Editing Hyperbolic motion (section)

== Motions on the hyperbolic plane ==
{{see also|transformation geometry}}
<!--Copied from [[hyperbolic geometry#Isometries of the hyperbolic plane]] -->
Every [[motion (geometry)|motion]] ([[Geometric transformation|transformation]] or [[isometry]]) of the hyperbolic plane to itself can be realized as the composition of at most three [[Reflection (mathematics)|reflections]]. In ''n''-dimensional hyperbolic space, up to ''n''+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)

All the isometries of the hyperbolic plane can be classified into these classes:
* Orientation preserving
** the [[Identity function|identity isometry]] &mdash; nothing moves; zero reflections; zero [[degrees of freedom]].
** [[Point reflection|inversion through a point (half turn)]] &mdash; two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point; two [[degrees of freedom]].
** [[Reflection (mathematics)|rotation]] around a normal point &mdash; two reflections through lines passing through the given point (includes inversion as a special case); points move on circles around the center; three degrees of freedom.
** "rotation" around an [[ideal point]] (horolation)  &mdash; two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom.
** translation along a straight line &mdash; two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom.
* Orientation reversing
** reflection through a line &mdash; one reflection; two degrees of freedom.
** combined reflection through a line and translation along the same line &mdash; the reflection and translation commute; three reflections required; three degrees of freedom.{{citation needed|date=July 2016}}