Bean Curve
There are a few plane curves known as "bean curves."
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
(OEIS A193505). The geometric centroid (x^_,y^_) of the interior by
and the area moment of inertia tensor of the interior by
(E. Weisstein, Feb. 3-5, 2018). The perimeter is given by
(OEIS A193506).
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
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
(cf. OEIS A244978). The geometric centroid (x^_,y^_) of the interior is given by
and the perimeter by
(OEIS A336501). The area moment of inertia tensor of the interior is given by
![画像: I=[(123pi)/(2048)a^4 -(9pi)/(1024)a^4; -(9pi)/(1024)a^4 (123pi)/(2048)a^4].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BeanCurve/NumberedEquation5.svg){kind=link}
See also
Bicuspid Curve, LimaçonExplore with Wolfram|Alpha
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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)." https://www.2dcurves.com/quartic/quarticbn.html.Wassenaar, J. "Mathematical Curves: Bean Curve (2)." https://www.2dcurves.com/quartic/quarticbn2.html.Referenced on Wolfram|Alpha
Bean CurveCite this as:
Weisstein, Eric W. "Bean Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BeanCurve.html