Poincaré disk model of hyperbolic plane

Question

Can someone please explain trigonometry in Equations (2) to (8) of:

PoincareDisk ?

Answer 1

It is just basic geometry (nothing hyperbolic)

• the boundary circle is the unit circle
• $\phi_1$ and $\phi_2$ are the angles from the x-axis to the ideal points of the hyperbolic line. (the points where the arc meets the boundary circle)

Then:

• $\phi$ is the angle from the x-axis to the centre of the orthogonal circle.

• $d\phi$ is the angle from $\phi_1$ or $\phi_2$ to the angle $\phi$.

• $r$ is the distance from $\phi_1$ or $\phi_2$ to the centre of the orthogonal circle. (and thus the radius of the orthogonal circle)

• $R$ is the distance from the origin to the centre of the orthogonal circle.

nothing really interesting I thought