What shape is a bean?

Question

How can I mathematically describe the shape of an idealised bean? (In two dimensions and in threes dimensions)

At the moment I'm calling the shape I refer to an ellipse/ellipsoid on a curved major axis.

EDIT

This seems to work for 2D: $$r \leq \sin^3\theta+\cos^3\theta$$

For some reason, this made me think of the shape from this question — davidlowryduda, Dec 12 '12 at 07:36
@Alex Youcis , A cardioid has a sharp indentation, a bean has a smooth indentation — Elements In Space, Dec 12 '12 at 07:46
Claim to fame: we should come up with one. In the spirit of the paraboloid and ellipsoid, I propose the legumenoid.
Edit: Oh, we're not actually naming this thing. Oops. Well it's homeomorphic to the unit ball, should we describe via homeomorphism? — Gyu Eun Lee, Dec 12 '12 at 07:50
It would be a surface homeomorphic to the ball with a saddle point? Would that be sufficient to characterize it? — Gyu Eun Lee, Dec 12 '12 at 07:55
this seems to work for 2D
$\quad r=\sin^3\theta+\cos^3\theta$ — Elements In Space, Dec 12 '12 at 08:05
Four years later., just curious... did you ever get your 3D, kidney-shaped bean? — uhoh, Jul 03 '17 at 16:50
Nope. I can't work out how to revolve it around a curved axis. — Elements In Space, May 14 '18 at 13:48
@BrianD — do you think the curve at the top of that page is the same as the curve in this post? — Elements In Space, Jan 27 '22 at 23:01

DrZ214 · Answer 1 · 2017-02-28T10:55:57.603

The problem is, we do not know exactly what you mean by "idealised bean". Can you describe the shape better? A closed curve shape with two opposite indentations? (To me that seems more like a peanut.)

Then you say in the comments that $r = \sin^3\theta + \cos^3\theta$ seems to work well.

What do you want to do with that curve now? Give it another indentation on the opposite side?

$r = \cos^2(\theta) + 1$

More indented with $r = 2\cos 2\theta + 4$

You also said "an ellipse/ellipsoid on a curved major axis." I'm not sure exactly what that looks like. If you can sketch the picture and show us, we'd have a better idea of the shape you're looking for.

BTW, to make 3D versions, usually we just rotate the 2D shape about some axis.

For reference, here are the equations for an ellipse:

$y = \pm b\sqrt{1 - \frac{x^2}{a^2}} \hspace{1cm}$ or $\hspace{1cm}r = \displaystyle \frac{ab}{\sqrt{a^2\sin^2\theta + b^2\cos^2\theta}}$ in polar coordinates.

What shape is a bean?

1 Answers1