Consider a sphere $\mathcal{S}$ of radius $R$, and $O\left(R,\theta_0,\phi_0\right)$ a point on this sphere.
I am trying to characterize the locus of points at a constant great circle distance from $O$. That is, given $0<r<R$ , is it possible to find a simple condition on $(\theta,\phi)$ so that $d\left(O~;~M(R,\theta,\phi)\right)=r$, where $d$ is the great circle distance.
Using the expression of this distance, this yields the following : $\cos(\theta)\cos(\theta_0)+\sin(\theta)\sin(\theta_0)\cos(\phi-\phi_0) = \cos(r/R)$. Is there a way to obtain a simpler relation? Ideally I would like to have an analytical solution, but something that could be easily solved numerically would interest me as well.