Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

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

(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)

Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.

A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.

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

This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space 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 axiomatic difference is the parallel postulate.
When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry.
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's ''Elements'', are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's ''Elements'' prove the existence of parallel/non-intersecting lines.

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

Lines

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

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

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 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:

: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 parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the ideal points 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 parallels make an angle ''θ'' with ''PB''; this angle depends only on the Gaussian curvature of the plane and the distance ''PB'' and is called the angle of parallelism.

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

Circles and disks

In hyperbolic geometry, the circumference of a circle of radius ''r'' is greater than $2 \pi r$.

Let $R = \frac$, where $K$ is the Gaussian curvature 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:

:$4\pi R^2 \sinh^2 \frac = 2\pi R^2 \left(\cosh \frac - 1\right) \,.$

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 curvature 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, a curve whose normal radii ( perpendicular lines) are all limiting parallel to each other (all converge asymptotically in one direction to the same ideal point, the centre of the horocycle).

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

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

The length of 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 curvature 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 radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians (180°, a straight angle). The difference is referred to as the defect.

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 in which all three angles are 0° is equal to this maximum.

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

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

Regular apeirogon

A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides.

In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 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 bisectors 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 horocycles.

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 polygons as faces.

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

Standardized Gaussian curvature

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

This results in some formulas becoming simpler. Some examples are:
* The area of a triangle is equal to its angle defect in radians.
* The area of a horocyclic sector is equal to the length of its horocyclic arc.
* An arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1.
* The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is ''e'' :1.

Cartesian-like coordinate systems

In hyperbolic geometry, the sum of the angles of a quadrilateral 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 through the origin centered around $(0, + \infty )$ 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

A Cartesian-like coordinate system (''x, y'') on the oriented hyperbolic plane is constructed as follows. Choose a line in the hyperbolic plane together with an orientation and an origin ''o'' on this line. Then:
*the ''x''-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin;
*the ''y''-coordinate is the signed distance from 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
$\operatorname (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname \left( \cosh y_1 \cosh (x_2 - x_1) \cosh y_2 - \sinh y_1 \sinh y_2 \right) \,.$

This formula can be derived from the formulas about hyperbolic triangles.

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

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

ultraparallel to the ''x''-axis
:$\tanh (y) = \tanh (y_0) \cosh (x - x_0)$
asymptotically parallel on the negative side
:$\tanh (y) = A \exp (x)$
asymptotically parallel on the positive side
:$\tanh (y) = A \exp (- x)$
intersecting perpendicularly
:$x = x_0$
intersecting at an angle ''α''
:$\tanh (y) = \tan (\alpha) \sinh (x - x_0)$
Generally, these equations will only hold in a bounded domain (of ''x'' values). At the edge of that domain, the value of ''y'' blows up to ±infinity. See also Coordinate systems for the hyperbolic plane#Polar coordinate system.

History

Since the publication of Euclid's Elements circa 300 BCE, many geometers made attempts to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám, Nasīr al-Dīn al-Tūsī, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, 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 quadrilaterals, 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 introduced the hyperbolic functions and computed the area of a hyperbolic triangle.

19th-century developments

In the 19th century, hyperbolic geometry was explored extensively by Nikolai Ivanovich Lobachevsky, János Bolyai, Carl Friedrich Gauss and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.
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" 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 (see below) of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was.

The term "hyperbolic geometry" was introduced by Felix Klein in 1871. Klein followed an initiative of Arthur Cayley to use the transformations of projective geometry to produce isometries. The idea used a conic section or quadric to define a region, and used cross ratio to define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor.

Philosophical consequences

The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers (for example Hobbes and Spinoza) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in ''Euclid's Elements''.

Kant in the ''Critique of Pure Reason'' came to the conclusion that space (in Euclidean geometry) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians", 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, analytical philosophy and logic. 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 of Sirius and treating Sirius as the ideal point of an angle of parallelism. 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 (, 10 parsec).
Some argue that his measurements were methodologically flawed.

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

The geometrization conjecture gives a complete list of eight possibilities for the