The Formula for the Curve of a Trumpet Bell

I'm guessing this problem belongs to integral calculus:
I've been tasked with making some anvils, on a metal lathe, for the repair of brass instrument bells. I need to calculate corresponding X-Y values along the curve of that bell.One side only as it's symmetrical. I can certainly take sample measurements at frequent intervals of the bell, but I don't know how to come up with the formula for the curve that will allow me to create a chart of the thousands of points I need to accurately machine it. Any ideas ? Salamat/Thanks.

Jhun Vert's picture

I am not sure if this will work, but the equation of Normal Distribution in Statistics is called a bell shaped curve due to the fact that it is in the form of a bell. The equation is

$$ f(x) = \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\dfrac{1}{2} \left( \dfrac{x - \mu}{\sigma} \right)^2} $$

You may randomly assign the value of $\sigma$ and see if the size of bell fits to what you desire.

2021 Update
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