Hyperbolic plane

The hyperbolic plane is the simply connected two-dimensional geometry of constant curvature minus one.

A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius $\rho$ has Euclidean radius $\tanh (\rho /2)$; an isometry transports this description to any other centre.

A horocycle in the Poincare disc model is a Euclidean circle internally tangent to the boundary circle. In the upper half-plane model, a horocycle is a Euclidean circle tangent to the real axis from above, or a horizontal line $y=h>0$. The common geometric feature is a single ideal centre of a horocycle.

Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.

The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.

A hyperbolic circle of centre $o$ and radius $R>0$ is the level set $\{x:d_{\mathbb{H}}(o,x)=R\}$ for the hyperbolic distance. Equally spaced directions from its centre give vertices of a regular geodesic polygon.

In the Poincare half-plane model, a hyperbolic circle of radius $R$ centred at $u+iv$ is the Euclidean circle $(x-u{)}^2+(y-v\cosh R{)}^2=v^2{\sinh }^{2}R$. Its Euclidean centre is shifted upward from its hyperbolic centre, and its Euclidean radius is $v\sinh R$. The lower height $v{e}^{-R}$ is positive. The formula follows by mapping the centre to zero in the Poincare disc model and using radial distance $2\mathrm{artanh}\rho$.

A hyperbolic line is a complete geodesic in the hyperbolic plane. In the Poincare half-plane model, hyperbolic lines are vertical rays and semicircles orthogonal to the boundary.

An ideal endpoint of a hyperbolic line is one of its two limiting points on the boundary at infinity. In the Poincare disc model these lie on the unit circle; in the upper half-plane model they lie on $\mathbb{R}\cup \{\infty \}$. These are boundary points, not points of the hyperbolic plane. Any two distinct ideal endpoints determine a unique hyperbolic line.

Two distinct hyperbolic lines are parallel, in the limiting-parallel convention, when they do not intersect in the hyperbolic plane and have exactly one common ideal endpoint. In the upper half-plane model, distinct vertical lines are parallel, sharing infinity. Disjoint lines with no common ideal endpoint are instead ultraparallel hyperbolic lines; only these have a common perpendicular of ultraparallel hyperbolic lines.

Two hyperbolic lines are ultraparallel when they are disjoint and have no common ideal endpoint. They possess a unique common perpendicular, which realizes the distance between them.

Hyperbolic distance is the Riemannian distance induced by the hyperbolic metric.

A horoball in the hyperbolic plane is a region bounded by a horocycle. In the upper-half-plane model, a horoball based at infinity is a Euclidean half-plane $\{y>c\}$; quotients by parabolic isometries give cusp neighborhoods.

A noncentral finite-order orientation-preserving hyperbolic isometry fixes an interior point and is elliptic.