Is it possible to elementarily parametrize a circle without using trigonometric functions? Just out of curiosity: Is it possible to parametrize a full circle or part of one with elementary functions but without using trigonometric functions? If so, what are advantages/disadvantages compared to the standard parametrizations using $\cos(t)$ and $\sin(t)$?

What about $f(x,\pm)=\pm\sqrt{1-x^2}$, where $f(\cdot,\cdot)$ has a discrete and continous parameter defined in $[-1,1]$...
You may also use $e^{it}=\cos(t)+i\sin(t)$ to represent a circle in the complex plane. With this calculating Fourier transforms becomes handy...

Just a comment to H.H. Rugh answer that needs graphical support:
His parametrisation is the stereograhic projection which has an application in Photography:
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