Inmathematics,hyperbolic geometry (also calledLobachevskian geometry orBolyai–Lobachevskian geometry) is anon-Euclidean geometry. Theparallel postulate ofEuclidean geometry is replaced with:

For any given lineR and pointP not onR, in the plane containing both lineR and pointP there are at least two distinct lines throughP that do not intersectR.

(Compare the above withPlayfair's axiom, the modern version ofEuclid'sparallel postulate.)

Thehyperbolic plane is aplane where every point is asaddle point. Hyperbolic planegeometry is also the geometry ofpseudospherical surfaces, surfaces with a constant negativeGaussian curvature.Saddle surfaces have negative Gaussian curvature in at least some regions, where theylocally resemble the hyperbolic plane.

Thehyperboloid model of hyperbolic geometry provides a representation ofevents one temporal unit into the future inMinkowski space, the basis ofspecial relativity. Each of these events corresponds to arapidity 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 namehyperbolic geometry to include it in the now rarely used sequenceelliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry.In theformer Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometerNikolai Lobachevsky.

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the onlyaxiomatic difference is theparallel postulate.When the parallel postulate is removed from Euclidean geometry the resulting geometry isabsolute geometry.There are two kinds of absolute geometry, Euclidean and hyperbolic.All theorems of absolute geometry, including the first 28 propositions of book one ofEuclid'sElements, are valid in Euclidean and hyperbolic geometry.Propositions 27 and 28 of Book One of Euclid'sElements 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 theangle of parallelism, hyperbolic geometry has anabsolute scale, a relation between distance and angle measurements.

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 aresupplementary.

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 themodel used, even if the lines may look radically different.

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines inEuclidean geometry:

For any lineR and any pointP which does not lie onR, in the plane containing lineR and pointP there are at least two distinct lines throughP that do not intersectR.

This implies that there are throughP an infinite number of coplanar lines that do not intersectR.

These non-intersecting lines are divided into two classes:

• Two of the lines (x andy in the diagram) arelimiting parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of theideal points at the "ends" ofR, asymptotically approachingR, always getting closer toR, 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 calledultraparallel,diverging parallel or sometimesnon-intersecting.

Some geometers simply use the phrase "parallel lines" to mean "limiting parallel lines", withultraparallel lines meaning justnon-intersecting.

Theselimiting parallels make an angleθ withPB; this angle depends only on theGaussian curvature of the plane and the distancePB and is called theangle of parallelism.

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

In hyperbolic geometry, the circumference of a circle of radiusr is greater than${\displaystyle 2\pi r}$ .

Let${\displaystyle R=\frac{1}{\sqrt{-K}}}$ , where${\displaystyle K}$  is theGaussian curvature of the plane. In hyperbolic geometry,${\displaystyle K}$  is negative, so the square root is of a positive number.

Then the circumference of a circle of radiusr is equal to:

${\displaystyle 2\pi R\sinh \frac{r}{R}.}$ 

And the area of the enclosed disk is:

${\displaystyle 4\pi R^2{\sinh }^2\frac{r}{2R}=2\pi R^2\left(\cosh \frac{r}{R}-1\right).}$ 

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

If the Gaussian curvature of the plane is −1 then thegeodesic curvature of a circle of radiusr is:${\displaystyle \frac{1}{\tanh (r)}}$ ^[1]

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 ahypercycle.

Another special curve is thehorocycle, whosenormal radii (perpendicular lines) are alllimiting parallel to each other (all converge asymptotically in one direction to the sameideal point, the centre of the horocycle).

Through every pair of points there are two horocycles. The centres of the horocycles are theideal points of theperpendicular bisector of the line-segment between them.

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

Thelength 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 thegeodesic curvature of a horocycle is 1 and that of a hypercycle is between 0 and 1.^[1]

Unlike Euclidean triangles, where the angles always add up to πradians (180°, astraight angle), in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians (180°). The difference is called thedefect. Generally, the defect of a convex hyperbolic polygon with${\displaystyle n}$  sides is its angle sum subtracted from${\displaystyle (n-2)\cdot {180}^{\circ }}$ .

The area of a hyperbolic triangle is given by its defect in radians multiplied byR², which is also true for all convex hyperbolic polygons.^[2] Therefore all hyperbolic triangles have an area less than or equal toR²π. The area of a hyperbolicideal triangle in which all three angles are 0° is equal to this maximum.

As inEuclidean geometry, each hyperbolic triangle has anincircle. In hyperbolic space, if all three of its vertices lie on ahorocycle orhypercycle, then the triangle has nocircumscribed circle.

As inspherical andelliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.

Special polygons in hyperbolic geometry are the regularapeirogon andpseudogonuniform polygons with an infinite number of sides.

InEuclidean 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 anglebisectors 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 concentrichorocycles.

If the bisectors are diverging parallel then it is a pseudogon and can be inscribed and circumscribed byhypercycles (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).

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane withregular polygons asfaces.

There are an infinite number of uniform tilings based on theSchwarz triangles (pqr) where 1/p + 1/q + 1/r < 1, wherep, q, r are each orders of reflection symmetry at three points of thefundamental domain triangle, the symmetry group is a hyperbolictriangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.^[3]

Though hyperbolic geometry applies for any surface with a constant negativeGaussian curvature, it is usual to assume a scale in which the curvatureK is −1.

This results in some formulas becoming simpler. Some examples are:

• The area of a triangle is equal to its angle defect inradians.
• The area of a horocyclic sector is equal to the length of its horocyclic arc.
• An arc of ahorocycle so that a line that is tangent at one endpoint islimiting parallel to the radius through the other endpoint has a length of 1.^[4]
• The ratio of the arc lengths between two radii of two concentrichorocycles where the horocycles are a distance 1 apart ise : 1.^[4]

Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of aquadrilateral 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 (thex-axis) and after that many choices exist.

The Lobachevsky coordinatesx andy are found by dropping a perpendicular onto thex-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 thehorocycle through the origin centered around${\displaystyle (0,+\mathrm{\infty })}$  and the length along this horocycle.^[5]

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

A Cartesian-like^{[citation needed]} 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 origino on this line. Then:

• thex-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;
• they-coordinate is the signeddistance 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]}$${\displaystyle \mathrm{dist}(⟨x_1,y_1⟩,⟨x_2,y_2⟩)=\mathrm{arcosh}\left(\cosh y_1\cosh (x_2-x_1)\cosh y_2-\sinh y_1\sinh y_2\right).}$$ 

This formula can be derived from the formulas abouthyperbolic triangles.

The corresponding metric tensor field is:${\displaystyle (\mathrm{d}s{)}^2={\cosh }^{2}y(\mathrm{d}x{)}^2+(\mathrm{d}y{)}^2}$ .

In this coordinate system, straight lines take one of these forms ((x,y) is a point on the line;x₀,y₀,A, andα are parameters):

ultraparallel to thex-axis

${\displaystyle \tanh (y)=\tanh (y_0)\cosh (x-x_0)}$ 

asymptotically parallel on the negative side

${\displaystyle \tanh (y)=A\exp (x)}$ 

asymptotically parallel on the positive side

${\displaystyle \tanh (y)=A\exp (-x)}$ 

intersecting perpendicularly

${\displaystyle x=x_0}$ 

intersecting at an angleα

${\displaystyle \tanh (y)=\tan (\alpha )\sinh (x-x_0)}$ 

Generally, these equations will only hold in a bounded domain (ofx values). At the edge of that domain, the value ofy blows up to ±infinity.

Since the publication ofEuclid'sElements circa 300BC, manygeometers tried to prove theparallel postulate. Some tried to prove it byassuming its negation and trying to derive a contradiction. Foremost among these wereProclus,Ibn al-Haytham (Alhacen),Omar Khayyám,^[6]Nasīr al-Dīn al-Tūsī,Witelo,Gersonides,Alfonso, and laterGiovanni Gerolamo Saccheri,John Wallis,Johann Heinrich Lambert, andLegendre.^[7]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 onquadrilaterals, including theIbn al-Haytham–Lambert quadrilateral andKhayyam–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.^[8]

In the 18th century,Johann Heinrich Lambert introduced thehyperbolic functions^[9] and computed the area of ahyperbolic triangle.^[10]

In the 19th century, hyperbolic geometry was explored extensively byNikolai Lobachevsky,János Bolyai,Carl Friedrich Gauss andFranz 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.^[11][12]

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"^[13] 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 consistentif and only if Euclidean geometry was.

The term "hyperbolic geometry" was introduced byFelix Klein in 1871.^[14] Klein followed an initiative ofArthur Cayley to use the transformations ofprojective geometry to produceisometries. The idea used aconic section orquadric to define a region, and usedcross ratio to define ametric. The projective transformations that leave the conic section or quadricstable are the isometries. "Klein showed that if theCayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."^[15]

The discovery of hyperbolic geometry had importantphilosophical consequences. Before its discovery many philosophers (such asHobbes andSpinoza) viewed philosophical rigor in terms of the "geometrical method", referring to the method of reasoning used inEuclid'sElements.

Kant inCritique of Pure Reason concluded that space (inEuclidean 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.^[16]

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of theBoeotians" (stereotyped as dullards by the ancient Athenians^[17]), which would ruin his status asprinceps mathematicorum (Latin, "the Prince of Mathematicians").^[18]The "uproar of the Boeotians" came and went, and gave an impetus to great improvements inmathematical rigour,analytical philosophy andlogic. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

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 theparallax ofSirius and treating Sirius as the ideal point of anangle of parallelism. He realized that his measurements werenot precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then theabsolute length is at least one million times the diameter ofEarth's orbit (2000000 AU, 10parsec).^[19]Some argue that his measurements were methodologically flawed.^[20]

Henri Poincaré, with hissphere-worldthought experiment, came to the conclusion that everyday experience does not necessarily rule out other geometries.

Thegeometrization 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.^[21]

Special relativity places space and time on equal footing, so that one considers the geometry of a unifiedspacetime instead of considering space and time separately.^[22][23]Minkowski geometry replacesGalilean geometry (which is the 3-dimensional Euclidean space with time ofGalilean relativity).^[24]

In relativity, rather than Euclidean, elliptic and hyperbolic geometry, the appropriate geometries to consider areMinkowski space,de Sitter space andanti-de Sitter space,^[25][26] corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity throughrapidity, which stands in forvelocity, and is expressed by ahyperbolic angle. The study of this velocity geometry has been calledkinematic 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).^[27]

There exist variouspseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.

ByHilbert's theorem, one cannot isometricallyimmerse a complete hyperbolic plane (a complete regular surface of constant negativeGaussian curvature) in a 3-D Euclidean space.

Other usefulmodels of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on thepseudosphere is due toWilliam Thurston.

The art ofcrochet has beenused to demonstrate hyperbolic planes, the first such demonstration having been made byDaina Taimiņa.^[28]

In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball" (more precisely, atruncated order-7 triangular tiling).^[29][30]

Instructions on how to make a hyperbolic quilt, designed byHelaman Ferguson,^[31] have been made available byJeff Weeks.^[32]

Variouspseudospheres – 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, thetractoid (or pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using acone orcylinder 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 fourmodels commonly used for hyperbolic geometry: theKlein model, thePoincaré disk model, thePoincaré half-plane model, and the Lorentz orhyperboloid 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 byBeltrami, not byPoincaré orKlein. All these models are extendable to more dimensions.

TheBeltrami–Klein model, also known as the projective disk model, Klein disk model andKlein model, is named afterEugenio Beltrami andFelix Klein.

For the two dimensions this model uses the interior of theunit circle for the complete hyperbolicplane, and thechords of this circle are the hyperbolic lines.

For higher dimensions this model uses the interior of theunit ball, and thechords of thisn-ball are the hyperbolic lines.

• This model has the advantage that lines are straight, but the disadvantage thatangles are distorted (the mapping is notconformal), and also circles are not represented as circles.
• The distance in this model is half the logarithm of thecross-ratio, which was introduced byArthur Cayley inprojective geometry.

ThePoincaré disk model, also known as the conformal disk model, also employs the interior of theunit circle, but lines are represented by arcs of circles that areorthogonal to the boundary circle, plus diameters of the boundary circle.

• This model preserves angles, and is therebyconformal. All isometries within this model are thereforeMöbius 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 aretangent to the boundary circle, minus the point of contact.
• Hypercycles are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.

ThePoincaré half-plane model takes one-half of the Euclidean plane, bounded by a lineB of the plane, to be a model of the hyperbolic plane. The lineB is not included in the model.

The Euclidean plane may be taken to be a plane with theCartesian coordinate system and thex-axis is taken as lineB and the half plane is the upper half (y > 0 ) of this plane.

• Hyperbolic lines are then either half-circles orthogonal toB or rays perpendicular toB.
• The length of an interval on a ray is given bylogarithmic measure so it is invariant under ahomothetic transformation${\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\lambda >0.}$ 
• Like the Poincaré disk model, this model preserves angles, and is thusconformal. All isometries within this model are thereforeMöbius transformations of the plane.
• The half-plane model is the limit of the Poincaré disk model whose boundary is tangent toB at the same point while the radius of the disk model goes to infinity.

Thehyperboloid model or Lorentz model employs a 2-dimensionalhyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensionalMinkowski space. This model is generally credited to Poincaré, but Reynolds^[33] says thatWilhelm Killing used this model in 1885

• This model has direct application tospecial relativity, as Minkowski 3-space is a model forspacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that variousinertially moving observers, starting from a common event, will reach in a fixedproper time.
• The hyperbolic distance between two points on the hyperboloid can then be identified with the relativerapidity 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.

Thehemisphere 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 theunit sphere:${\displaystyle x^2+y^2+z^2=1,z>0.}$ 

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

The hemisphere model is part of aRiemann sphere, and different projections give different models of the hyperbolic plane:

• Stereographic projection from${\displaystyle (0,0,-1)}$  onto the plane${\displaystyle z=0}$  projects corresponding points on thePoincaré disk model
• Stereographic projection from${\displaystyle (0,0,-1)}$  onto the surface${\displaystyle x^2+y^2-z^2=-1,z>0}$  projects corresponding points on thehyperboloid model
• Stereographic projection from${\displaystyle (-1,0,0)}$  onto the plane${\displaystyle x=1}$  projects corresponding points on thePoincaré half-plane model
• Orthographic projection onto a plane${\displaystyle z=C}$  projects corresponding points on theBeltrami–Klein model.
• Central projection from the centre of the sphere onto the plane${\displaystyle z=1}$  projects corresponding points on theGans Model

All models essentially describe the same structure. The difference between them is that they represent differentcoordinate charts laid down on the samemetric space, namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negativeGaussian curvature, which is indifferent to the coordinate chart used. Thegeodesics 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.^[34]

Once we choose a coordinate chart (one of the "models"), we can alwaysembed 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:

• the Beltrami–Klein model's relation to the hyperboloid model,
• the Beltrami–Klein model's relation to the Poincaré disk model,
• andthe Poincaré disk model's relation to the hyperboloid model.

In 1966 David Gans proposed aflattened hyperboloid model in the journalAmerican Mathematical Monthly.^[35] It is anorthographic 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 entireEuclidean plane.
• The lines in this model are represented as branches of ahyperbola.^[36]

The conformal square model of the hyperbolic plane arises from usingSchwarz-Christoffel mapping to convert thePoincaré disk into a square.^[37] 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 employs a portion of the Euclidean plane between two parallel lines.^[38] Distance is preserved along one line through the middle of the band. Assuming the band is given by , the metric is given by${\displaystyle |dz|\sec (\mathrm{Im}z)}$ .

Everyisometry (transformation ormotion) of the hyperbolic plane to itself can be realized as the composition of at most threereflections. Inn-dimensional hyperbolic space, up ton+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
  • theidentity isometry — nothing moves; zero reflections; zerodegrees of freedom.
  • 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; twodegrees of freedom.
  • 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 anideal 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]}

M. C. Escher's famous printsCircle Limit III andCircle Limit IVillustrate the conformal disc model (Poincaré disk model) quite well. The white lines inIII are not quite geodesics (they arehypercycles), but are close to them. It is also possible to see quite plainly the negativecurvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

For example, inCircle 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 isexponential growth. InCircle Limit III, for example, one can see that the number of fishes within a distance ofn from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radiusn must rise exponentially inn.

The art ofcrochet hasbeen used to demonstrate hyperbolic planes (pictured above) with the first being made byDaina Taimiņa,^[28] whose bookCrocheting Adventures with Hyperbolic Planes won the 2009Bookseller/Diagram Prize for Oddest Title of the Year.^[39]

HyperRogue is aroguelike game set on various tilings of thehyperbolic plane.

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

Hyperbolic space of dimensionn is a special case of a Riemanniansymmetric space of noncompact type, as it isisomorphic to the quotient

${\displaystyle \mathrm{O}(1,n)/(\mathrm{O}(1)\times \mathrm{O}(n)).}$ 

Theorthogonal groupO(1,n)acts by norm-preserving transformations onMinkowski spaceR^1,n, and it actstransitively 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 hyperbolicn-space. Thestabilizer of any particular line is isomorphic to theproduct 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 inlinear 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 ofLie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphismsSO⁺(1, 2) ≅ PSL(2,R) ≅ PSU(1, 1) allow one to interpret the upper half plane model as the quotientSL(2,R)/SO(2) and the Poincaré disc model as the quotientSU(1, 1)/U(1). In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers inPGL(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 ofPGL(2,C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphismO⁺(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 byunipotentupper triangular matrices.

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• 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)Hyperbolic Geometry, MSRI Publications, volume 31.

• Javascript freeware for creating sketches in the Poincaré Disk Model of Hyperbolic Geometry University of New Mexico
• "The Hyperbolic Geometry Song" A short music video about the basics of Hyperbolic Geometry available at YouTube.
• "Lobachevskii geometry",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weisstein, Eric W."Gauss–Bolyai–Lobachevsky Space".MathWorld.
• Weisstein, Eric W."Hyperbolic Geometry".MathWorld.
• More on hyperbolic geometry, including movies and equations for conversion between the different models University of Illinois at Urbana-Champaign
• Hyperbolic Voronoi diagrams made easy, Frank Nielsen
• Stothers, Wilson (2000)."Hyperbolic geometry".maths.gla.ac.uk.University of Glasgow., interactive instructional website.
• Hyperbolic Planar Tesselations
• Models of the Hyperbolic Plane