Poincaré half-plane model  In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H formula_1 , 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 (due to Bernhard Riemann), 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 formula_2 dimensional hyperbolic space by replacing the real number "x" by a vector in an "n" dimensional Euclidean vector space. The metric of the model on the half-plane, formula_3 is:   where "s" measures the length along a (possibly curved) line. The "straight lines" in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the "x"-axis (half-circles whose origin is on the "x"-axis) and straight vertical rays perpendicular to the "x"-axis. In general, the "distance" between two points measured in this metric along such a geodesic is:  where "arcosh" and "arsinh" are inverse hyperbolic functions  Some special cases can be simplified:  Another way to calculate the distance between two points that are on a (Euclidean) half circle is:  where formula_11 are the points where the halfcircles meet the boundary line and formula_12is the euclidean length of the line segment connecting the points "P" and "Q" in the model. A Euclidean circle with center formula_19 and radius formula_20 represents:   Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane.