Editing Hyperbolic motion (section)

{{Short description|Isometric automorphisms of a hyperbolic space}}
{{for|hyperbolic motion in physics|hyperbolic motion (relativity)}}

In [[geometry]], '''hyperbolic motions''' are [[isometry|isometric]] [[automorphism]]s of a [[hyperbolic space]]. Under composition of mappings, the hyperbolic motions form a [[continuous group]]. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by [[Felix Klein]] in his [[Erlangen program]]. The idea of reducing geometry to its characteristic group was developed particularly by [[Mario Pieri]] in his reduction of the [[primitive notion]]s of geometry to merely [[point (geometry)|point]] and ''motion''.

Hyperbolic motions are often taken from [[inversive geometry]]: these are mappings composed of reflections in a line or a circle (or in a [[hyperplane]] or a [[hypersphere]] for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the [[Cayley absolute|absolute]]. The proviso is that the absolute must be an [[invariant (mathematics)#Invariant set|invariant set]] of all hyperbolic motions. The absolute divides the plane into two [[Connected component (topology)|connected components]], and hyperbolic motions must ''not'' permute these components.

One of the most prevalent contexts for inversive geometry and hyperbolic motions is in the study of mappings of the [[complex plane]] by [[Möbius transformation]]s. Textbooks on [[complex function]]s often mention two common models of hyperbolic geometry: the [[Poincaré half-plane model]] where the absolute is the real line on the complex plane, and the [[Poincaré disk model]] where the absolute is the [[unit circle]] in the complex plane.
Hyperbolic motions can also be described on the [[hyperboloid model]] of hyperbolic geometry.<ref>[[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', §3.11 Hyperbolic motions, [[Cambridge University Press]], {{ISBN|0-521-61325-6}}, {{MathSciNet|id=2194744}}</ref>

This article exhibits these examples of the use of hyperbolic motions: the extension of the metric <math>d(a,b) = |{\log(b/a)}|</math> to the half-plane and the [[unit disk]].