Where does the Poincaré metric come from?

According to Poincaré disk model - Wikipedia, the free encyclopedia the transformation map from points on the Poincaré disk to points on the hyperboloid is . I verified the calculation, this transformation seems to be correct (without considering the hight t). However, the metric yields

which is not the metric mentionned in the article, that is:

This is known as the Poincaré disk metric and I would really like to know how to calculate it geometrically from the hyberboloid model as explained in the Wikipedia article. So can anyone explain me what's wrong here? Do I need to still apply another transformation to get the right metric? I tried but this does not seem to be an easy task...