In non-Euclidean geometry

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

, the Poincaré half-plane model is the upper half-plane (denoted below as H), together with a metric, the Poincaré metric

Poincaré metric

In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.There are three equivalent...

, that makes it a model of two-dimensional hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

.

It is named after Henri Poincaré

Henri Poincaré

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

, but originated with Eugenio Beltrami

Eugenio Beltrami

Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...

, who used it, along with the Klein model

Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...

 and the Poincaré disk model

Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...

 (due to Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

),
to show that hyperbolic geometry was equiconsistent

Equiconsistency

In mathematical logic, two theories are equiconsistent if, roughly speaking, they are "as consistent as each other".It is not in general possible to prove the absolute consistency of a theory T...

 with Euclidean geometry. The disk model and the half-plane model are isomorphic under a conformal mapping.

Metric

The metric

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 of the model on the half-plane

is given by

where s measures length along a possibly curved line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e. curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose origin is on the x-axis) and straight vertical lines ending on the x-axis. The distance between two points measured in this metric along such a geodesic is

This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.

This can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.

Special curves

In addition to the straight lines mentioned above, there are other special curves on the hyperbolic plane which can be modeled in the Euclidean half-plane:

• A circle (curve equidistant from a central point) with center and radius is modeled by a circle with center and radius
• A curve equidistant from a straight line is modeled by either a circular arc which intersects the x-axis at the same two points as the half-circle which models the given line, or by a straight line which intersects the x-axis at the same point as the vertical line which models the given line.
• An oricircle (a curve which is like a circle with an infinite radius) is modeled by a circle tangent to the x-axis (but excluding the point of intersection), or by a line parallel to the x-axis. An oricircle is a limit on one side of a sequence of ever larger circles tangent at the same point to a given line; on the other side it is a limit of a sequence of curves equidistant from straight lines which are ever further away.

Constructing the curves

Here is how one can use compass and straightedge constructions in the model to achieve the effect of the basic constructions in the hyperbolic plane

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

. For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points.

• Creating the line through two existing points:

Draw the line segment between the two points. Construct the perpendicular bisector of the line segment. Find its intersection with the x-axis. Draw the circle around the intersection which passes through the given points. Erase the part which is on or below the x-axis.

Or in the special case where the two given points lie on a vertical line, draw that vertical line through the two points and erase the part which is on or below the x-axis.

• Creating the circle through one point with center another point:

Draw the radial line (half-circle) between the two given points as in the previous case. Construct a tangent to that line at the non-central point. Drop a perpendicular from the given center point to the x-axis. Find the intersection of these two lines to get the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.

Or if the two given points lie on a vertical line and the given center is above the other given point, then draw a horizontal line through the non-central point. Draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Construct the tangent to that circle at its intersection with the horizontal line. The midpoint between the intersection of the tangent with the vertical line and the given non-central point is the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.

Or if the two given points lie on a vertical line and the given center is below the other given point, then draw a circle around the intersection of the vertical line and the x-axis which passes through the given central point. Draw a line tangent to the circle which passes through the given non-central point. Draw a horizontal line through that point of tangency and find its intersection with the vertical line. The midpoint between that intersection and the given non-central point is the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.

• Creating the point which is the intersection of two existing lines, if they intersect:

Find the intersection of the two given semicircles (or vertical lines).

• Creating the one or two points in the intersection of a line and a circle (if they intersect):

Find the intersection of the given semicircle (or vertical line) with the given circle.

• Creating the one or two points in the intersection of two circles (if they intersect):

Find the intersection of the two given circles.

Symmetry groups

The projective linear group

Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...

 PGL(2,C) acts on the Riemann sphere by the Möbius transformations. The subgroup that maps the upper half-plane, H, onto itself is PSL(2,R), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space

Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

.

There are four closely related Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.

• The special linear group
  Special linear group
  In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
  
   SL(2,R) which consists of the set of 2×2 matrices with real entries whose determinant equals +1. Note that many texts (including Wikipedia) often say SL(2,R) when they really mean PSL(2,R).
• The group S*L(2,R) consisting of the set of 2×2 matrices with real entries whose determinant equals +1 or −1. Note that SL(2,R) is a subgroup of this group.
• The projective special linear group PSL(2,R) = SL(2,R)/{±I}, consisting of the matrices in SL(2,R) modulo plus or minus the identity matrix.
• The group PS^*L(2,R) = S^*L(2,R)/{±I}=PGL(2,R) is again a projective group, and again, modulo plus or minus the identity matrix. PSL(2,R) is contained as an index-two normal subgroup, the other coset being the set of 2×2 matrices with real entries whose determinant equals −1, modulo plus or minus the identity.

The relationship of these groups to the Poincaré model is as follows:

• The group of all isometries
  Isometry
  In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
  
   of H, sometimes denoted as Isom(H), is isomorphic to PS^*L(2,R). This includes both the orientation preserving and the orientation-reversing isometries. The orientation-reversing map (the mirror map) is .
• The group of orientation-preserving isometries of H, sometimes denoted as Isom⁺(H), is isomorphic to PSL(2,R).

Important subgroups of the isometry group are the Fuchsian group

Fuchsian group

In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...

s.

One also frequently sees the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

 SL(2,Z). This group is important in two ways. First, it is a symmetry group of the square 2x2 lattice

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

 of points. Thus, functions that are periodic on a square grid, such as modular form

Modular form

In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

s and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid. Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area.

Isometric symmetry

The group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of the projective special linear group PSL(2,R) on H is defined by


Note that the action is transitive, in that for any , there exists a such that . It is also faithful, in that if for all z in H, then g=e.

The stabilizer

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 or isotropy subgroup of an element z in H is the set of which leave z unchanged: gz=z. The stabilizer of i is the rotation group


Since any element z in H is mapped to i by an element of PSL(2,R), this means that the isotropy subgroup of any z is isomorphic to SO(2). Thus, H = PSL(2,R)/SO(2). Alternatively, the bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

 of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle

Unit tangent bundle

In Riemannian geometry, a branch of mathematics, the unit tangent bundle of a Riemannian manifold , denoted by UT or simply UTM, is the unit sphere bundle for the tangent bundle T...

, is isomorphic to PSL(2,R).

The upper half-plane is tessellated into free regular set

Free regular set

In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action.To be more precise, let X be a topological space. Let G be a group of homeomorphisms from X to X...

s by the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

 SL(2,Z).

Geodesics

The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.

The unit-speed geodesic going up vertically, through the point i is given by


Because PSL(2,R) acts transitively by isometries of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL(2,R). Thus, the general unit-speed geodesic is given by


This provides the complete description of the geodesic flow on the unit-length tangent bundle (complex line bundle

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...

) on the upper half-plane.