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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: $$r \leq \sin^3\theta+\cos^3\theta$$

bean

  • 0
    I bet you can make a cardoid look like a bean.2012-12-12
  • 1
    For some reason, this made me think of the shape from this [question](http://math.stackexchange.com/q/70320/9754)2012-12-12
  • 0
    @Alex Youcis , A cardioid has a sharp indentation, a bean has a smooth indentation2012-12-12
  • 6
    It is bean-shaped.2012-12-12
  • 3
    What kind of bean? There are a number of different types.2012-12-12
  • 0
    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?2012-12-12
  • 0
    It would be a surface homeomorphic to the ball with a saddle point? Would that be sufficient to characterize it?2012-12-12
  • 0
    @Alex: I see that beans are close to your heart... :-)2012-12-12
  • 3
    this seems to work for 2D $\quad r=\sin^3\theta+\cos^3\theta$2012-12-12
  • 0
    I am not sure how to get the 3D curve.2013-01-01
  • 0
    What does bean-shaped mean @QiaochuYuan?2013-01-11
  • 0
    Four years later., just curious... did you ever get your 3D, kidney-shaped bean?2017-07-03
  • 0
    Nope. I can't work out how to revolve it around a curved axis.2018-05-14

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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.

enter image description here

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

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

enter image description here

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

enter image description here

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.