Mathematical description of a corncob

I'm trying to figure out how I can make a paper model of the corncob water tower in Rochester, Minnesota for my N-scale train layout.

The best I can find is this: http://www.korthalsaltes.com/model.php?name_en=hebdomicontadissaedron

Is there an equation for a corncob-like closed quadric surface?

I suppose I could do trial and error, using a vector graphics package to play around elongating the example above and such, but I'm wondering if there is a mathematical way I can do it?

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That's a hairy question! Hopefully someone will be able to crunch through it. –  Bruno Joyal Jan 17 '12 at 19:52
With or without the niblets? –  Ｊ. Ｍ. Apr 14 '13 at 12:34

If we have to use a single curve to represent a corn when looking sideways, then we want to find something increases very fast at the beginning, after reaching its maximum, decreases really slow.

So we could use: $$y = \frac{1}{a}x^{\alpha} (b-x)^{\beta},\quad \text{ for }0 < x < b.$$ Product rule yields: $$y' = \frac{1}{a}x^{\alpha-1}(b-x)^{\beta-1} (\alpha b- x (\alpha+\beta)),$$ we can see that if $\alpha,\beta<1$, $y'\to \infty$ as $x\to 0^+, b^-$, and $y' = 0$ when $x = \alpha b/(\alpha+\beta)$.

Let $a = b/2, b = 5, \alpha = 0.3, \beta = 0.7$, this is the corn curve we have:

Now we revolve this curve around the $x$-axis by $2\pi$, which is done by replacing $y$ by $\pm\sqrt{y^2+z^2}$. A surface will be obtained, and the upper half of the corn surface looks as follows:

In the picture you gave, my guess is that the bottom part and the upper part are two revolutions of two quadratic curves respectively.

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