Hyperbolic motion
Introduction of metric in the Poincaré half-plane model
Introduction of metric in the Poincaré half\-plane model
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[thumb\|right\|alt\=semi\-circles as hyperbolic lines\|For some hyperbolic motions in the half\-plane see the [Ultraparallel theorem](/wiki/Ultraparallel_theorem "Ultraparallel theorem").](/wiki/Image:Ultraparallel_theorem.svg "Ultraparallel theorem.svg")

The points of the [Poincaré half\-plane model](/wiki/Poincar%C3%A9_half-plane_model "Poincaré half-plane model") HP are given in [Cartesian coordinates](/wiki/Cartesian_coordinates "Cartesian coordinates") as {(*x*,*y*): *y* \> 0} or in [polar coordinates](/wiki/Polar_coordinates "Polar coordinates") as {(*r* cos *a*, *r* sin *a*): 0 \< *a* \< π, *r* \> 0 }.
The hyperbolic motions will be taken to be a [composition](/wiki/Function_composition "Function 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 ([dilation](/wiki/Homothetic_transformation "Homothetic transformation"))
 *p* → *q* \= ( *r* −1 cos *a*, *r* −1 sin *a* ) ([inversion in unit semicircle](/wiki/Inversive_geometry%23Circle_inversion "Inversive geometry#Circle inversion")).
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 \+ tan2*a* \= sec2*a*, the length of the triangle hypotenuse is sec *a*, where sec denotes the [secant](/wiki/Trigonometric_functions%23Reciprocal_functions "Trigonometric functions#Reciprocal functions") function. Set *r* \= sec *a* and apply the third fundamental hyperbolic motion to obtain *q* \= (*r* cos *a*, *r* sin *a*) where *r* \= sec−1*a* \= cos *a*. Now
\|*q* – (½, 0\)\|2 \= (cos2*a* – ½)2 \+cos2*a* sin2*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](/wiki/Logarithmic_measure "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](/wiki/Perpendicular_bisector "Perpendicular bisector") of the line segment connecting *m* and *n*. If *b* is parallel to the [abscissa](/wiki/Abscissa "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](/wiki/Metric_space "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](/wiki/Non-Euclidean_geometry "Non-Euclidean geometry"); nevertheless, the construction of the line and distance concepts for HP relies heavily on the original geometry of Euclid.


How do these motions relate to inversive geometry?