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.
Essentially, your question boils down to asking for equations whose solution set is bounded (since you can just divide by the sup, and multiply by the radius of your circle). There isn't a good answer, unless you can further restrict the functions that you're looking at - must they be continuous functions? polynomials?