Yes, usually a stereographic projection takes a sphere (minus a point) to the plane, but the principle generalizes straightforwardly to other dimensions. It then projects the $n$-sphere minus a point to $\mathbb R^n$.
For example you can restrict the usual stereographic projection to a (great or small) circle containing the missing point, which will give you a projection from a circle missing one point to an infinite line. It will be something like
$$p: (-\pi, \pi)\to\mathbb R, \qquad p(\theta)=\tan(\theta/2)$$
where the argument to $p$ is a point on the circle expressed in radians.
If you restrict the 2-dimensional projection to a circle on the sphere that doesn't contain the antipodal point, it projects into a circle or ellipse in the plane, which isn't much fun. (If the circle is a latitude line, you gain nothing more than a linear scaling).