Editing Pseudosphere (section)

== Universal covering space ==
[[Image:Geodesics on the pseudosphere and three other models of hyperbolic geometry.png|right|thumb|The pseudosphere and its relation to three other models of hyperbolic geometry]]
The half pseudosphere of curvature −1 is [[covering space|covered]] by the interior of a [[horocycle]]. In the [[Poincaré half-plane model]] one convenient choice is the portion of the half-plane with {{math|''y'' ≥ 1}}.<ref>{{citation|first=William|last=Thurston|title=Three-dimensional geometry and topology|volume=1|publisher=Princeton University Press|page=62}}.</ref>  Then the covering map is periodic in the {{mvar|x}} direction of period 2{{pi}}, and takes the horocycles {{math|1=''y'' = ''c''}} to the meridians of the pseudosphere and the vertical geodesics {{math|1=''x'' = ''c''}} to the tractrices that generate the pseudosphere.  This mapping is a local isometry, and thus exhibits the portion {{math|''y'' ≥ 1}} of the upper half-plane as the [[universal covering space]] of the pseudosphere.  The precise mapping is
: <math>(x,y)\mapsto \big(v(\operatorname{arcosh} y)\cos x, v(\operatorname{arcosh} y) \sin x, u(\operatorname{arcosh} y)\big) ,</math>
where
: <math>t\mapsto \big(u(t) = t - \operatorname{tanh} t,v(t) =  \operatorname{sech} t\big)</math>
is the parametrization of the tractrix above.