Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H ${\displaystyle =\{\langle x,y\rangle \mid y>0;x,y\in \mathbb {R} \}}$, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive.

The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.

This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.

The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model.

This model can be generalized to model an ${\displaystyle n+1}$ dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.