I'm confused about the following. Consider a circle $C$ on the Riemann sphere with the point at infinity in its interior but not at its center. Let's say $C_1$ is its center, and $C_2$ is the center of its exterior. I'm confused about the images of $C$, $C_1$ and $C_2$ if you would apply stereographic projection on this sphere (if needed assume the sphere has radius 1 and is placed on top of the origin of the complex plane with no rotation).
More explicitly, I'm confused about this: I think that what we would see now after stereographic projection is some circle in the complex plane with center the image of $C_2$. The exterior of this image circle seen in the complex plane would be the image of the interior of $C$ seen on the Riemann sphere, but then where in the complex plane is the image of $C_1$? And what even is this center? By how we defined $C$ it can't be the point at infinity (as that is stereographically projected to itself), but how could it be any other number? Seen from the complex plane we just have some circle and ask what the center of the exterior is, which could only be the point at infinity as far as I know.
I hope I made clear what the problem is. Thanks in advance!