 in non-euclidean geometry, the poincarÃ© half-plane model is the upper half-plane (denoted below as h), together with a metric, the poincarÃ© metric, that makes it a model of two-dimensional hyperbolic geometry. it is named after henri poincarÃ©, but originated with eugenio beltrami, who used it, along with the klein model and the poincarÃ© disk model (due to riemann), to show that hyperbolic geometry was equiconsistent with euclidean geometry. the disk model and the half-plane model are isomorphic under a conformal mapping. ==metric== the metric of the model on the half-plane :\{ \langle x, y \rangle | y > 0 \} \, is given by :(d s)^2 = \frac{(d x)^2 + (d y)^2}{y^2} \, where s measures 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 lines ending on the x-axis. the distance between two points measured in this metric along such a geodesic is :\operatorname{dist} (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname{arcosh} \left( 1 + \frac{ {(x_2 - x_1)}^2 + {(y_2 - y_1)}^2 }{ 2 y_1 y_2 } \right) \,. 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. this can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional euclidean vector space. ==special curves== in addition to the straight lines mentioned above, there are other special curves on the hyperbolic plane which can be modeled in the euclidean half-plane: * a circle (curve equidistant from a central point) with center \langle x, y \rangle \, and radius r \, is modeled by a circle with center \langle x, y \cosh r \rangle \, and radius y \sinh r \,. * a curve equidistant from a straight line is modeled by either a circular arc which intersects the x-axis at the same two points as the half-circle which models the given line, or by a straight line which intersects the x-axis at the same point as the vertical line which models the given line. * an horocycle (a curve which is like a circle with an infinite radius) is modeled by a circle tangent to the x-axis (but excluding the point of intersection), or by a line parallel to the x-axis. an horocycle is a limit on one side of a sequence of ever larger circles tangent at the same point to a given line; on the other side it is a limit of a se
Conformal_geometry Hyperbolic_geometry