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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:

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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Only the architect could answer this question accurately. But there are considerations that suggest what the shape is. First, it should be a nice smooth shape, and an elongated half-ellipsoid would do well. Architects' mathematical education probably includes ellipsoids, but more exotic surfaces would likely require more research effort than such a corny concept calls for. The radii of the support rings in the underlying steel framework would be easy to calculate for an elliptical shape. Even if the shape is something else, I bet that an ellipsoidal model will come impressively close if you choose the overall width-to-height ratio reasonably accurately. The bottom seems to be a quite different surface, but this could well be a (flattish) half-ellipsoid too. The panels representing the corn grains appear to have been fitted in an ad hoc and somewhat irregular way, suggesting that the designer was not over-concerned with mathematical purity in this respect.

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