Editing Unit disk (section)

==Hyperbolic plane==
The open unit disk forms the set of points for the [[Poincaré disk model]] of the hyperbolic plane. [[Circular arc]]s perpendicular to the unit circle form the "lines" in this model. The unit circle is the [[Cayley absolute]] that determines a [[metric (mathematics)|metric]] on the disk through use of [[cross-ratio]] in the style of the [[Cayley–Klein metric]]. In the language of differential geometry, the circular arcs perpendicular to the unit circle are [[geodesic]]s that show the shortest distance between points in the model. The model includes [[motion (geometry)|motion]]s which are expressed by the special unitary group [[SU(1,1)]]. The disk model can be transformed to the [[Poincaré half-plane model]] by the mapping ''g'' given above.

Both the Poincaré disk and the Poincaré half-plane are ''conformal'' models of the hyperbolic plane, which is to say that angles between intersecting curves are preserved by motions of their isometry groups.

Another model of hyperbolic space is also built on the open unit disk: the [[Beltrami–Klein model]]. It is ''not conformal'', but has the property that the geodesics are straight lines.