Consider that one regular hexagon and one square cut a sphere of radius $r$ as shown in the attached figure. A circular arc of length $s_1$ and a circular arc of length $s_2$ are formed (blue arcs). The arc $AB$ is a part of a small circle centered at a given point $O_1$, while the arc $AC$ is an arbitrary arc of a circle centered at the point $O_2$.
Assuming that the coordinates of the points $A, B, C$ with respect to a common coordinate system are known, how can we compute the geodesic curvature of these circular arcs?
Any tip/help would be useful.
My approach: Given the formula presented in Geodesic curvature of sphere parallels, the small circle geodesic curvature is given by $\kappa_g = \frac{\sqrt{r^2 - r_1^2}}{rr_1}$, where $r_1$ denotes the radius of the small circle centered at $O_1$.
Similarly, how can we compute $\kappa_g$ for the other arbitrary circular arc?
That seems an exceedingly cumbersome formula and I haven't checked it. It is a standard formula (which can be checked with a tiny bit of trig, or by using the formula $\kappa_g = \pm\sqrt{\kappa^2-\kappa_n^2}$) that a geodesic circle of radius $R$ on a sphere of radius $r$ has $\kappa_g = \dfrac1r\cot R$. Does this help you? (Your circular arcs are arcs of geodesic circles centered at $O_1$ and $O_2$.)