(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?
Best Answer
Intersect the spherical circle with a meridian of the sphere that passes through its center. Find the two intersection points (if you have the location of the center in spherical coordinates, that is just a matter of adding and subtracting the radius from the polar coordinate/latitude/zenith distance). Project the two intersection points to the plane. By symmetry they have to be diametrically opposite points of the image circle, so the apparent center of the image circle is halfway between them.