Geodesic in the Poincare half-plane model
= Geodesic in the Poincare half-plane model
{c}
= Hyperbolic geodesic
{synonym}
The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand $\sqrt{1+y'^2}/y$ has the <Beltrami identity>
$$
\frac1{y\sqrt{1+y'^2}}=\frac1R,
$$
which integrates to $(x-c)^2+y^2=R^2$. Swapping the coordinate axes gives circles centred on the vertical boundary for the metric $(dx^2+dy^2)/x^2$.