Editing Hyperbolic geometry (section)

== Standardized Gaussian curvature ==

Though hyperbolic geometry applies for any surface with a constant negative [[Gaussian curvature]], it is usual to assume a scale in which the curvature ''K'' is −1.

This results in some formulas becoming simpler. Some examples are:
* The area of a triangle is equal to its angle defect in [[radian]]s.
* The area of a horocyclic sector is equal to the length of its horocyclic arc.
* An arc of a [[horocycle]] so that a line that is tangent at one endpoint is [[limiting parallel]] to the radius through the other endpoint has a length of 1.<ref name="Sommerville2005">{{cite book|last1=Sommerville|first1=D.M.Y.|title=The elements of non-Euclidean geometry|date=2005|publisher=Dover Publications|location=Mineola, N.Y.|isbn=0-486-44222-5|page=58|edition=Unabr. and unaltered republ.}}</ref>
* The ratio of the arc lengths between two radii of two concentric [[horocycle]]s where the horocycles are a distance 1 apart is [[e (mathematical constant)|''e'']]&thinsp;:{{Hair space}}1.<ref name="Sommerville2005"/>

===Cartesian-like coordinate systems===

{{Main article|Coordinate systems for the hyperbolic plane}}
<!-- Still in draft, feel free to add, but it is not ready for public yet ::

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.

=== Distance ===
{{main|Coordinate systems for the hyperbolic plane#Polar coordinate system}}

A Cartesian-like{{citation needed|date=September 2022}} coordinate system (''x, y'') on the oriented hyperbolic plane is constructed as follows. Choose a line in the hyperbolic plane together with an orientation and an origin ''o'' on this line. Then:
*the ''x''-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin; 
*the ''y''-coordinate is the signed [[distance from a point to a line|distance]] from the point to the line, with the sign according to whether the point is on the positive or negative side of the oriented line. 

The distance between two points represented by (''x_i, y_i''), ''i=1,2'' in this coordinate system is{{citation needed|date=December 2018}}
<math display=block>\operatorname{dist} (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname{arcosh} \left( \cosh y_1 \cosh (x_2 - x_1) \cosh y_2 - \sinh y_1 \sinh y_2 \right) \,.</math>

This formula can be derived from the formulas about [[hyperbolic triangle]]s.

The corresponding metric tensor field is: <math> (\mathrm{d} s)^2 = \cosh^2 y \, (\mathrm{d} x)^2 + (\mathrm{d} y)^2 </math>.

In this coordinate system, straight lines take one of these forms ((''x'', ''y'') is a point on the line; ''x''<sub>0</sub>, ''y''<sub>0</sub>, ''A'', and ''&alpha;'' are parameters):

ultraparallel to the ''x''-axis
:<math> \tanh (y) = \tanh (y_0) \cosh (x - x_0) </math>
asymptotically parallel on the negative side
:<math> \tanh (y) = A \exp (x) </math>
asymptotically parallel on the positive side
:<math> \tanh (y) = A \exp (- x) </math>
intersecting perpendicularly
:<math> x = x_0 </math>
intersecting at an angle ''&alpha;''
:<math> \tanh (y) = \tan (\alpha) \sinh (x - x_0) </math>
Generally, these equations will only hold in a bounded domain (of ''x'' values). At the edge of that domain, the value of ''y'' blows up to ±infinity.