Upon reading up on the hyperboloid model of the hyperbolic space I have found a pretty way of obtaining geodesic line segments on the hyperboloid of two sheets connecting two given points. I became interested with finding geodesic circles of a given radius R and center (x, y).. Browsing through various books and scientific articles I have found a way to do this in this thesis. It takes the circle of given radius R around the pole (0,0,1) of the hyperboloid and applies to it the same transformation that moves the pole into the given center (x, y). While this method works I am interested learn if there's another one similar to the way a geodesic is described in the wikipedia page of hyperboloid model.
I am aware that a geodesic line is the non-empty intersection of the hyperboloid and a 2-dimensional subspace containing the origin whereas a geodesic line is the non-empty elliptic intersection of the hyperboloid and a 2-dimensional subspace not containing the origin with slope less than one. Can such a compact form of the circle be found?
If $B$ denotes the Minkowski bilinear form, the points $P$ on the circle around $Q$ with radius $r$ satisfy $B(P,Q) = \cosh(r)$, which is a linear equation in $P$ and thus defines a plane intersecting the hyperboloid.