Distance between points in hyperbolic disk models I was puzzeling with the distance between points in hyperbolic geometry and found that the same formula is used for calculating the length in the Poincare disk model as for the Beltrami-Klein model the formula 
$$ d(PQ)=\frac{1}{2} \left| \log \left(\frac{|QA||BP|}{|PA||BQ|}\right)\right| $$
where A and B are the idealpoints (extremities) of the line  (in the Beltrami-Klein model )  or the circle or diameter (in the Poincare disk model) that contains P and Q while  PA, PB, QA, QB be the euclidean distances between them. (but see below for an extra question) 
But let P and Q for simplicity be points on a diameter, then by going from a Beltrami-Klein model to a Poincare disk model the points P and Q get closer to the centre while the end points stay on the same points so the euclidean distances change, and the formula could give a different value.
Therefore (I think) the formula cannot be correct for both models, and so my question for which model is this equation and what is the formula for the other model. 
ADDED LATER: 
A more worked out example: (Schweikart Constant, altitude of the largest orthogonal isocleses triangle) 
Let r be the radius of the disk
Then $ A = ( - \frac{1}{2}  r \sqrt{2} ,  - \frac{1}{2}  r \sqrt{2} ) $ , 
$ B = (  \frac{1}{2}  r \sqrt{2} ,   \frac{1}{2}  r \sqrt{2} ) $ , 
 P = (0,0) and Q is on the line x=y
and the hypothenuse is the hyperbolic line between (r,0) and (0,r) 
The euclidean lengths for PQ are:
For the Poincare Disk model: $ PQ = r ( \sqrt2 - 1 ) $
For the Beltrami-Klein model: $ PQ= \frac{1}{2}  r \sqrt{2} $
What gives for the altitude:
For the Poincare Disk model: 
$ d(PQ)= \frac{1}{2} | \log ( 1 + \sqrt{2} | $
And for the Beltrami-Klein model: 
$ d(PQ)= \frac{1}{2} | \log ( 3 + 2 \sqrt{2} ) | = \log ( 1 + \sqrt{2})  $
What is right way to calculate the Schweikart Constant? 
The Schweikart Constant is $ \log ( 1 + \sqrt{2})  $ , so it looks like the value in the  Beltrami-Klein model is correct, but what is the correct formula for the Poincare Disk model?
Additional Question :
For the lengths in the Poincare disk models:
If the hyperbolic line is an euclidean circle are the euclidean lengths measured as the segment-lengths or as arc-lengths (along the circle)? 

There are two common versions of the Poincaré disk model:


*

*Constant curvature $-1$; metric element $\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}$

*Constant curvature $-4$; metric element $\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$ 


The difference is where you stick that annoying factor of $4$. Wikipedia uses  version 1, and its description of the relation with Beltrami-Klein model is also based on version 1. 
Let's compute the distance from $(0,0)$ to $(x,0)$, $x>0$, in each version:


*

*$\int_0^x \frac{2\,dt}{ 1-t^2 } = \log\frac{1+x}{1-x}$

*$\int_0^x \frac{  dt}{ 1-t^2 } = \frac12\log\frac{1+x}{1-x}$


As you can see, it's the second version that has $\frac12$ in front of the logarithm. 
The transformation $s=\frac{2u}{1+u^2}$, stated in Wikipedia under "relation to the Poincaré model", exactly doubles the Poincaré model distance from the origin, because 
$$\log\frac{1+s}{1-s}=2\log\frac{1+u}{1-u}$$
