I'm out of my depth playing with non-euclidean geometry, so if terminology is off or I'm missing critical things, leave a comment and I'll edit.
Given the radii ($r_1, r_2, r_3$) and centerpoints of 3 circles $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$ (polar coordinates work too, I assume they'd be easier but I don't have any experience using them) and the minor and major semiperimeters of an ellipsoid (I think that's enough information to define it), assuming the circles all intersect at one point, how do you find the intersection of the 3 circles?
On the surface of a sphere, solving for intersection seems a fair bit simpler. To my understanding, you just take the 3 circle equations (defined by the center points and the chords of the radii) and use algebra to solve. Please correct me if this understanding is wrong.