Assume a sphere of radius $r$. Consider that one hexagon and one square cut the sphere as shown in the attached figure. One circle inscribed within the hexagon with given radius $r_1$ and one circle inscribed within the square with given radius $r_2$, are formed. The two circles are tangent. The centers of these circles are $O_1, O_2$, respectively. Also, assume that the cartesian coordinates of all points $A, B, C, O_1, O_2, O$ are known. The arc lengths $s_1, s_2$ with respect to $O_1, O_2$ are also known.
We are interested in calculating the geodesic curvature of the two blue circular arcs with respect to the center $O$ of the sphere.
Any help/tip would be useful.
EDIT: This a kind of a generalization of the question presented in Geodesic curvature of sphere parallels . In our case, the circles are not necessarily in parallel with a great circle.
