Exponential map of Beltrami-Klein model of hyperbolic geometry In the Betrami-Klein model of hyperbolic geometry, geodesics are represented as straight lines. Hence the exponential map of a tangent vector $\mathbf{v}$ at a point $\mathbf{p}$ is $\mathbf{p} + \lambda \mathbf{v}$, where $\lambda$ is a scalar that depends on $\mathbf{p}$ and $\mathbf{v}$. For example, suppose $\mathbf{p} = 0$. Then the exponential map is
$$ \exp_\mathbf{p}(\mathbf{v}) = \hat{\mathbf{v}} \tanh \lvert \mathbf{v} \rvert $$
Notice that $\tanh \lvert \mathbf{v} \rvert < 1$ always, since we cannot reach the circle at infinity with any finite $\mathbf{v}$.
Now consider the diagram below:

where $\mathbf{q} = \mathbf{p} + \lambda \mathbf{v}$. The hyperbolic distance between $\mathbf{p}$ and $\mathbf{q}$ must be equal to the length of the tangent vector, that is, $d(\mathbf{p}, \mathbf{q}) = \lvert \mathbf{v} \rvert$. This hyperbolic distance is
$$ d(\mathbf{p}, \mathbf{q}) = \frac{1}{2} \log \frac{\lvert\mathbf{p} - \mathbf{r_1}\rvert \lvert\mathbf{q} - \mathbf{r_2}\rvert}{\lvert\mathbf{p} - \mathbf{r_2}\rvert \lvert\mathbf{q} - \mathbf{r_1}\rvert}$$
where $\mathbf{r_1}$ and $\mathbf{r_2}$ are the two ideal points intersected by the straight line connecting $\mathbf{p}$ and $\mathbf{q}$. We can find these ideal points as follows:
\begin{align}
1 &= \mathbf{r} \cdot \mathbf{r} \\
&= (\mathbf{p} + \mu \mathbf{v}) \cdot (\mathbf{p} + \mu \mathbf{v}) \\
&= \mathbf{p} \cdot \mathbf{p} + 2 \mu \mathbf{p} \cdot \mathbf{v} + \mu^2 \mathbf{v} \cdot \mathbf{v} 
\end{align}
Hence
\begin{align}
\mu &= \frac{-2 \mathbf{p} \cdot \mathbf{v} \pm \sqrt{(2 \mathbf{p} \cdot \mathbf{v})^2 - 4 (\mathbf{v} \cdot \mathbf{v})(\mathbf{p} \cdot \mathbf{p} - 1)}}{2 \mathbf{v} \cdot \mathbf{v}} \\
&= \frac{-\mathbf{p} \cdot \mathbf{v} \pm \sqrt{(\mathbf{p} \cdot \mathbf{v})^2 - (\mathbf{v} \cdot \mathbf{v}) (\mathbf{p} \cdot \mathbf{p} - 1)}}{\mathbf{v} \cdot \mathbf{v}}
\end{align}
so that $\mathbf{r} = \mathbf{p} + \mu \mathbf{v}$ for each root. Hence we have
\begin{align}
\lvert \mathbf{v} \rvert &= \frac{1}{2} \log \frac{\lvert\mathbf{p} - (\mathbf{p} + \mu_1 \mathbf{v})\rvert \lvert(\mathbf{p} + \lambda \mathbf{v}) - (\mathbf{p} + \mu_2 \mathbf{v})\rvert}{\lvert\mathbf{p} - (\mathbf{p} + \mu_2 \mathbf{v})\rvert \lvert(\mathbf{p} + \lambda \mathbf{v}) - (\mathbf{p} + \mu_1 \mathbf{v})\rvert}
\\
&= \frac{1}{2} \log \frac{\lvert\mu_1 \mathbf{v}\rvert \lvert(\lambda - \mu_2) \mathbf{v}\rvert}{\lvert\mu_2 \mathbf{v}\rvert \lvert(\lambda - \mu_1) \mathbf{v}\rvert}
\\
&= \frac{1}{2} \log \frac{\lvert\mu_1\rvert \lvert\lambda - \mu_2\rvert}{\lvert\mu_2\rvert \lvert\lambda - \mu_1\rvert}
\end{align}
or equivalently
\begin{align}
\left\lvert \frac{\lambda - \mu_2}{\lambda - \mu_1} \right\rvert = \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert}
\\
\frac{\lambda - \mu_2}{\lambda - \mu_1} = \pm \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert}
\\
\lambda - \mu_2 = \pm \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert} (\lambda - \mu_1)
\\
\lambda \left( 1 \mp \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert} \right) = \mu_2 \mp \mu_1 \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert}
\\
\lambda = \frac{\mu_2 \mp \mu_1 \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert}}{1 \mp \left\lvert \frac{\mu_2}{\mu_1} \right\rvert \mathrm{e}^{2 \lvert \mathbf{v} \rvert}}
\end{align}
We pick the root $\lambda > 0$. Is this result correct? Can it be simplified? Can anyone find a source for it?

I did some calculation and so and if 
$P =  (p_x,p_y)$ and $Q=(p_x+v_x,p_y+v_y)$ are two points on a chord the ideal points (points on the unit circle are:
$$I_1 = \left ( \frac{v_y \  c + v_x \sqrt{v_x^2 +v_y^2 -c^2}}
{v_x^2 +v_y^2} \ , \
 \frac{-v_x \ c  + v_y \sqrt{v_x^2 +v_y^2 -c^2} }{v_x^2 +v_y^2} \right) $$
$$I_2 = \left ( \frac{v_y \  c - v_x \sqrt{v_x^2 +v_y^2 -c^2}}
{v_x^2 +v_y^2} \ , \
 \frac{-v_x \ c  - v_y \sqrt{v_x^2 +v_y^2 -c^2} }{v_x^2 +v_y^2} \right) $$
with $ c =p_x v_y - v_x p_y $
I don't know if this will help you and also I don't know how to write this as a formula of complex numbers (but will make a seperate question for that :)
