I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) and I have some questions about pseudospherical circles at page 10: the author says that Beltrami finds the equation of a circle on the pseudosphere and then deduces that
From Gauss Lemma, such circles are orthogonal to the geodesics issuing from $(u_0, v_0)$.
I don't understand what lemma he is referring to and why the circles are orthogonal to the geodesics.
A few lines below, he talks about generalized metric circles $\gamma_C$:
The curves $\gamma_C$ corresponding to the same $(u_0, v_0)$, but to different values of $C$ are equidistant as it is easy to show. When the center $(u_0, v_0)$ is ideal, the value $C= 0$ is admissible and $\gamma_0$ is a geodesic [..] For $C \neq 0$ then $\gamma_C$ is a set of points having a fixed distance from the geodesic $\gamma_0$ and $\gamma_C$ is not a geodesic
Why are the curves $\gamma_C$ equidistant? In other words, how can you compute the distance between two geodesics on a pseudosphere? Also, why isn't $\gamma_C$ a geodesic on a pseudosphere? What exactly defines a geodesic on a pseudosphere?