If I'm asked how to cumpute a geodesic circle on a parametrized surface $x:U\rightarrow S$ centered at a point $p \in S$ my approach would be to
$1$. Compute all the geodesics $\gamma_\theta(t)$ such that $\gamma_\theta(0) = p$ and $\gamma_\theta'(0) = \cos(\theta)x_u + \sin(\theta)x_v$ using the geodesic differential equations.
$2$. If I want $r$ to be the radius of the geodesic circle, setting $C(\theta) = \gamma_\theta(r)$ as the geodesic circle.
This process though feels like a lot of work, is there a simpler way of finding a geodesic circle centered at a point?