# Poincaré disk construction

I am trying to understand how the Poincaré disk is constructed using the stereographic projection for the hyperboloid $$x^2+y^2-z^2=-1$$.

So I want to project a line from the fixed point $$(0,0,-1)$$ to a point on the hyperboloid.

The parametric equation of a line between $$(0,0,-1)$$ and a point $$(x,y,z)$$ is given by $$\vec{r}(t)=t(x,y,z)+(1-t)(0,0,-1)=(tx,ty,tz+t-1)$$, where $$t \in [0,1]$$.

Now I want to find where this line intersects the hyperboloid. So

$$(tx)^2+(ty)^2-(tz+t-1)^2=-1 \iff$$ $$(x^2+y^2-z^2-2z-1)t^2+(2z+2)t=0$$

Now $$x^2+y^2-z^2=-1$$ because I want to find the intersection. So I get the equation

$$-(2z+2)t^2+(2z+2)t=0,$$

whose only solutions are $$t=0$$ or $$t=1$$.

Shouldn't I obtain an expression for $$t$$ in terms of $$x,y,z$$?

You should be projecting from the hyperboloid (i.e., a point such that $$x^2+y^2-z^2=-1$$) to the plane $$z=0$$. What you are doing now is you are projecting the hyperboloid ($$x^2+y^2-z^2=-1$$) to itself ($$x^2+y^2-z^2=-1$$), so that is why nothing changes (for $$t=1$$).
Now I want to find where this line intersects the hyperboloid => Now I want to find where this line intersects the plane $$z=0$$.
The correct $$t$$ is given by $$tz+t-1=0$$.