parametrizing a circle near the origin In the book "Using Algebraic Geometry" by Cox and co-workers, i encountered the following example: consider the circle with center $(-1,0)$ and radius $1$, given by the equation $x^2+2x+y^2 = 0$. Then the authors say "a parametrization of the circle near the origin is given by $x = \frac{-2t^2}{1+t^2}, y=\frac{2t}{1+t^2}$". I can readily see that if we substitute for $x,y$ then the equation is still satisfied, and so every point given by this parametrization is a point of the circle. However, what does it mean to be "a parametrization of the circle near the origin"? How does that connect to the fact that both of the rational functions that give the parametrization are elements of the local ring $k[t]_{(t)}$? I am looking for some argument that connects the algebra with the geometry.
 A: The "near the origin" comment presumably comes from the fact that the parameterisation doesn't cover the entire circle. Specifically, there is no value of $t$ that yields the point $(-2,0)$ on the circle. This is easy to see algebraically: $y=0$ implies $t=0$, which gives $x=0$. The origin is diametrically opposite this problematic point, so "near the origin" is a safe region that's far away from the trouble. 
It would have been much clearer if they had said that the given functions parameterize the given circle minus the point $(-2,0)$.
The connection between the algebra and the geometry is that the formulae for $x$ and $y$ arise from stereographic projection of a vertical line onto the circle, using the point$(-2,0)$ as the center of projection. This explains geometrically why the point $(-2,0)$ gets omitted -- it is the image of "points at infinity" under the stereographic projection.
Sorry, but I don't know anything about local rings.
A: Well, the parametrization covers a neighborhood of the origin, but not the entire circle, since it leaves out $(-2,0)$. Maybe that's all the authors mean?
The top Related question has some good insights: Why is there no polynomial parametrization for the circle?
