Geodesics in the upper half plane

Question

There is a nice description of the geodesics on the upper sheet of the hyperboloid $H^2 \subset \mathbb{R}^3$ in terms of hyperbolic sine and cosine as $$ \lambda(t)= (\cosh(t))x + (\sinh(t))y $$ where $x,y$ are Lorentz othonormal vectors in $\mathbb{R}^3$.

The geodesics in the upper half-plane $U \subset \mathbb{C}$ are Euclidean lines and circles that intersect the real line orthogonally. For example ,the imaginary line $\lambda(t)=it$, for $t \in \mathbb{R}$ is a geodesic in $U$.

I am curious how to describe geodesics in $U$ in terms of hyperbolic functions.

Is there a way of parameterizing $\lambda(t) = it$ in terms of hyperbolic functions?

( I am asking about this because I am reading a paper that is describing geodesics in the upper half-plane in terms of hyperbolic tangent and hyperbolic secant, as $$ \lambda(t) = c_1 \left( \tanh \omega(t+ t_0) + i \operatorname{sech} \omega(t + t_0) \right) $$)

Answer 1

The best parametrization of geodesics of the upper-half plane (in the hyperbolic metric) follows from that $PSL_2(\Bbb{R})$ acts transitively and isometrically on those, so any geodesic is of the form $\{ \gamma (i e^t), t\in (-\infty,\infty)\}$ for some $\gamma \in PSL_2(\Bbb{R})$, which reduces to a quotient of $\sinh,\cosh,\exp$.

(or $PSL_2(\Bbb{R})$ quotiented by the diagonal matrices, leaving the imaginary axis fixed)

where $i e^t,t\in (-\infty,\infty)$ is the parametrization of the imaginary axis such that $hyperbolicDistance(ie^t,ie^u)=|t-u|$.

https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model#Geodesics

Answer 2

The natural parameterization (that is, one that can be viewed as the pointwise limit of suitably translated semicircular geodesics as $c_1\to\infty$, keeping $\omega$ constant but varying $t_0$ to make $\lambda(0)$ follow a horocycle perpendicular to all the geodesics) would be $$ \lambda(t) = ie^{\omega(t+t_0)} $$

However, if you want hyperbolic functions in particular, you're free to write that as

$$ \lambda(t) = i(\cosh \omega(t+t_0) + \sinh \omega(t+t_0)) $$

Alternatively just declare that $e^x$ is a hyperbolic function already.