Editing Hyperbolic geometry (section)

==Properties==

===Relation to Euclidean geometry===
{{comparison_of_geometries.svg}}
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only [[axiom]]atic difference is the [[parallel postulate]].
When the parallel postulate is removed from Euclidean geometry the resulting geometry is [[absolute geometry]].
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of [[Euclid's Elements|Euclid's ''Elements'']], are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's ''Elements'' prove the existence of parallel/non-intersecting lines.

This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the [[angle of parallelism]], hyperbolic geometry has an [[absolute scale]], a relation between distance and angle measurements.

===Lines===
Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended.

Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are [[supplementary angles|supplementary]].

When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines.

These properties are all independent of the [[#Models of the hyperbolic plane|model]] used, even if the lines may look radically different.

====Non-intersecting / parallel lines====
[[File:Hyperbolic.svg|frame|right|Lines through a given point ''P'' and asymptotic to line ''R'']]

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in [[Euclidean geometry]]:

:For any line ''R'' and any point ''P'' which does not lie on ''R'', in the plane containing line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.

This implies that there are through ''P'' an infinite number of coplanar lines that do not intersect ''R''.

These non-intersecting lines are divided into two classes:
* Two of the lines (''x'' and ''y'' in the diagram) are [[limiting parallel]]s (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the [[ideal point]]s at the "ends" of ''R'', asymptotically approaching ''R'', always getting closer to ''R'', but never meeting it.
* All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ''ultraparallel'', ''diverging parallel'' or sometimes ''non-intersecting.''
Some geometers simply use the phrase "''parallel'' lines" to mean "''limiting parallel'' lines", with ''ultraparallel'' lines meaning just ''non-intersecting''.

These [[limiting parallel]]s make an angle ''θ'' with ''PB''; this angle depends only on the [[Gaussian curvature]] of the plane and the distance ''PB'' and is called the [[angle of parallelism]].

For ultraparallel lines, the [[ultraparallel theorem]] states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.

=== Circles and disks ===

In hyperbolic geometry, the circumference of a circle of radius ''r'' is greater than <math> 2 \pi r </math>.

Let <math> R =  \frac{1}{\sqrt{-K}}  </math>, where <math> K </math> is the [[Gaussian curvature]] of the plane.  In hyperbolic geometry, <math>K</math> is negative, so the square root is of a positive number.

Then the circumference of a circle of radius ''r'' is equal to:

:<math>2\pi R \sinh \frac{r}{R} \,.</math>

And the area of the enclosed disk is:

:<math>4\pi R^2 \sinh^2 \frac{r}{2R} = 2\pi R^2 \left(\cosh \frac{r}{R} - 1\right) \,.</math>

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than <math> 2\pi </math>, though it can be made arbitrarily close by selecting a small enough circle.

If the Gaussian curvature of the plane is −1 then the [[geodesic curvature]] of a circle of radius ''r''  is: <math> \frac{1}{\tanh(r)} </math><ref name="auto">{{cite web|url=https://math.stackexchange.com/q/2430495/88985|website=math [[stackexchange]]|title= Curvature of curves on the hyperbolic plane|access-date=24 September 2017}}</ref>

=== Hypercycles and horocycles ===
[[File:Hyperbolic pseudogon example0.png|thumb|Hypercycle and pseudogon in the [[Poincare disk model]] ]]
{{main article|Hypercycle (hyperbolic geometry)|horocycle}}

In hyperbolic geometry, there is no line whose points are all equidistant from another line. Instead, the points that are all the same distance from a given line lie on a curve called a [[hypercycle (hyperbolic geometry)|hypercycle]].

Another special curve is the [[horocycle]], whose [[normal (geometry)|normal]] radii ([[perpendicular]] lines) are all [[limiting parallel]] to each other (all converge asymptotically in one direction to the same [[ideal point]], the centre of the horocycle).

Through every pair of points there are two horocycles. The centres of the horocycles are the [[ideal point]]s of the [[perpendicular bisector]] of the line-segment between them.

Given any three distinct points, they all lie on either a line, hypercycle, [[horocycle]], or circle.

The '''length''' of a line-segment is the shortest length between two points.

The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of the arc horocycle, connecting the same two points.

The lengths of the arcs of both horocycles connecting two points are equal, and are longer than the arclength of any hypercycle connecting the points and shorter than the arc of any circle connecting the two points.

If the Gaussian curvature of the plane is −1, then the [[geodesic curvature]] of a horocycle is 1 and that of a hypercycle is between 0 and 1.<ref name="auto"/>

=== Triangles ===
{{main article|Hyperbolic triangle}}

Unlike Euclidean triangles, where the angles always add up to π [[radian]]s (180°, a [[straight angle]]), in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians (180°). The difference is called the [[Angular defect|defect]]. Generally, the defect of a convex hyperbolic polygon with <math>n</math> sides is its angle sum subtracted from <math>(n - 2) \cdot 180^\circ</math>. 

The area of a hyperbolic triangle is given by its defect in radians multiplied by ''R''{{sup|2}}, which is also true for all convex hyperbolic polygons.<ref>{{Cite book |last=Thorgeirsson |first=Sverrir |url=https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503 |title=Hyperbolic geometry: history, models, and axioms |date=2014}}</ref> Therefore all hyperbolic triangles have an area less than or equal to ''R''{{sup|2}}π. The area of a hyperbolic [[ideal triangle]] in which all three angles are 0° is equal to this maximum.

As in [[Euclidean geometry]], each hyperbolic triangle has an [[incircle]]. In hyperbolic space, if all three of its vertices lie on a [[horocycle]] or [[hypercycle (hyperbolic geometry)|hypercycle]], then the triangle has no [[circumscribed circle]].

As in [[spherical geometry|spherical]] and [[elliptical geometry]], in hyperbolic geometry if two triangles are similar, they must be congruent.

===Regular apeirogon and pseudogon===
[[File:Hyperbolic apeirogon example.png|thumb|An [[apeirogon]] and circumscribed [[horocycle]] in the [[Poincaré disk model]].]]
{{main article|Apeirogon#Hyperbolic geometry}}

Special polygons in hyperbolic geometry are the regular [[apeirogon]] and '''pseudogon''' [[uniform polygon]]s with an infinite number of sides.

In [[Euclidean geometry]], the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180° and the apeirogon approaches a straight line.

However, in hyperbolic geometry, a regular apeirogon or pseudogon has sides of any length (i.e., it remains a polygon with noticeable sides).

The side and angle [[bisection|bisectors]] will, depending on the side length and the angle between the sides, be limiting or diverging parallel. If the bisectors are limiting parallel then it is an apeirogon and can be inscribed and circumscribed by concentric [[horocycle]]s.

If the bisectors are diverging parallel then it is a  pseudogon and can be inscribed and circumscribed by [[hypercycle (geometry)|hypercycles]] (since all its vertices are the same distance from a line, the axis, and the midpoints of its sides are also equidistant from that same axis).

=== Tessellations ===
{{main article|Uniform tilings in hyperbolic plane}}
{{see also|Regular hyperbolic tiling}}

[[File:Rhombitriheptagonal tiling.svg|thumb|[[Rhombitriheptagonal tiling]] of the hyperbolic plane, seen in the [[Poincaré disk model]] ]]
Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with [[regular polygon]]s as [[Face (geometry)|faces]].

There are an infinite number of uniform tilings based on the [[Schwarz triangles]] (''p'' ''q'' ''r'') where 1/''p'' + 1/''q'' + 1/''r'' < 1, where ''p'',&thinsp;''q'',&thinsp;''r'' are each orders of reflection symmetry at three points of the [[fundamental domain triangle]], the symmetry group is a hyperbolic [[triangle group]]. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.<ref>{{Cite journal | doi=10.1140/epjb/e2003-00032-8|title = Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings| journal=The European Physical Journal B | volume=31| issue=2| pages=273–284|year = 2003|last1 = Hyde|first1 = S.T.| last2=Ramsden| first2=S.|bibcode = 2003EPJB...31..273H| citeseerx=10.1.1.720.5527|s2cid = 41146796}}</ref>