what are the geodesics in the hyperbolic upper half plane? In the upper half-plane $H = \{(x, y) \in \mathbb{R}^2 \mid y > 0\}$.
The distance between the two points (a,A) and (b,B) is set by the shortest curvature in metric $F(y) = \int_a^b \frac{\sqrt{1 + (y')^2}}{y}dx,\; y(a) = A, y(b) = B$.
What is the geodesic in this metric?

Hyperbolic metric is :
$$ ds^2 = \dfrac{dx^2 + dy^2}{y^2} $$
Hyperbolic distance
$$ = \int_a ^b \frac{\sqrt{1+y'^{2}} }{y} dx $$
that integrates to semi-circle geodesics centered on x-axis. It is $ \log \tan ( \pi/2 + \phi/2) $ reckoned to a point slope $\phi$ from top.
This is Poincaré half-plane model, geodesics have vanishing hyperbolic curvature and minimum length between points $a,b$ in the hyperbolic plane.
