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{{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 ''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) = \vert \log(b/a) \vert</math> to the half-plane, and in the location of a quasi-sphere of a hypercomplex number system.
 
==Introduction of metric in upper half-plane==
[[Image:Ultraparallel theorem.svg|thumb|right|alt=semi-circles as hyperbolic lines|For some hyperbolic motions in the half-plane see the [[Ultraparallel theorem]].]]
The points of the [[upper half-plane]] model HP are given in [[Cartesian coordinates]] as {(''x'',''y''): ''y'' > 0} or in [[polar coordinates]] as {(''r'' cos ''a'', ''r'' sin ''a''): 0 < ''a'' < π, ''r'' > 0 }.The hyperbolic motions will be taken to be a [[function composition|composition]] of three fundamental hyperbolic motions.
Let  ''p'' = (''x,y'')  or  ''p'' = (''r'' cos ''a'', ''r'' sin ''a''), ''p'' ∈ HP.  The fundamental motions are:
: ''p'' → ''q'' = (''x'' + ''c'', ''y'' ), ''c'' ∈ '''R''' (left or right shift)
: ''p'' → ''q'' = (''sx'', ''sy'' ),  ''s'' > 0  ([[Homothetic transformation|dilation]])
: ''p'' → ''q'' = ( ''r''<sup> &minus;1</sup> cos ''a'',  ''r''<sup> &minus;1</sup> sin ''a'' ) ([[inversive geometry#Circle inversion|inversion in unit semicircle]]).
Note: the shift and dilation are mappings from inversive geometry composed of a pair of reflections in vertical lines or concentric circles respectively.

===Use of semi-circle Z===
Consider the triangle {(0,0),(1,0),(1,tan ''a'')}. Since 1 + tan<sup>2</sup>''a'' = sec<sup>2</sup>''a'', the length of the triangle hypotenuse is sec ''a'', where sec denotes the [[Trigonometric functions#Reciprocal functions|secant]] function. Set ''r'' = sec ''a'' and apply the third fundamental hyperbolic motion to obtain  ''q'' = (''r'' cos ''a'', ''r'' sin ''a'') where  ''r'' = sec<sup>&minus;1</sup>''a'' = cos ''a''.  Now
:|''q'' &ndash; (½, 0)|<sup>2</sup> = (cos<sup>2</sup>''a'' &ndash; ½)<sup>2</sup> +cos<sup>2</sup>''a'' sin<sup>2</sup>''a'' = ¼

so that ''q'' lies on the semicircle ''Z'' of radius ½ and center (½, 0). Thus the tangent ray at (1, 0) gets mapped to ''Z'' by the third fundamental hyperbolic motion.  Any semicircle can be re-sized by a dilation to radius ½ and shifted to ''Z'', then the inversion carries it to the tangent ray. So the collection  of hyperbolic motions permutes the semicircles with diameters on ''y'' = 0 sometimes with vertical rays, and vice versa.  Suppose one agrees to measure length on vertical rays by using the [[logarithm]] function:
:''d''((''x'',''y''),(''x'',''z'')) = |log(''z''/''y'')|.

Then by means of hyperbolic motions one can measure distances between points on semicircles too: first move the points to ''Z'' with appropriate shift and dilation, then place them by inversion on the tangent ray where the logarithmic distance is known.

For  ''m'' and ''n'' in HP, let  ''b'' be the [[perpendicular bisector]] of the line segment connecting ''m'' and ''n''. If ''b'' is parallel to the [[abscissa]], then ''m'' and ''n'' are connected by a vertical ray, otherwise ''b'' intersects the abscissa so there is a semicircle centered at this intersection that passes through ''m'' and ''n''. The set HP becomes a [[metric space]] when equipped with the distance  ''d''(''m'',''n'') for  ''m'',''n'' ∈ HP as found on the vertical ray or semicircle. One calls the vertical rays and semicircles the ''hyperbolic lines'' in HP.
The geometry of points and hyperbolic lines in HP is an example of a [[non-Euclidean geometry]]; nevertheless, the construction of the line and distance concepts for HP relies heavily on the original geometry of Euclid.

==Disk model motions==
Consider the disk D = {''z'' ∈ '''C''' : ''z z''* < 1 } in the [[complex plane]] '''C'''. The geometric plane of [[Lobachevsky]] can be displayed in D with circular arcs perpendicular to the boundary of D signifying ''hyperbolic lines''. Using the arithmetic and geometry of complex numbers, and [[Möbius transformation]]s, there is the [[Poincaré disc model]] of the hyperbolic plane:

Suppose ''a'' and ''b'' are complex numbers with ''a a''*&nbsp;&minus;&nbsp;''b b''* = 1. Note that
:|''bz'' + ''a''*|<sup>2</sup> &minus; |''az'' + ''b''*|<sup>2</sup> = (''aa''* &minus; ''bb''*)(1 &minus; |''z''|<sup>2</sup>),
so that |''z''| < 1 implies |(''a''z + ''b''*)/(''bz'' + ''a''*)| < 1 . Hence the disk D is an [[invariant (mathematics)#Invariant set|invariant set]] of the Möbius transformation
:f(''z'') = (''az'' + ''b''*)/(''bz'' + ''a''*).
Since it also permutes the hyperbolic lines, we see that these transformations are '''motions''' of the D model of [[hyperbolic geometry]]. A complex matrix 
:<math>q = \begin{pmatrix} a & b \\ b^* & a^* \end{pmatrix}</math>
with ''aa''* &minus; ''bb''* = 1, which represents the Möbius transformation from the [[complex projective line|projective viewpoint]], can be considered to be on the [[hyperboloid#Relation to the sphere|unit quasi-sphere]] in the [[ring (mathematics)|ring]] of [[coquaternion]]s.

==References==
{{Reflist}}

* [[Lars Ahlfors]] (1967) "Hyperbolic motions", ''Nagoya Mathematical Journal'' 29:163&ndash;5.
* Francis Bonahon (2009) ''Low-dimensional geometry : from euclidean surfaces to hyperbolic knots'', Chapter 2 "The Hyperbolic Plane", pages 11&ndash;39, [[American Mathematical Society]]: ''Student Mathematical Library'', volume 49 ISBN 978-0-8218-4816-6 .
* Victor V. Prasolov & VM Tikhomirov (1997,2001) ''Geometry'', [[American Mathematical Society]]: ''Translations of Mathematical Monographs'', volume 200, ISBN 0-8218-2038-9 .
* A.S. Smogorzhevsky (1982) ''Lobachevskian Geometry'', [[Mir Publishers]], Moscow.

[[Category:Inversive geometry]]
[[Category:Hyperbolic geometry]]