TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, hyperbolic geometry (also called Lobachevskian geometry or BolyaiLobachevskian geometry) is a
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
. The
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
, the modern version of
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

.) Hyperbolic plane
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

is also the geometry of
saddle surface In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and , surfaces with a constant negative
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

. A modern use of hyperbolic geometry is in the theory of
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
, particularly the
Minkowski model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski is a model of ''n''-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
. 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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence
elliptic geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...
(
spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...
), parabolic geometry (
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
), and hyperbolic geometry. In the
former Soviet Union The post-Soviet states, also known as the former Soviet Union (FSU), the former Soviet Republics and in Russia as the near abroad (russian: links=no, ближнее зарубежье, blizhneye zarubezhye), are the 15 sovereign state A sovere ...
, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kno ...
. This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See
hyperbolic space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for more information on hyperbolic geometry extended to three and more dimensions.

Properties

Relation to Euclidean geometry

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

atic difference is the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

. When the parallel postulate is removed from Euclidean geometry the resulting geometry is
absolute geometry Absolute geometry is a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with pro ...
. 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'', 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 In hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
, hyperbolic geometry has an
absolute scale An absolute scale is a system of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects ...
, 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 . 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
model In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
used, even if the lines may look radically different.

Non-intersecting / parallel lines

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
: :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 frame, The two lines through a given point ''P'' and limiting parallel to line ''R''. In neutral or absolute geometry, and in hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
s (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the
ideal point 200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space. Giv ...
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 frame, The two lines through a given point ''P'' and limiting parallel to line ''R''. In neutral or absolute geometry, and in hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
s make an angle ''θ'' with ''PB''; this angle depends only on the
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

of the plane and the distance ''PB'' and is called the
angle of parallelism In hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
. For ultraparallel lines, the 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 $2 \pi r$. Let $R = \frac$, where $K$ is the
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

of the plane. In hyperbolic geometry, $K$ is negative, so the square root is of a positive number. Then the circumference of a circle of radius ''r'' is equal to: :$2\pi R \sinh \frac \,.$ And the area of the enclosed disk is: :$4\pi R^2 \sinh^2 \frac = 2\pi R^2 \left\left(\cosh \frac - 1\right\right) \,.$ Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than $2\pi$, 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 curvatureIn Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
of a circle of radius ''r'' is: $\frac$

Hypercycles and horocycles

In hyperbolic geometry, there is no line all of whose points are equidistant from another line. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle. Another special curve is the
horocycle In hyperbolic geometry 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 ...
, a curve whose normal radii (
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

lines) are all
limiting parallel frame, The two lines through a given point ''P'' and limiting parallel to line ''R''. In neutral or absolute geometry, and in hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
to each other (all converge asymptotically in one direction to the same
ideal point 200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space. Giv ...
, the centre of the horocycle). Through every pair of points there are two horocycles. The centres of the horocycles are the
ideal point 200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space. Giv ...
s of the
perpendicular bisector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of the line-segment between them. Given any three distinct points, they all lie on either a line, hypercycle,
horocycle In hyperbolic geometry 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 ...
, or circle. The length of the 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 a horocycle, connecting the same two points. The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger than the arc-length of a horocycle connecting two points. If the Gaussian curvature of the plane is −1 then the
geodesic curvatureIn Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
of a horocycle is 1 and of a hypercycle is between 0 and 1.

Triangles

Unlike Euclidean triangles, where the angles always add up to π
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s (180°, a
straight angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s (180°, a
straight angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
). The difference is referred to as the
defect A defect is a physical, functional, or aesthetic attribute of a product or service that exhibits that the product or service failed to meet one of the desired specifications. Defect, defects or defected may also refer to: Examples * Angular defec ...
. The area of a hyperbolic triangle is given by its defect in radians multiplied by ''R''2. As a consequence, all hyperbolic triangles have an area that is less than or equal to ''R''2π. The area of a hyperbolic
ideal triangle Image:IdealTriangle HalfPlane.svg, 200px, Two ideal triangles in the Poincaré half-plane model In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ...
in which all three angles are 0° is equal to this maximum. As in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, each hyperbolic triangle has an
incircle (I), excircles, excenters (J_A, J_B, J_C), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt: ...

. In hyperbolic geometry, if all three of its vertices lie on a
horocycle In hyperbolic geometry 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 ...
or hypercycle, then the triangle has no
circumscribed circle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
. As in
spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values ...
and
elliptical geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...
, in hyperbolic geometry if two triangles are similar, they must be congruent.

Regular apeirogon

A special polygon in hyperbolic geometry is the regular
apeirogon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
, a uniform polygon with an infinite number of sides. In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, 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 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of any length (i.e., it remains a polygon). The side and angle will, depending on the side length and the angle between the sides, be limiting or diverging parallel (see lines above). If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric
horocycle In hyperbolic geometry 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 ...
s. If the bisectors are diverging parallel then a pseudogon (distinctly different from an apeirogon) can be inscribed in hypercycles (all vertices are the same distance of a line, the axis, also the midpoint of the side segments are all equidistant to the same axis.)

Tessellations

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with
regular polygon In , a regular polygon is a that is (all angles are equal in measure) and (all sides have the same length). Regular polygons may be either or . In the , a sequence of regular polygons with an increasing number of sides approximates a , if the ...
s as
faces The face is the front of an animal's head that features three of the head's Sense, sense organs, the eyes, nose, and mouth, and through which animals express many of their Emotion, emotions. The face is crucial for human Personal identity, ident ...
. 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'', ''q'', ''r'' are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic
triangle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.

Standardized Gaussian curvature

Though hyperbolic geometry applies for any surface with a constant negative
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

, 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 The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s. * The area of a horocyclic sector is equal to the length of its horocyclic arc. * An arc of a
horocycle In hyperbolic geometry 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 ...
so that a line that is tangent at one endpoint is
limiting parallel frame, The two lines through a given point ''P'' and limiting parallel to line ''R''. In neutral or absolute geometry, and in hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
to the radius through the other endpoint has a length of 1. * The ratio of the arc lengths between two radii of two concentric
horocycle In hyperbolic geometry 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 ...
s where the
horocycle In hyperbolic geometry 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 ...
s are a distance 1 apart is ''e'' :1.

Cartesian-like coordinate systems

In hyperbolic geometry, the sum of the angles of a
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and ...

is always less than 360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems. 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 Lobachevski 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 In hyperbolic geometry 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 ...
through the origin centered around $\left(0, + \infty \right)$ and the length along this horocycle. Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

Distance

Construct a Cartesian-like coordinate system as follows. Choose a line (the ''x''-axis) in the hyperbolic plane (with a standardized curvature of −1) and label the points on it by their distance from an origin (''x''=0) point on the ''x''-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates ''x'' and ''y'' 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). Then the distance between two such points will be :$\operatorname \left(\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle\right) = \operatorname \left\left( \cosh y_1 \cosh \left(x_2 - x_1\right) \cosh y_2 - \sinh y_1 \sinh y_2 \right\right) \,.$ This formula can be derived from the formulas about
hyperbolic triangle In hyperbolic geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...

s. The corresponding metric tensor is: $\left(\mathrm s\right)^2 = \cosh^2 y \, \left(\mathrm x\right)^2 + \left(\mathrm y\right)^2$. In this coordinate system, straight lines are either perpendicular to the ''x''-axis (with equation ''x'' = a constant) or described by equations of the form :$\tanh y = A \cosh x + B \sinh x \quad \text \quad A^2 < 1 + B^2$ where ''A'' and ''B'' are real parameters which characterize the straight line.

History

Since the publication of
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
circa 300 BCE, many
geometers A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * M ...
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were
Proclus Proclus Lycius (; 410/411/ 7 Feb. or 8 Feb. 412 –17 April 485 AD), called Proclus the Successor, Proclus the Platonic Successor, or Proclus of Athens (Greek: Προκλου Διαδοχου ''Próklos Diádochos'', ''"''in some Manuscript ...
,
Ibn al-Haytham Ḥasan Ibn al-Haytham (Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s and ...

(Alhacen),
Omar Khayyám Omar Khayyam (; fa, عمر خیّام ; 18 May 1048 – 4 December 1131) was a Persians, Persian polymath, Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, philosopher, and Persian poetry, ...
,
Nasīr al-Dīn al-Tūsī Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the ...
,
Witelo Vitello (or Witelo; fl. c. 1270–1285) was a Silesian friar A friar is a brother and a member of one of the mendicant orders founded in the twelfth or thirteenth century; the term distinguishes the mendicants' itinerant apostolic characte ...
,
Gersonides Levi ben Gershon (1288 – 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew Hebrew (, , or ) is a Northwest Semitic languages, Northwest Semitic language of the Afroasia ...
,
Alfonso Alphons (Latinized ''Alphonsus'', ''Adelphonsus'', or ''Adefonsus'') is a male given name recorded from the 8th century ( Alfonso I of Asturias, r. 739-757) in the Christian successor states of the Visigothic kingdom in the Iberian peninsula. I ...
, and later
Giovanni Gerolamo Saccheri Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ...
,
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

,
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a Swiss Swiss may refer to: * the adjectival form of Switzerland , french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Sv ...
, and Legendre. 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 A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and ...

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. In the 18th century,
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a Swiss Swiss may refer to: * the adjectival form of Switzerland , french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Sv ...
introduced the
hyperbolic functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

and computed the area of a
hyperbolic triangle In hyperbolic geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...

.

19th-century developments

In the 19th century, hyperbolic geometry was explored extensively by
Nikolai Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russia Russia (russian: link=no, ...
,
János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian people, Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discover ...
,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

and
Franz Taurinus Franz Adolph Taurinus (15 November 1794 – 13 February 1874) was a Germans, German mathematician who is known for his work on non-Euclidean geometry. Life Franz Taurinus was the son of Julius Ephraim Taurinus, a court official of the Count of ...

. 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. Gauss wrote in an 1824 letter to
Franz Taurinus Franz Adolph Taurinus (15 November 1794 – 13 February 1874) was a Germans, German mathematician who is known for his work on non-Euclidean geometry. Life Franz Taurinus was the son of Julius Ephraim Taurinus, a court official of the Count of ...

that he had constructed it, but Gauss did not publish his work. Gauss called it "
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
" 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 Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italy, Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the f ...
provided models (see below) of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
Euclidean geometry was. The term "hyperbolic geometry" was introduced by
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
in 1871. Klein followed an initiative of
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

to use the transformations of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
to produce
isometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The idea used a
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
or
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that departme ...
to define a region, and used
cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, thei ...
to define a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geomet ...
. The projective transformations that leave the conic section or quadric
stable A stable is a building in which livestock Livestock are the domesticated Domestication is a sustained multi-generational relationship in which one group of organisms assumes a significant degree of influence over the reproduction and c ...
are the isometries. "Klein showed that if the
Cayley absoluteCayley may refer to: * Cayley (surname) *Cayley, Alberta, Canada, a hamlet *Mount Cayley, a volcano in southwestern British Columbia, Canada *Cayley (crater), a lunar crater *Cayley computer algebra system, designed to solve mathematical problems, p ...
is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..." For more history, see article on
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...
, and the references
Coxeter Harold Scott MacDonald "Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington ...
and
Milnor John Willard Milnor (born February 20, 1931) is an United States of America , American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and o ...
.

Philosophical consequences

The discovery of hyperbolic geometry had important
philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence, reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning ...

consequences. Before its discovery many philosophers (for example
Hobbes Thomas Hobbes ( ; sometimes known as Thomas Hobbes of Malmesbury; 5 April 1588 – 4 December 1679) was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first sp ...
and
Spinoza Baruch (de) Spinoza (; ; ; born Baruch Espinosa; later as an author and a correspondent Benedictus de Spinoza, anglicized to Benedict de Spinoza; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Spanish and Portuguese Jews, Port ...

) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in ''
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
''.
Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about r ...

in the ''Critique of Pure Reason'' came to the conclusion that space (in
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
) 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. It is said that
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

did not publish anything about hyperbolic geometry out of fear of the "uproar of the
Boeotians Boeotia, sometimes alternatively Latinised Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...
", which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians"). The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in
mathematical rigour Rigour (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...
,
analytical philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of ...
and
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

Geometry of the universe (spatial dimensions only)

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 Parallax is a displacement or difference in the apparent positionThe apparent place of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the ...

of
Sirius Sirius () is the list of brightest stars, brightest star in the night sky. Its name is derived from the Ancient Greek language, Greek word (, 'glowing' or 'scorching'). The star is designated α Canis Majoris, Latinisation of name ...

and treating Sirius as the ideal point of an
angle of parallelism In hyperbolic geometry In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
. He realised that his measurements were 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 absolute length is at least one million times the diameter of the
earth's orbit Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Eart ...
(, 10
parsec The parsec (symbol: pc) is a unit of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every ...

). Some argue that his measurements were methodologically flawed.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French people, French mathematician, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as ...
, with his
sphere-world The idea of a sphere-world was constructed by Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or ...
thought experiment A thought experiment is a hypothetical situation in which a hypothesis A hypothesis (plural hypotheses) is a proposed explanation An explanation is a set of statements usually constructed to describe a set of facts which clarifies the ...
, came to the conclusion that everyday experience does not necessarily rule out other geometries. The
geometrization conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...
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.

Geometry of the universe (special relativity)

Special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
places space and time on equal footing, so that one considers the geometry of a unified
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
instead of considering space and time separately. Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of
Galilean relativity Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using Galileo's ship, ...
). In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
,
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its cano ...
and
anti-de Sitter space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, corresponding to zero, positive and negative curvature respectively. Hyperbolic geometry enters special relativity through
rapidity In Theory of relativity, relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being a ...
, which stands in for
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

, and is expressed by a
hyperbolic angle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

. The study of this velocity geometry has been called 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).

Physical realizations of the hyperbolic plane

The hyperbolic plane is a plane where every point is a
saddle point In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

. There exist various
pseudosphereIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

s in Euclidean space that have a finite area of constant negative Gaussian curvature. By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative
Gaussian curvature ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

) in a three-dimensional Euclidean space. Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the
pseudosphereIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

is due to
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

. The art of
crochet Crochet (; ) is a process of creating textile A textile is a flexible material made by creating an interlocking bundle of yarn Yarn is a long continuous length of interlocked fibres, suitable for use in the production of textiles, se ...

has been used (see ) to demonstrate hyperbolic planes, the first such demonstration having been made by Daina Taimiņa. In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "
hyperbolic soccerball In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex (geometry), vertex, forming a pattern similar ...
" (more precisely, a ). Instructions on how to make a hyperbolic quilt, designed by
Helaman Ferguson Helaman Rolfe Pratt Ferguson (born 1940 in Salt Lake City Salt Lake City (often shortened to Salt Lake and abbreviated as SLC) is the Capital (political), capital and List of cities and towns in Utah, most populous city of the U.S. state of Uta ...
, have been made available by Jeff Weeks.

Models of the hyperbolic plane

There are different pseudospherical surfaces that have for a large area a constant negative Gaussian curvature, the
pseudosphereIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

being the best well known of them. But it is easier to do hyperbolic geometry on other models. There are four Mathematical model, models 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

The Beltrami–Klein model, also known as the projective disk model, Klein disk model and Klein model, is named after
Eugenio Beltrami Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italy, Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the f ...
and
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
. 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 angles 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 Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
.

The 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 transformations. * 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. * Horocycles are circles within the disk which are tangent to the boundary circle, minus the point of contact. * Hypercycle (hyperbolic geometry), Hypercycles 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

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 $\left(x, y\right) \mapsto \left(x, \lambda y\right),\quad \lambda > 0 .$ * Like the Poincaré disk model, this model preserves angles, and is thus conformal map, conformal. All isometries within this model are therefore Möbius transformations 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

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 In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
. This model is generally credited to Poincaré, but Reynolds says that Wilhelm Killing used this model in 1885 * This model has direct application to
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
, as Minkowski 3-space is a model for
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time. * The hyperbolic distance between two points on the hyperboloid can then be identified with the relative
rapidity In Theory of relativity, relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being a ...
between the two corresponding observers. * The model generalizes directly to an additional dimension, where 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 visualising transformations between the other models. The hemisphere model uses the upper half of the unit sphere: $x^2 + y^2 +z^2 = 1 , z > 0.$ 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 $\left(0,0, -1\right)$ onto the plane $z = 0$ projects corresponding points on the Poincaré disk model * Stereographic projection from $\left(0,0, -1\right)$ onto the surface $x^2 + y^2 - z^2 = -1 , z > 0$ projects corresponding points on the hyperboloid model * Stereographic projection from $\left(-1,0,0\right)$ onto the plane $x=1$ projects corresponding points on the Poincaré half-plane model * Orthographic projection onto a plane $z = C$ projects corresponding points on the Beltrami–Klein model. * Central projection from the centre of the sphere onto the plane $z = 1$ projects corresponding points on the Gans Model See further: #Connection between the models, Connection between the models (below)

The Gans model

In 1966 David Gans proposed a flattened hyperboloid model in the journal American Mathematical Monthly. 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.

The band model

The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by $\$, the metric is given by $, dz, \sec \left(\operatorname z\right)$.

Connection between the models

::: 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 ), a surface of zero Gaussian curvature (cylinder A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric sh ...

, which is indifferent to the coordinate chart used. The geodesics are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane. 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.

Isometries of the hyperbolic plane

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 the 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 200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space. Giv ...
(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.

Hyperbolic geometry 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 quite plainly 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 Crochet (; ) is a process of creating textile A textile is a flexible material made by creating an interlocking bundle of yarn Yarn is a long continuous length of interlocked fibres, suitable for use in the production of textiles, se ...

has Mathematics and fiber arts#Knitting and crochet, been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, whose book ''Crocheting Adventures with Hyperbolic Planes'' won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year. HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.

Higher dimensions

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 :: $\mathrm\left(1,n\right)/\left(\mathrm\left(1\right) \times \mathrm\left(n\right)\right).$ The orthogonal group Group action (mathematics), acts by norm-preserving transformations on
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
R1,''n'', 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 algebraic 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 groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms allow one to interpret the upper half plane model as the quotient and the Poincaré disc model as the quotient . In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in 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 on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism . 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.

* Constructions in hyperbolic geometry * Hyperbolic 3-manifold * Hyperbolic manifold * Hyperbolic set * Hjelmslev transformation * Hyperbolic tree * Kleinian group * Lambert quadrilateral * Open universe * Poincaré metric * Saccheri quadrilateral * Systolic geometry * Uniform tilings in hyperbolic plane * δ-hyperbolic space * Band model

References

* A'Campo, Norbert and Papadopoulos, Athanase, (2012) ''Notes on hyperbolic geometry'', in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN , DOI 10.4171/105. * Harold Scott MacDonald Coxeter, Coxeter, H. S. M., (1942) ''Non-Euclidean geometry'', University of Toronto Press, Toronto * * * 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, /hbk * John Milnor, Milnor, John W., (1982)
Hyperbolic geometry: The first 150 years
', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24. * Reynolds, William F., (1993) ''Hyperbolic Geometry on a Hyperboloid'', American Mathematical Monthly 100:442–455. * * Samuels, David, (March 2006) ''Knit Theory'' Discover Magazine, volume 27, Number 3. * James W. Anderson, ''Hyperbolic Geometry'', Springer 2005, * James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997)
Hyperbolic Geometry
', MSRI Publications, volume 31.