I am trying to deduce the relation of the coordinates on a poincare disk with respect to the corresponding coordinate on a hyperboloid (described by $x^2+y^2-z^2=-1$) in Minkowski space.
In Euclidian space, for example when stereographically mapping a sphere, it is possible to parametrize the line between two points and calculate the intersection of this line with a plane, like they do in this example: http://www.ams.org/samplings/feature-column/fc-2014-02
I am wondering if in Minkowski space, where $\Delta s^2=\Delta x^2+\Delta y^2- \Delta z^2$, the same procedure is possible, because the distance between two lines is defined differently. So essentially my question is how you would parametrize the line between two points in Minkowski space in order to find it's intersection with a plane. Or maybe this is just a bad idea, and the approach should be entirely different.
If all you want to do is describe the line through two points $A$ and $B$ in $\mathbb R^{2,1}$, then all that matters is that you can view this as a three-dimensional $\mathbb R$-vectorspace. The metric is unimportant. So just thing $\mathbb R^3$ and define that line as $\{A+\lambda(B-A)\mid\lambda\in\mathbb R\}$ or equivalently (and more symmetrically) as $\{\lambda A+\mu B\mid\lambda,\mu\in\mathbb R,\lambda+\mu=1\}$. If you are dealing with homogeneous coordinates, you can even drop the requirement $\lambda+\mu=1$ and just require $(\lambda,\mu)\neq(0,0)$ instead, so at least one of them has to be non-zero.
The metric becomes relevant if you need a constant speed parametrization, i.e. need distances along the line to be proportional to the change in parameter. Or perhaps even unit speed, so that increasing the parameter by one adds one to the distance. Then you can't use the formulas above as they are, but may have to convert between metrics first.
If you want to use the lines to perform a stereographic projection between the Poincaré disk and the hyperboloid model, the speed of the parametrization is irrelevant, though. All you need is incidence: joining points to form lines, and intersecting lines with the plane of the disk or the sheet of the hyperboloid. You'd get projections like this:
(Figure taken from section 2.1.5 of my dissertation.)