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.


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


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    $\begingroup$ I bet you can make a cardoid look like a bean. $\endgroup$ – Alex Youcis Dec 12 '12 at 7:32
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    $\begingroup$ For some reason, this made me think of the shape from this question $\endgroup$ – davidlowryduda Dec 12 '12 at 7:36
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    $\begingroup$ It is bean-shaped. $\endgroup$ – Qiaochu Yuan Dec 12 '12 at 7:47
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    $\begingroup$ What kind of bean? There are a number of different types. $\endgroup$ – The_Sympathizer Dec 12 '12 at 7:50
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    $\begingroup$ this seems to work for 2D $\quad r=\sin^3\theta+\cos^3\theta$ $\endgroup$ – Elements in Space Dec 12 '12 at 8:05

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.


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