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I have come upon the curve with the following parametric equations:

$x(t)=\log(2+2\cos(t))/2$ $y(t)=t/2$

for $-\pi. It gives the image in the complex plane under $\log(1+z)$ of the unit circle. Does anyone know whether it has a name? It seems like this must have been studied at some point before.

Greg

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    For a non-parametric form, observe $x = \log(2+2\cos t)/2=\log(4 \cos^2(t/2))/2=\log|2\cos y|$. The bounds on $t$ make $\cos y$ non-negative, so that we can drop the absolute value, and we can go on to write $e^x = 2\cos y$.2011-07-28

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This is more or less a reflected and shifted version of the so-called "catenary of equal resistance" (en français, sorry). Here is the paper where they were first studied.

Wikipedia gives a derivation for the equation of the catenary of equal resistance; in some references, this is also called the "catenary of uniform strength". See this for instance.

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    @Willie: I will admit to spending (wasting?) the better part of my teenage years studying plane curves (mucking around in BASIC/Logo, scrounging books, etc.)... :) ...so yes, I still have those names from memory. I only went to that French site to check if my memory was correct.2011-07-28