I want to understand the equation of a line through two points $z_1$ and $z_2$ in the poincare half-disc plane defined as $\mathcal{H} = \left\{(x,y):x^2+y^2 < 1\right\}$, i.e. set of points inside the unit circle.
I'm told that a line in $\mathcal{H}$ is either a diameter $ax+by = 0$ or the arc of a circle that intersects the unit circle at right angles, with an equation $x^2+y^2+2dx+2ey+1=0$ where $d^2+e^2 >1$.
I haven't really been given an example of how this works. How does one determine for example the lines between $z_1 = \frac{1}{2}$ and $z_2 = \frac{1}{2}i$ or see if they lie on a diameter or not?
EDIT: Is it $x = \Re(z_1)$, $y= \Im (z_1)$ and $x = \Re(z_2)$, $y= \Im (z_2)$ and we just plug these to the given equations? I think I'm starting to sort of understand if that's the case, but I'd still appreciate some explanation. Thanks.