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I am looking for the technical name for the horn shape which is created by repeating circles while increasing the radius size varying with an exponential function. Any references that can help me find the name, and all related terminology would be great. Thanks.

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The horn you describe seems to be a particular example of a surface of revolution. But I don't know if this particular example has a special name attached to it. There's a very famous horn though, called Gabriel's horn. – Adrián Barquero Oct 4 '12 at 14:24
Try googling horn angle and Lebesgue spine. – Dave L. Renfro Jan 31 '13 at 22:57

I think Adrián Barquero's comment is exactly to the point, so I am reposting it CW.

In general, a surface made by concatenating circles of various radii is called a surface of revolution. The idea is that instead of thinking of it as a stack of circles, you think of it as the result of revolving some curve around an axis. Each point on the curve traces out a circle as it revolves.

If you revolve a circle about its own diameter, you get a sphere; if you revolve a circle about a line that passes outside the circle, you get a torus.

In your case, you want the result of revolving an exponential curve about some line that passes outside the curve. For example, you might revolve the curve defined by $y=e^x$ around the line $y=0$ to get a horn shape; the horn has a circular cross-section, and the diameter of the cross-section at $x=x_0$ is $e^{x_0}$.

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