Suppose I have a sphere of points with two coordinates (two angles), all points are located on a unit sphere, so radius of the sphere is one.
Now my problem is, I want to find empty circles, or rather the center of circles, on that sphere, that fit between points with a radius that is greater than a specified angular radius, for example 0.1 rad. And with empty is meant, that inside that circle there are no other points.
As a practical example, think of the points on the sphere as stars limited by magnitude, and the circle describes an empty, dark spot for doing some background measurements with an instrument, whose field of view should be smaller than the diameter of the circle.
My idea is first to calculate all great arc distances between three close stars, or formulated differently, to span a net of spherical triangles onto that sphere.
And after that trying to calculate the in-circle in the triangles.
This is where I am stuck, I don't know how to do this on a sphere, in cartesian 2D-Coordinates this would be easier.
I suppose I can't use the formulas for an in-circle in cartesian coordinates for spherical coordinates, or only for a really rough estimation.
Do you have some ideas that would push me into the right direction?