(forgive my drawing skills, everything is out of scale, and I wish my sphere looked more like a sphere)
If I hava a circle on a sphere, and then map the sphere to a plane using a stereographic projection, the circle will still be a circle on the projection.
The blue thing below is the sphere.
Then projected to a plane from the point opposite to the circle center:
But depending on the projection point I choose on the sphere, the resulting circle is scaled and its original center is no longer the circle center in the projection:
So, given the circle in a radius 1 sphere (I have the position of the center in a rotation matrix, and the arc radius in radians), how can I find the center of the circle in the projection plane?