Editing Hyperbolic geometry (section)

===Cartesian-like coordinate systems===

{{Main article|Coordinate systems for the hyperbolic plane}}
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In Euclidean geometry the most widely used [[coordinate system]] is the [[Cartesian coordinate system]]. this coordinate system has many advantages:
1. RxR
2. distance to axis
3. axial perpendiculars
4. path first x then y gives same point as path first y then x
5. easy equations (implicit) of lines
6. maybe more
(maybe reshuffle)

In hyperbolic geometry it is not that simple:
* In hyperbolic geometry the sum of the angles of any quadrilateral is [[lambert quadrilateral|always less than 360 degrees]], so condition 2 and 3 are incompatible
* etc.

-- Is there a Hyperbolic coordinate system that does give easy equations (implicit) of lines?

-- something about polar coordinates

end of draft -->

Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a [[quadrilateral]] is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes it to rotate when it returns to the origin; etc.

There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the ''x''-axis) and after that many choices exist.

The Lobachevsky coordinates ''x'' and ''y'' are found by dropping a perpendicular onto the ''x''-axis. ''x'' will be the label of the foot of the perpendicular. ''y'' will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other).

Another coordinate system measures the distance from the point to the [[horocycle]] through the origin centered around <math> (0, + \infty )</math> and the length along this horocycle.<ref>{{cite book|last1=Ramsay|first1=Arlan|last2=Richtmyer|first2=Robert D.|title=Introduction to hyperbolic geometry|url=https://archive.org/details/introductiontohy0000rams|url-access=registration|date=1995|publisher=Springer-Verlag|location=New York|isbn=0387943390|pages=[https://archive.org/details/introductiontohy0000rams/page/97 97–103]}}</ref>

Other coordinate systems use the Klein model or the Poincaré disk model described below, and take the Euclidean coordinates as hyperbolic.