Polar coordinates and parametric equation of circle in hyperbolic plane

Question

We know that the parametric equation of circles of radius r and center at origin is given by $\gamma (t) = (r\cos t, r \sin t)$ and any arbitrary closed curve can be written in the form $\gamma (t) = (r(t)\cos t, r(t) \sin t)$. Can someone help me please to write parametric equation of circle or any other closed curve in hyperbolic plane? I was thinking about $\gamma (t) = (r(t)\cosh t, r(t) \sinh t)$, but it will not work as it has different radius than the Euclidean circle. and also origin cannot be the centre