Editing Hyperbolic geometry

{{More citations needed|date=June 2023}}
{{Short description|Type of non-Euclidean geometry}}
{{Other uses|Hyperbolic (disambiguation)}}
[[File:Hyperbolic.svg|frame|right|Lines through a given point ''P'' and asymptotic to line ''R'']]
{{General geometry |branches}}
[[File:Hyperbolic triangle.svg|thumb|250px|right|A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), along with two diverging ultra-parallel lines]]

In [[mathematics]], '''hyperbolic geometry''' (also called '''Lobachevskian geometry''' or '''[[János Bolyai|Bolyai]]–[[Nikolai Lobachevsky|Lobachevskian]] geometry''') is a [[non-Euclidean geometry]]. The [[parallel postulate]] of [[Euclidean geometry]] is replaced with:
:For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.
(Compare the above with [[Playfair's axiom]], the modern version of [[Euclid]]'s [[parallel postulate]].)

The '''hyperbolic plane''' is a [[plane (mathematics)|plane]] where every point is a [[saddle point]]. 
Hyperbolic plane [[geometry]] is also the geometry of [[pseudosphere|pseudospherical surfaces]], surfaces with a constant negative [[Gaussian curvature]]. [[Saddle surface]]s have negative Gaussian curvature in at least some regions, where they [[local property|locally]] resemble the hyperbolic plane.

The [[hyperboloid model]] of hyperbolic geometry provides a representation of [[event (relativity)|event]]s one temporal unit into the future in [[Minkowski space]], the basis of [[special relativity]]. Each of these events corresponds to a [[rapidity]] in some direction.

When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; [[Felix Klein]] finally gave the subject the name '''hyperbolic geometry''' to include it in the now rarely used sequence [[elliptic geometry]] ([[spherical geometry]]), parabolic geometry ([[Euclidean geometry]]), and hyperbolic geometry.
In the [[Post-Soviet states|former Soviet Union]], it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer [[Nikolai Lobachevsky]].

==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>

== 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.

== History ==
{{see also|Non-euclidean Geometry}}
Since the publication of [[Euclid's Elements|Euclid's ''Elements'']] circa 300BC, many [[geometers]] tried to prove the [[parallel postulate]]. Some tried to prove it by [[Proof by contradiction|assuming its negation and trying to derive a contradiction]]. Foremost among these were [[Proclus]], [[Ibn al-Haytham]] (Alhacen), [[Omar Khayyám]],<ref>See for instance, {{cite web|url=http://www.resonancepub.com/omarkhayyam.htm|title=Omar Khayyam 1048–1131|access-date=2008-01-05|archive-date=2007-09-28|archive-url=https://web.archive.org/web/20070928084550/http://www.resonancepub.com/omarkhayyam.htm|url-status=dead}}</ref> [[Nasīr al-Dīn al-Tūsī]], [[Witelo]], [[Gersonides]], [[Abner of Burgos|Alfonso]], and later [[Giovanni Gerolamo Saccheri]], [[John Wallis]], [[Johann Heinrich Lambert]], and [[Adrien-Marie Legendre|Legendre]].<ref>{{cite web|url=http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html|title=Non-Euclidean Geometry Seminar|website=Math.columbia.edu|access-date=21 January 2018}}</ref>
Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tūsī on [[quadrilateral]]s, including the [[Ibn al-Haytham–Lambert quadrilateral]] and [[Khayyam–Saccheri quadrilateral]], were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447–494 [470], [[Routledge]], London and New York: {{blockquote|"Three scientists, Ibn al-Haytham, Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's ''[[Book of Optics]]'' (''Kitab al-Manazir'') – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar [[Gersonides|Levi ben Gerson]], who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that ''Pseudo-Tusi's Exposition of Euclid'' had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."}}</ref>

In the 18th century, [[Johann Heinrich Lambert]] introduced the [[hyperbolic functions]]<ref>{{citation|title=Foundations and Fundamental Concepts of Mathematics|first=Howard|last=Eves|publisher=Courier Dover Publications|year=2012|isbn=9780486132204|page=59|url=https://books.google.com/books?id=J9QcmFHj8EwC&pg=PA59|quote=We also owe to Lambert the first systematic development of the theory of hyperbolic functions and, indeed, our present notation for these functions.}}</ref> and computed the area of a [[hyperbolic triangle]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=Graduate Texts in Mathematics|first=John|last=Ratcliffe|publisher=Springer|year=2006|isbn=9780387331973|page=99|url=https://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99|quote=That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph ''Theorie der Parallellinien'', which was published posthumously in 1786.}}</ref>

===19th-century developments===

In the 19th century, hyperbolic geometry was explored extensively by [[Nikolai Lobachevsky]], [[János Bolyai]], [[Carl Friedrich Gauss]] and [[Franz Taurinus]]. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.<ref>{{Cite book|author=Bonola, R.|title=Non-Euclidean geometry: A critical and historical study of its development|year=1912|location=Chicago|publisher=Open Court|url=https://archive.org/details/noneuclideangeom00bono}}</ref><ref>{{cite book|author-link1=Marvin Greenberg|last1=Greenberg|first1=Marvin Jay|title=Euclidean and non-Euclidean geometries: development and history|url=https://archive.org/details/euclideannoneucl00gree_304|url-access=limited|date=2003|publisher=Freeman|location=New York|isbn=0716724464|page=[https://archive.org/details/euclideannoneucl00gree_304/page/n194 177]|edition=3rd|quote=Out of nothing I have created a strange new universe. JÁNOS BOLYAI}}</ref>

Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "[[non-Euclidean geometry]]"<ref>Felix Klein, ''Elementary Mathematics from an Advanced Standpoint: Geometry'', Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176</ref> causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

In 1868, [[Eugenio Beltrami]] provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent [[if and only if]] Euclidean geometry was.

The term "hyperbolic geometry" was introduced by [[Felix Klein]] in 1871.<ref>F. Klein. "Über die sogenannte Nicht-Euklidische Geometrie". ''Math. Ann.'' 4, 573–625 (also in ''Gesammelte Mathematische Abhandlungen'' 1, 244–350).</ref> Klein followed an initiative of [[Arthur Cayley]] to use the transformations of [[projective geometry]] to produce [[isometries]]. The idea used a [[conic section]] or [[quadric]] to define a region, and used [[cross ratio]] to define a [[metric (mathematics)|metric]]. The projective transformations that leave the conic section or quadric [[invariant (mathematics)#Invariant set|stable]] are the isometries. "Klein showed that if the [[Cayley absolute]] is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."<ref>Rosenfeld, B.A. (1988) ''A History of Non-Euclidean Geometry'', page 236, Springer-Verlag {{ISBN|0-387-96458-4}}</ref>

=== Philosophical consequences ===

The discovery of hyperbolic geometry had important [[philosophical]] consequences. Before its discovery many philosophers (such as [[Hobbes]] and [[Spinoza]]) viewed philosophical rigor in terms of the "geometrical method", referring to the method of reasoning used in [[Euclid's Elements|Euclid's ''Elements'']].

[[Kant]] in [[Critique of Pure Reason#Space and time|''Critique of Pure Reason'']] concluded that space (in [[Euclidean geometry]]) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.<ref>{{cite book |last=Lucas |first=John Randolph |author-link=John Lucas (philosopher) |title= Space, Time and Causality |year=1984 |page=149 |publisher=Clarendon Press |isbn=0-19-875057-9}}</ref>

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the [[Boeotia]]ns" (stereotyped as dullards by the ancient Athenians<ref>{{cite journal
 | last = Wood | first = Donald
 | date = April 1959
 | doi = 10.1177/030639685900100207
 | issue = 2
 | journal = Race
 | pages = 65–71
 | title = Some Greek stereotypes of other peoples
 | volume = 1}}</ref>), which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians").<ref>{{cite book|last1=Torretti|first1=Roberto|title=Philosophy of Geometry from Riemann to Poincare|date=1978|publisher=Reidel|location=Dordrecht Holland|page=255}}</ref>
The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in [[mathematical rigour]], [[analytical philosophy]] and [[logic]]. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

=== Geometry of the universe (spatial dimensions only) ===
{{main article|Philosophy of space and time}}
{{see also|Shape of the universe#Curvature of the universe}}

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?

Lobachevsky had already tried to measure the curvature of the universe by measuring the [[parallax]] of [[Sirius]] and treating Sirius as the ideal point of an [[angle of parallelism]]. He realized that his measurements were [[margin of error|not precise enough]] to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the [[#Standardized Gaussian curvature|absolute length]] is at least one million times the diameter of [[Earth's orbit]] ({{val|2000000|ul=AU}}, 10 [[parsec]]).<ref>{{cite book|last1=Bonola|first1=Roberto|title=Non-Euclidean geometry : a critical and historical study of its developments|url=https://archive.org/details/noneuclideangeom0000bono|url-access=registration|date=1955|publisher=Dover|location=New York, NY|isbn=0486600270|page=[https://archive.org/details/noneuclideangeom0000bono/page/95 95]|edition=Unabridged and unaltered republ. of the 1. English translation 1912.}}</ref>
Some argue that his measurements were methodologically flawed.<ref>{{cite book|last1=Richtmyer|first1=Arlan Ramsay, Robert D.|title=Introduction to hyperbolic geometry|date=1995|publisher=Springer-Verlag|location=New York|isbn=0387943390|pages=[https://archive.org/details/introductiontohy0000rams/page/118 118–120]|url=https://archive.org/details/introductiontohy0000rams/page/118}}</ref>

[[Henri Poincaré]], with his [[sphere-world]] [[thought experiment]], came to the conclusion that everyday experience does not necessarily rule out other geometries.

The [[geometrization conjecture]] gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.<ref>{{cite web|url=http://www.learner.org/courses/mathilluminated/units/8/textbook/08.php|title=Mathematics Illuminated - Unit 8 - 8.8 Geometrization Conjecture|website=Learner.org|access-date=21 January 2018}}</ref>

=== Geometry of the universe (special relativity) ===
[[Special relativity]] places space and time on equal footing, so that one considers the geometry of a unified [[spacetime]] instead of considering space and time separately.<ref>{{cite book|title=Classical Theory of Fields|author1=L. D. Landau |author2=E. M. Lifshitz |series=[[Course of Theoretical Physics]]|edition=4th|volume=2|publisher=Butterworth Heinemann|isbn=978-0-7506-2768-9|year=1973|pages=1–4}}</ref><ref>{{cite book|title=[[Feynman Lectures on Physics]]|author1=R. P. Feynman |author2=R. B. Leighton |author3=M. Sands |volume=1|isbn=0-201-02116-1|year=1963|publisher=Addison Wesley|page=(17-1)–(17-3)}}</ref> [[Minkowski space|Minkowski geometry]] replaces [[Galilean geometry]] (which is the 3-dimensional Euclidean space with time of [[Galilean relativity]]).<ref>{{cite book|title=Dynamics and Relativity|url=https://archive.org/details/dynamicsrelativi00fors|url-access=limited|author1=J. R. Forshaw |author-link1=Jeff Forshaw|author2=A. G. Smith |series=Manchester physics series|isbn=978-0-470-01460-8|year=2008|publisher=Wiley|pages=[https://archive.org/details/dynamicsrelativi00fors/page/n260 246]–248}}</ref>

In relativity, rather than Euclidean, elliptic and hyperbolic geometry, the appropriate geometries to consider are [[Minkowski space]], [[de Sitter space]] and [[anti-de Sitter space]],<ref>{{cite book |author1=Misner |author2=Thorne |author3=Wheeler |date=1973 |title=Gravitation |url=https://archive.org/details/gravitation00cwmi |url-access=limited |pages=[https://archive.org/details/gravitation00cwmi/page/n53 21], 758 }}</ref><ref>{{cite book |author1=John K. Beem|author2=Paul Ehrlich |author3=Kevin Easley |date=1996 |title=Global Lorentzian Geometry |edition=Second}}</ref> corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity through [[rapidity]], which stands in for [[velocity]], and is expressed by a [[hyperbolic angle]]. The study of this velocity geometry has been called [[Non-Euclidean geometry#Kinematic geometries|kinematic geometry]]. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).<ref>{{cite book|title=Classical Theory of Fields|author1=L. D. Landau |author2=E. M. Lifshitz |series=[[Course of Theoretical Physics]]|edition=4th|volume=2|publisher=Butterworth Heinemann|isbn=978-0-7506-2768-9|year=1973|page=38}}</ref>

== Physical realizations of the hyperbolic plane ==
[[File:Crochet hyperbolic kelp.jpg|thumb|right|A collection of crocheted hyperbolic planes, in imitation of a coral reef, by [[Institute For Figuring]]]]
[[File:Hyperbolicsoccerball.jpg|thumb|right|The "hyperbolic soccerball", a paper model which approximates (part of) the hyperbolic plane as a [[truncated icosahedron]] approximates the sphere.]]

There exist various [[pseudosphere]]s in Euclidean space that have a finite area of constant negative Gaussian curvature.

By [[Hilbert's theorem (differential geometry)|Hilbert's theorem]], one cannot isometrically [[Immersion (mathematics)|immerse]] a complete hyperbolic plane (a complete regular surface of constant negative [[Gaussian curvature]]) in a 3-D Euclidean space.

Other useful [[#Models of the hyperbolic plane|models]] of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the [[pseudosphere]] is due to [[William Thurston]].

The art of [[crochet]] has been [[Mathematics and fiber arts#Knitting and crochet|used]] to demonstrate hyperbolic planes, the first such demonstration having been made by [[Daina Taimiņa]].<ref name="hyperbolicspace">{{cite web | date = December 21, 2006 | url = http://theiff.org/oexhibits/oe1e.html | title = Hyperbolic Space | work = The Institute for Figuring | access-date = January 15, 2007}}</ref>

In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "[[hyperbolic soccerball]]" (more precisely, a [[truncated order-7 triangular tiling]]).<ref>{{cite web|url=http://www.theiff.org/images/IFF_HypSoccerBall.pdf|title=How to Build your own Hyperbolic Soccer Ball|website=Theiff.org|access-date=21 January 2018}}</ref><ref>{{cite web|url=http://www.math.tamu.edu/~sottile/research/stories/hyperbolic_football/index.html|title=Hyperbolic Football|website=Math.tamu.edu|access-date=21 January 2018}}</ref>

Instructions on how to make a hyperbolic quilt, designed by [[Helaman Ferguson]],<ref>{{cite web|url=http://www.helasculpt.com/gallery/hyperbolicquilt/ |title=Helaman Ferguson, Hyperbolic Quilt |url-status=dead |archive-url=https://web.archive.org/web/20110711162245/http://www.helasculpt.com/gallery/hyperbolicquilt/ |archive-date=2011-07-11 }}</ref> have been made available by [[Jeffrey Weeks (mathematician)|Jeff Weeks]].<ref>{{cite web | url = http://www.geometrygames.org/HyperbolicBlanket/index.html | title = How to sew a Hyperbolic Blanket|website=Geometrygames.org|access-date=21 January 2018}}</ref>

== Models of the hyperbolic plane ==
Various [[pseudosphere]]s – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean metric, and so can be made into tangible models. Of these, the [[Pseudosphere#Tractroid|tractoid]] (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a [[cone]] or [[cylinder]] as a model of the Euclidean plane. However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly exploring hyperbolic geometry.

There are four [[Mathematical model|model]]s commonly used for hyperbolic geometry: the [[Klein model]], the [[Poincaré disk model]], the [[Poincaré half-plane model]], and the Lorentz or [[hyperboloid model]]. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by [[Eugenio Beltrami|Beltrami]], not by [[Henri Poincaré|Poincaré]] or [[Felix Klein|Klein]]. All these models are extendable to more dimensions.

===The Beltrami–Klein model===
{{main article|Beltrami–Klein model}}

The [[Beltrami–Klein model]], also known as the projective disk model, Klein disk model and [[Klein model]], is named after [[Eugenio Beltrami]] and [[Felix Klein]].

For the two dimensions this model uses the interior of the [[unit circle]] for the complete hyperbolic [[plane (mathematics)|plane]], and the [[chord (geometry)|chords]] of this circle are the hyperbolic lines.

For higher dimensions this model uses the interior of the [[unit ball]], and the [[chord (geometry)|chords]] of this ''n''-ball are the hyperbolic lines.
* This model has the advantage that lines are straight, but the disadvantage that [[angle]]s are distorted (the mapping is not [[Conformal map|conformal]]), and also circles are not represented as circles.
* The distance in this model is half the logarithm of the [[cross-ratio]], which was introduced by [[Arthur Cayley]] in [[projective geometry]].

=== The Poincaré disk model ===
[[File:Hyperbolic tiling omnitruncated 3-7.png|thumb|Poincaré disk model with [[truncated triheptagonal tiling]]]]
{{main article|Poincaré disk model}}

The [[Poincaré disk model]], also known as the conformal disk model, also employs the interior of the [[unit circle]], but lines are represented by arcs of circles that are [[orthogonal]] to the boundary circle, plus diameters of the boundary circle.
* This model preserves angles, and is thereby [[conformal map|conformal]]. All isometries within this model are therefore [[Möbius transformation]]s.
* Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle.
* [[Horocycle]]s are circles within the disk which are [[tangent]] to the boundary circle, minus the point of contact.
* [[Hypercycle (hyperbolic geometry)|Hypercycle]]s are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.

=== The Poincaré half-plane model ===

{{main article|Poincaré half-plane model}}

The [[Poincaré half-plane model]] takes one-half of the Euclidean plane, bounded by a line ''B'' of the plane, to be a model of the hyperbolic plane.  The line ''B'' is not included in the model.

The Euclidean plane may be taken to be a plane with the [[Cartesian coordinate system]] and the [[x-axis]] is taken as line ''B'' and the half plane is the upper half (''y'' > 0 )  of this plane.
* Hyperbolic lines are then either half-circles orthogonal to ''B'' or rays perpendicular to ''B''.
* The length of an interval on a ray is given by [[logarithmic measure]] so it is invariant under a [[homothetic transformation]] <math>(x, y) \mapsto (\lambda x, \lambda y),\quad \lambda > 0 .</math>
* Like the Poincaré disk model, this model preserves angles, and is thus [[conformal map|conformal]]. All isometries within this model are therefore [[Möbius transformation]]s of the plane.
* The half-plane model is the limit of the Poincaré disk model whose boundary is tangent to ''B'' at the same point while the radius of the disk model goes to infinity.

=== The hyperboloid model ===
{{main article|hyperboloid model}}

The [[hyperboloid model]] or Lorentz model employs a 2-dimensional [[hyperboloid]] of revolution (of two sheets, but using one) embedded in 3-dimensional [[Minkowski space]]. This model is generally credited to Poincaré, but Reynolds<ref>{{aut|Reynolds, William F.}}, (1993) ''Hyperbolic Geometry on a Hyperboloid'', [[American Mathematical Monthly]] 100:442–455.</ref> says that [[Wilhelm Killing]] used this model in 1885
* This model has direct application to [[special relativity]], as Minkowski 3-space is a model for [[spacetime]], suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that various [[Inertial frame of reference|inertially]] moving observers, starting from a common event, will reach in a fixed [[proper time]].
* The hyperbolic distance between two points on the hyperboloid can then be identified with the relative [[rapidity]] between the two corresponding observers.
* The model generalizes directly to an additional dimension: a hyperbolic 3-space  three-dimensional hyperbolic geometry relates to Minkowski 4-space.

=== The hemisphere model ===

The [[Sphere#Hemisphere|hemisphere]] model is not often used as model by itself, but it functions as a useful tool for visualizing transformations between the other models.

The hemisphere model uses the upper half of the [[unit sphere]]:
<math> x^2 + y^2 +z^2 = 1 , z > 0. </math>

The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere.

The hemisphere model is part of a [[Riemann sphere]], and different projections give different models of the hyperbolic plane:
* [[Stereographic projection]] from  <math> (0,0, -1) </math> onto the plane <math> z = 0 </math>  projects corresponding points on the [[Poincaré disk model]]
* [[Stereographic projection]] from  <math> (0,0, -1) </math> onto the surface <math> x^2 + y^2 - z^2  = -1 , z > 0  </math>  projects corresponding points on the [[hyperboloid model]]
* [[Stereographic projection]] from <math> (-1,0,0) </math> onto the plane <math> x=1 </math> projects corresponding points on the [[Poincaré half-plane model]]
* [[Orthographic projection]] onto a plane <math> z = C </math> projects corresponding points on the [[Beltrami–Klein model]].
* [[Central projection]] from the centre of the sphere onto the plane <math> z = 1 </math> projects corresponding points on the [[Gans Model]]

===Connection between the models===
[[File:Relation5models.png|thumb|upright=1.2|Poincaré disk, hemispherical and hyperboloid models are related by [[stereographic projection]] from −1. [[Beltrami–Klein model]] is [[orthographic projection]] from hemispherical model. [[Poincaré half-plane model]] here projected from the hemispherical model by rays from left end of Poincaré disk model.]]

All models essentially describe the same structure. The difference between them is that they represent different [[Atlas (topology)|coordinate charts]] laid down on the same [[metric space]], namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negative [[Gaussian curvature]], which is indifferent to the coordinate chart used. The [[geodesic]]s are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry is generally introduced in terms of the geodesics and their intersections on the hyperbolic plane.<ref>Arlan Ramsay, Robert D. Richtmyer, ''Introduction to Hyperbolic Geometry'', Springer; 1 edition (December 16, 1995)</ref>

Once we choose a coordinate chart (one of the "models"), we can always [[Immersion (mathematics)|embed]] it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.

Since the four models describe the same metric space, each can be transformed into the other.

See, for example:
* [[Beltrami–Klein model#Relation to the hyperboloid model|the Beltrami–Klein model's relation to the hyperboloid model]],
* [[Beltrami–Klein model#Relation to the Poincaré disk model|the Beltrami–Klein model's relation to the Poincaré disk model]],
* and [[Poincaré disk model#Relation to the hyperboloid model|the Poincaré disk model's relation to the hyperboloid model]].



=== Other models of hyperbolic geometry ===

==== The Gans model ====
In 1966 David Gans proposed a [[flattened hyperboloid model]] in the journal ''[[American Mathematical Monthly]]''.<ref>{{cite journal|author=Gans David |title=A New Model of the Hyperbolic Plane |journal=American Mathematical Monthly |volume= 73 |issue= 3 |date=March 1966 |doi=10.2307/2315350 |pages=291–295|jstor=2315350 }}</ref> It is an [[orthographic projection]] of the hyperboloid model onto the xy-plane.
This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.
* Unlike the Klein or the Poincaré models, this model utilizes the entire [[Euclidean plane]].
* The lines in this model are represented as branches of a [[hyperbola]].<ref>{{cite web|url=http://www.d.umn.edu/cs/thesis/kedar_bhumkar_ms.pdf|title=Computer Science Department|last=vcoit|date=8 May 2015}}</ref>

==== The conformal square model ====
[[File:Omnitruncated tiling on conformal square.png|thumb|Conformal square model with [[truncated triheptagonal tiling]]]]

The conformal square model of the hyperbolic plane arises from using [[Schwarz–Christoffel mapping]] to convert the [[Poincaré disk model|Poincaré disk]] into a square.<ref>{{cite conference|url=http://archive.bridgesmathart.org/2016/bridges2016-179.pdf|title=The Conformal Hyperbolic Square and Its Ilk |last=Fong| first = C.|year=2016 |conference=Bridges Finland Conference Proceedings}}</ref> This model has finite extent, like the Poincaré disk. However, all of the points are inside a square. This model is conformal, which makes it suitable for artistic applications.

==== The band model ====
{{main article|Band model}}
The band model employs a portion of the Euclidean plane between two parallel lines.<ref>{{cite book|url=http://matrixeditions.com/TeichmullerVol1.html|title=Teichmüller theory and applications to geometry, topology, and dynamics|year=2006–2016 |publisher=Matrix Editions|others=Hubbard, John Hamal |isbn=9780971576629|location=Ithaca, NY|oclc=57965863|chapter=2|chapter-url=http://matrixeditions.com/TVol1.Chap2.pdf|page=25}}</ref> Distance is preserved along one line through the middle of the band. Assuming the band is given by <math>\{z \in \mathbb C:|\operatorname {Im} z| < \pi / 2\}</math>, the metric is given by <math>|dz| \sec (\operatorname{Im} z)</math>.

== Isometries of the hyperbolic plane ==
{{see also|Hyperbolic motion|transformation geometry}}

Every [[isometry]] ([[Geometric transformation|transformation]] or [[motion (geometry)|motion]]) 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 isometries of the hyperbolic plane can be classified into these classes:
* Orientation preserving
** the [[Identity function|identity isometry]] – nothing moves; zero reflections; zero [[degrees of freedom]].
** [[Point reflection|inversion through a point (half turn)]] – 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 – 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) – 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 – 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 – one reflection; two degrees of freedom.
** combined reflection through a line and translation along the same line – the reflection and translation commute; three reflections required; three degrees of freedom.{{citation needed|date=July 2016}}
<!-- are all hyperbolic reflections in 3 lines glide reflections? -->
<!-- also wondering should this section not be moved to properties, and [[hyperbolic motions]] needs improving -->

== In art ==

[[M. C. Escher]]'s famous prints ''[[Circle Limit III]]'' and ''Circle Limit IV''
illustrate the conformal disc model ([[Poincaré disk model]]) quite well. The white lines in ''III'' are not quite geodesics (they are [[hypercycle (hyperbolic geometry)|hypercycles]]), but are close to them. It is also possible to see the negative [[curvature]] of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

For example, in ''Circle Limit III'' every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is [[exponential growth]]. In ''Circle Limit III'', for example, one can see that the number of fishes within a distance of ''n'' from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius ''n'' must rise exponentially in ''n''.

The art of [[crochet]] has [[Mathematics and fiber arts#Knitting and crochet|been used]] to demonstrate hyperbolic planes (pictured above) with the first being made by [[Daina Taimiņa]],<ref name="hyperbolicspace" /> whose book ''[[Crocheting Adventures with Hyperbolic Planes]]'' won the 2009 [[Bookseller/Diagram Prize for Oddest Title of the Year|''Bookseller''/Diagram Prize for Oddest Title of the Year]].<ref>{{Cite journal
 | last = Bloxham | first = Andy
 | date = March 26, 2010
 | journal = [[The Daily Telegraph|The Telegraph]]
 | title = Crocheting Adventures with Hyperbolic Planes wins oddest book title award
 | url = https://www.telegraph.co.uk/culture/books/bookprizes/7520047/Crocheting-Adventures-with-Hyperbolic-Planes-wins-oddest-book-title-award.html
 }}</ref>

''[[HyperRogue]]'' is a [[roguelike]] game set on various tilings of the hyperbolic plane.

==Higher dimensions==
{{main article|Hyperbolic space}}

Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions.

==Homogeneous structure==
[[Hyperbolic space]] of dimension ''n'' is a special case of a Riemannian [[symmetric space]] of noncompact type, as it is [[isomorphic]] to the quotient
:: <math>\mathrm{O}(1,n)/(\mathrm{O}(1) \times \mathrm{O}(n)).</math>
The [[orthogonal group]] {{nowrap|O(1, ''n'')}} [[Group action (mathematics)|acts]] by norm-preserving transformations on [[Minkowski space]] '''R'''<sup>1,''n''</sup>, and it acts [[Group action (mathematics)#Types of actions|transitively]] on the two-sheet hyperboloid of norm 1 vectors.  Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic ''n''-space.  The [[Stabilizer subgroup|stabilizer]] of any particular line is isomorphic to the [[Direct product of groups|product]] of the orthogonal groups O(''n'') and O(1), where O(''n'') acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin.  Many of the elementary concepts in hyperbolic geometry can be described in [[linear algebra]]ic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.

In small dimensions, there are exceptional isomorphisms of [[Lie group]]s that yield additional ways to consider symmetries of hyperbolic spaces.  For example, in dimension 2, the isomorphisms {{nowrap|SO<sup>+</sup>(1, 2) ≅ PSL(2, '''R''') ≅ PSU(1, 1)}} allow one to interpret the upper half plane model as the quotient {{nowrap|SL(2, '''R''')/SO(2)}} and the Poincaré disc model as the quotient {{nowrap|SU(1, 1)/U(1)}}.  In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in {{nowrap|PGL(2, '''C''')}} of the respective subspaces of the Riemann sphere.  The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group.  In dimension 3, the fractional linear action of {{nowrap|PGL(2, '''C''')}} on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism {{nowrap|O<sup>+</sup>(1, 3) ≅ PGL(2, '''C''')}}. This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices.  For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by [[unipotent]] [[Triangular matrix|upper triangular]] matrices.

== See also ==
{{div col|colwidth=30em}}
* [[Band model]]
* [[Constructions in hyperbolic geometry]]
* [[Hjelmslev transformation]]
* [[Hyperbolic 3-manifold]]
* [[Hyperbolic manifold]]
* [[Hyperbolic set]]
* [[Hyperbolic tree]]
* [[Kleinian group]]
* [[Lambert quadrilateral]]
* [[Open universe]]
* [[Poincaré metric]]
* [[Saccheri quadrilateral]]
* [[Systolic geometry]]
* [[Uniform tilings in hyperbolic plane]]
* [[δ-hyperbolic space]]
{{div col end}}

==Notes==
{{Reflist}}

==Bibliography==
* A'Campo, Norbert and Papadopoulos, Athanase, (2012) ''Notes on hyperbolic geometry'', in: Strasbourg Master class on Geometry, pp.&nbsp;1–182,  IRMA Lectures in Mathematics and Theoretical Physics,  Vol. 18,  Zürich: European Mathematical Society (EMS), 461 pages, SBN {{isbn|978-3-03719-105-7}}, DOI 10.4171–105.
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]], (1942) ''Non-Euclidean geometry'', University of Toronto Press, Toronto
* {{cite book
 | last=Fenchel
 | first=Werner |author-link=Werner Fenchel
 | title=Elementary geometry in hyperbolic space
 | series=De Gruyter Studies in mathematics
 | volume=11
 | publisher=Walter de Gruyter & Co.
 | location=Berlin-New York
 | year=1989
}}
* {{cite book
 | last= Fenchel
 | first=Werner |author-link=Werner Fenchel
 |author2=[[Jakob Nielsen (mathematician)|Nielsen, Jakob]] |editor=Asmus L. Schmidt
 | title=Discontinuous groups of isometries in the hyperbolic plane
 | series=De Gruyter Studies in mathematics
 | volume=29
 | publisher=Walter de Gruyter & Co.
 | location=Berlin
 | year=2003
}}
* Lobachevsky, Nikolai I., (2010) ''Pangeometry'', Edited and translated by Athanase Papadopoulos,  Heritage of European Mathematics, Vol. 4. Zürich: European Mathematical Society (EMS). xii, 310~p, {{isbn|978-3-03719-087-6}}/hbk
* [[John Milnor|Milnor, John W.]], (1982) ''[http://projecteuclid.org/euclid.bams/1183548588 Hyperbolic geometry: The first 150 years]'', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp.&nbsp;9–24.
* Reynolds, William F., (1993) ''Hyperbolic Geometry on a Hyperboloid'', [[American Mathematical Monthly]] 100:442–455.
* {{Cite book | last1=Stillwell | first1=John | author1-link=John Stillwell | title=Sources of hyperbolic geometry | url=https://books.google.com/books?id=ZQjBXxxQsucC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=History of Mathematics | isbn=978-0-8218-0529-9 | year=1996 | volume=10 | mr=1402697}}
* Samuels, David, (March 2006) ''Knit Theory'' Discover Magazine, volume 27, Number 3.
* James W. Anderson, ''Hyperbolic Geometry'', Springer 2005, {{isbn|1-85233-934-9}}
* James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997) ''[http://www.msri.org/communications/books/Book31/files/cannon.pdf Hyperbolic Geometry]'', MSRI Publications, volume 31.

==External links==
{{commonscat|Hyperbolic geometry}}
*[http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html Javascript freeware for creating sketches in the Poincaré Disk Model of Hyperbolic Geometry] University of New Mexico
*[https://www.youtube.com/watch?v=B16YjC9OS0k&mode=user&search= "The Hyperbolic Geometry Song"] A short music video about the basics of Hyperbolic Geometry available at YouTube.
*{{springer|title=Lobachevskii geometry|id=p/l060030}}
*{{mathworld|urlname=Gauss-Bolyai-LobachevskySpace|title=Gauss–Bolyai–Lobachevsky Space}}
*{{mathworld|urlname=HyperbolicGeometry|title=Hyperbolic Geometry}}
*[https://web.archive.org/web/20140303111006/http://www.geom.uiuc.edu/~crobles/hyperbolic/ More on hyperbolic geometry, including movies and equations for conversion between the different models] University of Illinois at Urbana-Champaign
*[https://arxiv.org/abs/0903.3287 Hyperbolic Voronoi diagrams made easy, Frank Nielsen]
*{{Cite journal|first=Wilson|last=Stothers|title=Hyperbolic geometry|url=http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html|publisher=[[University of Glasgow]]|year=2000 |website=maths.gla.ac.uk}}, interactive instructional website.
*[http://www.plunk.org/~hatch/HyperbolicTesselations/ Hyperbolic Planar Tesselations]
*[http://www.roguetemple.com/z/hyper/models.php Models of the Hyperbolic Plane]

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