Recently, I came across a simple problem whose solution involved showing that a line intersects the unit circle in at most two points. It's easy to find algebraic curves that intersect in the circle in many points (e.g. $4xy=1$ intersects the unit circle at four points).
I am interested to know whether there are algebraic curves that match the unit circle completely in some region, but do not in others. For example:
Is there an algebraic curve (the zero set of a polynomial in two variables $x$ and $y$) which matches the unit circle (where $x^2+y^2=1$) for all positive values of $x$ and $y$?
The only examples I can find match the unit circle completely for all inputs, and are obtained by multiplying factors in the original equation by $x^2+y^2$. For example, $x^2+y^2(x^2+y^2)=1$ or $x^4+2x^2y^2+y^4=x^2+y^2$.
For other equations, I can find curves which match each other in one region, but differ in another, for example $(x^3+y^3)^2=1$ contains all solutions to $x^3+y^3=1$, as well as distinct points.
Is there an algebraic curve other than the unit circle, which matches the unit circle for all $x,y>0$?