# Center of circle on a stereographic projection

(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?

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In principle, you can do something like the following: Use the rotation matrix to move your circle-center to something nice like (1,0,0). From here, you can compute a parametrization of the circle using trigonometric functions. Then, use the inverse rotation matrix to move everything (including your circle parametrization) back. Finally, apply the formula for stereographic projection to get a parametrization of a circle in the plane. Then take three points on the circle and solve for the unique equidistant point. Alternatively, take the three points before you project. –  Charles Staats Apr 11 '12 at 21:51

In Euclidean coordinates, it is probably easiest to work in the plane containing the projection axis and the center. Then, if the spherical radius of the circle is $\rho$, the Euclidean distance between the center and a point on the circle is $2\cos \frac \rho 2$, and so you can find the Euclidean coordinates of the two intersection points by intersecting two plane circles (i.e. just a quadratic equation or two). –  Henning Makholm Apr 12 '12 at 10:08