Editing Hyperbolic motion (section)

==Introduction of metric in the Poincaré half-plane model==
[[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 [[Poincaré 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]]{{Broken anchor|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=Trigonometric functions#Reciprocal functions|reason= The anchor (Reciprocal functions) [[Special:Diff/779343428|has been deleted]].}} 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 [[logarithmic measure]]:
:''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.