# equation for the region inside a circle

What equation or group of equations fill the entire or part of a region inside a circle without using inequalities?

Update

I don't know if this problem is already solved, I'm trying to find the "length" of the region inside the circle. The function sould be continuos and with integrable length of arc.

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From your figure, it looks like you're looking for a parametric curve in polar coordinates, $r = f(\theta)$, whose range for $\theta\in[-\infty,\infty]$ is the entire circle/annulus, is that right? – Rahul Jan 1 '13 at 9:14
@JasperLoy I've updated the question for clarification – rraallvv Jan 1 '13 at 15:46
@RahulNarain I was thinking about a parametric function with correspondence one to one (binjective function), but my guess is that it can't be bijective. – rraallvv Jan 1 '13 at 15:47

@rraallvv Is the parametric function going to be continuous (do you know what this word means?) How about $f: [ 0 , 1 ] \rightarrow \mathbb{R}^2$ given by $f (t) = \begin{cases} t & t \in \mathbb{Q} \\ -t & t \not \in \mathbb{Q}\\ \end{cases}$. – Calvin Lin Jan 1 '13 at 6:57
There is such a thing as a space-filling curve, so you can get continuous solutions. But you won't find one where the arc length is finite, because a curve of finite arc length has Hausdorff dimension $1$, while the region inside a circle or an annulus has Hausdorff dimension $2$. – Robert Israel Jan 1 '13 at 9:28