The Poincaré half-plane model represents the hyperbolic plane by the upper half-plane
 |
with hyperbolic metric
 |
Its hyperbolic lines are the vertical Euclidean lines and the Euclidean semicircles orthogonal to the real axis. The real fractional linear transformations with positive determinant act by orientation-preserving isometries of this metric.
The fractional linear transformation 
carries the half-plane conformally onto the
Poincaré hyperbolic disk.
