Going from Metric to Distance Function in the Poincaré Half Plane

Question

Let the Poincaré Half Plane be the set $\{(x, y) \in \mathbb{R}^2 : y > 0\}$. It is a known result that the the metric

$ds^2 = \frac{dx^2 + dy^2}{y^2}$

yields a distance function $f$ such that its output is the length of the geodesic between two points on the Poincaré Half Plane. Through some process that I do not understand, it is possible to prove that

$f((x_1, y_1), (x_2, y_2)) = \operatorname{arcosh} \left( 1 + \frac{ {(x_2 - x_1)}^2 + {(y_2 - y_1)}^2 }{ 2 y_1 y_2 } \right)$.

How does one prove the previous equality from the given metric?

Answer 1

One way is to take advantage of symmetry.

• Prove that vertical lines $y=\text{constant}$ are geodesics, and that your formula holds for any pair of points lying on a vertical line.
• Prove that the metric $ds^2$ is invariant under the group of fractional linear transformations $x+iy \mapsto \frac{a(x+iy) + b}{c(x+iy)+d}$.
• Prove that your formula is invariant under the group of fractional linear transformations.
• Prove that the set of images of the vertical lines, under the action of the group of fractional linear transformations, are the semicircles with diameter on the $x$-axis. Hence, your formula holds for any two points lying on such a semicircle.
• Prove that any two points lie on such a semicircle.