I am currently covering a geodesics section in my Differential Geometry course and hyperbolic space came up. The model that I am using is the upper half plane model, i.e., $\mathbb{H}^2= \{(x,y)\in \mathbb{R}^2 \vert y>0 \}$, and the First Fundamental Form is defined as $I= \frac{dx^2 +dy^2}{y^2}$. So my problem is the following:
Give points $P=(0,1), R=(1,1), Q=(0,2)$ find the lenghts and angles in $\mathbb{H}^2$.
$\vec{PR}= ?$
$\vec{PQ}= \int_1^2 \sqrt{I}= \int_1^2 \frac{dy}{y}=ln(2)$
$\vec{RQ}= ?$
So the geodesic between $P=(0,1)$ and $R=(1,1)$ is the segment of the circle $C_1 :(x-1/2)^2+y^2 = 5/4$, thus solving for the length of the curve originating at $R$ and ending at $P$ is done by considering the trasformation $\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 1+ \sqrt{5}/2 cos\theta \\ \sqrt{5}/2 sin\theta \end{pmatrix}$ and then solving the integral $L_{C_1}= \int_R^P \sqrt{\frac{dx^2+dy^2}{y^2}}=\int_{\theta_1}^{\theta_2}\sqrt{csc^2\theta \cdot d\theta^2}=\frac{\sqrt{5}}{2}ln\lvert csc \theta - cot \theta \lvert$. Solving for $csc\theta_1$ is done by considering the $(1/2 , 1, \sqrt{5}/2)$ ,triangle generated by the center of $C_1, R,$ and the projection of $R$ onto the $x-$axis. Thus, $csc \theta_1 = \frac{\sqrt{5}}{2}$ and $cot \theta_1 = \frac{1}{2}$ and similarly, $csc \theta_2 = \frac{\sqrt{5}}{2}$ and $cot \theta_2 = \frac{1}{2}$. The issue I am having is that then when plugged back in to solve for the integral, I end up with value $0$, which I know is not correct. What am I doing incorrectly?
And I do not know how to approach the angles question. I know it will involve the first fundamental from, but I do not know how to apply it. Any suggestions are helpful.
The full geodesics in the upper half plane model of the hyperbolic plane are Euclidean circles with their diameters on the real axis (with vertical lines as degenerate cases).
Therefore, to determine the distance between $(1,1)$ and $(0,2)$, you need to find the radius and center of the appropriate circle they lie on, then parametrize the arc between them as $(x(t),y(t))$ (say with unit speed in the Euclidean metric so the $dx^2+dy^2=(\dot{x}^2+\dot{y}^2)dt^2$ will just be $dt^2$), and then integrate $dt/y$.
Also, it looks like your calculation of the distance between $(0,1)$ and $(1,1)$ used the Euclidean line segment straight between them - that is not the shortest path betweeen them in the hyperbolic plane, so you'll need to apply the circle idea to that calculation as well.
As for hyperbolic angles, they match Euclidean angles in this model, so you need to examine the tangent vectors of the parametrizations for the arcs you find.