

and the area moment of inertia tensor of the interior by

I_(xx)=(113pi)/(1728sqrt(3))a^4
(7)
I_(xy)=0
(8)
I_(yy)=(23pi)/(108sqrt(3))a^4
(9)

(E. Weisstein, Feb. 3-5, 2018). The perimeter is given by

s=2aint_0^1sqrt(1+((1-2x+(1+3x-6x^2)/(sqrt(1+2x-3x^2)))^2)/(8x(1-x+sqrt(1+2x-3x^2))))dx
(10)
=3.75021364...a
(11)

(OEIS A193506).

LimaBeanCurve

A second bean curve that more closely resembles an actual bean (in particular, a lima bean), here called the "lima bean curve," is given by the simple polar equation

 r=a(sin^3theta+cos^3theta)
(12)

(Wassenaar; left figure above). It also is a quartic curve and has Cartesian equation

 (x^2+y^2)^2=a(x^3+y^3).
(13)

If the lima bean is rotated so that it appears entirely in the y>0 half-plane and is oriented symmetrically about the x-axis (right figure above), its Cartesian equation becomes

 sqrt(2)(x^2+y^2)^2=ay(3x^2+y^2).
(14)

The parametric equations of the original polar curve are

x=acost(sin^3t+cos^3t)
(15)
y=asint(sin^3t+cos^3t).
(16)

This curve has maximum values x_(max)=y_(max)=1 and minimum values x_(min)=y_(min)=r, where r=-0.28288... is the real root of 27-27x-288x^2+512x^3=0. The area enclosed by the curve is

A=5/(16)pia^2
(17)
=0.98174770...a^2
(18)

(cf. OEIS A244978). The geometric centroid (x^_,y^_) of the interior is given by

x^_=3/(10)a
(19)
y^_=3/(10)a,
(20)

and the perimeter by

s=aint_0^pisqrt(1+3/2sin^2(2theta)-2sin^3(2theta))dtheta
(21)
=1/2aint_0^(2pi)sqrt(1+3/2sin^2x-2sin^3x)dx
(22)
=aint_(-1)^1sqrt((2+3y^2-4y^3)/(2(1-y^2)))dy
(23)
=3.93170280...a
(24)

(OEIS A336501). The area moment of inertia tensor of the interior is given by

 I=[(123pi)/(2048)a^4 -(9pi)/(1024)a^4; -(9pi)/(1024)a^4 (123pi)/(2048)a^4].
(25)

See also

Bicuspid Curve, Limaçon

Explore with Wolfram|Alpha

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Sloane, N. J. A. Sequences A193505, A193506, A244978, and A336501 in "The On-Line Encyclopedia of Integer Sequences."Wassenaar, J. "Mathematical Curves: Bean Curve (1)." http://www.2dcurves.com/quartic/quarticbn.htmlWassenaar, J. "Mathematical Curves: Bean Curve (2)." http://www.2dcurves.com/quartic/quarticbn2.html

Cite this as:

Weisstein, Eric W. "Bean Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BeanCurve.html

Subject classifications