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In
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 ge ...
, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with coordinates whose coordinate is greater than zero, the
upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
, and a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
(definition of distance) called the Poincaré metric is adopted, in which the local scale is inversely proportional to the coordinate. Points on the -axis, whose coordinate is equal to zero, represent ideal points (points at infinity), which are outside the hyperbolic plane proper. Sometimes the points of the half-plane model are considered to lie in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
with positive
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. Using this interpretation, each point in the hyperbolic plane is associated with a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. The half-plane model can be thought of as a
map projection In cartography, a map projection is any of a broad set of Transformation (function) , transformations employed to represent the curved two-dimensional Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In a map projection, ...
from the curved hyperbolic plane to the flat Euclidean plane. From the
hyperboloid 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 points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
(a representation of the hyperbolic plane on a hyperboloid of two sheets embedded in 3-dimensional
Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...
, analogous to the
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
embedded in 3-dimensional Euclidean space), the half-plane model is obtained by
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Plane (mathematics), two dimensions. Orthographic projection is a form of parallel projection in ...
in a direction parallel to a
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...
, which can also be thought of as a kind of
stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
centered on an ideal point. The projection is conformal, meaning that it preserves angles, and like the stereographic projection of the sphere it projects generalized circles (
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s, hypercycles, horocycles, and circles) in the hyperbolic plane to generalized circles (lines or circles) in the plane. In particular, geodesics (analogous to straight lines), project to either half-circles whose center has coordinate zero, or to vertical straight lines of constant coordinate, hypercycles project to circles crossing the -axis, horocycles project to either circles tangent to the -axis or to horizontal lines of constant coordinate, and circles project to circles contained entirely in the half-plane. Hyperbolic motions, the distance-preserving
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function wh ...
s from the hyperbolic plane to itself, are represented in the Poincaré half-plane by the subset of
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s of the plane which preserve the half-plane; these are conformal, circle-preserving transformations which send the -axis to itself without changing its orientation. When points in the plane are taken to be complex numbers, any Möbius transformation is represented by a
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional t ...
of complex numbers, and the hyperbolic motions are represented by elements of the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
. The
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
provides an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
between the half-plane model and the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
, which is a stereographic projection of the hyperboloid centered on any ordinary point in the hyperbolic plane, which maps the hyperbolic plane onto a disk in the Euclidean plane, and also shares the properties of conformality and mapping generalized circles to generalized circles. The Poincaré half-plane model is named after
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
, but it originated with Eugenio Beltrami who used it, along with the Klein model and the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
, to show that hyperbolic geometry was equiconsistent with
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. The half-plane model can be generalized to the Poincaré half-space model of -dimensional
hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...
by replacing the single coordinate by distinct coordinates.


Metric

The
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
of the model on the half-plane, \, is: :(d s)^2 = \frac where ''s'' measures the length along a (possibly curved) line. The ''straight lines'' in the hyperbolic plane (
geodesics In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to the ''x''-axis (half-circles whose centers are on the ''x''-axis) and straight vertical rays perpendicular to the ''x''-axis.


Distance calculation

If p_1 = \langle x_1, y_1 \rangle and p_2 = \langle x_2, y_2 \rangle are two points in the half-plane y > 0 and \tilde p_1 = \langle x_1, -y_1 \rangle is the reflection of p_1 across the ''x''-axis into the lower half plane, the ''distance'' between the two points under the hyperbolic-plane metric is: \begin \operatorname (p_1, p_2) &= 2\operatorname \frac \\ 0mu&= 2\operatorname \frac \\ 0mu&= 2\ln \frac , \end where \, p_2 - p_1\, = \sqrt is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
between points and , \vphantom\Big) \operatornamex = \textstyle \ln \bigl(x + \sqrt \bigr) is the inverse hyperbolic sine, and \operatorname x \vphantom\big) = \tfrac12\ln\bigl((1+x)/(1-x)\bigr) is the inverse hyperbolic tangent. This 2\operatorname formula can be thought of as coming from the chord length in the Minkowski metric between points in the
hyperboloid 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 points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
, \operatorname(p_1,\, p_2) = \vphantom analogous to finding arclength on a sphere in terms of chord length. This 2\operatorname formula can be thought of as coming from Euclidean distance in the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
with one point at the origin, analogous to finding arclength on the sphere by taking a stereographic projection centered on one point and measuring the Euclidean distance in the plane from the origin to the other point. If the two points p_1 and p_2 are on a hyperbolic line (Euclidean half-circle) which intersects the ''x''-axis at the ideal points p_0 = \langle x_0, 0 \rangle and p_3 = \langle x_3, 0 \rangle, the distance from p_1 to p_2 is: \operatorname(p_1, p_2) = \left, \, \ln \frac \. Cf.
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 , , , on a line, their cross ratio is defin ...
. Some special cases can be simplified. Two points with the same coordinate: \bigl(\langle x, y_1 \rangle, \langle x, y_2 \rangle \bigr) =\left, \, \ln\frac \ = , \ln y_2 - \ln y_1 , . Two points with the same y coordinate: \bigl( \langle x_1,\, y \rangle, \langle x_2,\, y \rangle \bigr) = 2 \operatorname \frac. One point at the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...
of the semicircle (x - x_1)^2 + y^2 = r^2, and another point at a central angle of . \bigl( \langle x_1, r \rangle, \langle x_1 \pm r\sin\phi, r\cos\phi \rangle \bigr) = \bigl( \bigr) = \operatorname^ \phi, where is the inverse
Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...
.


Special points and curves

Ideal points (points at infinity) in the Poincaré half-plane model are of two kinds: the points on the -axis, and one imaginary point at y = \infty which is the ideal point to which all lines
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
to the -axis converge. Straight lines, geodesics (the shortest path between the points contained within it) are modeled by either half-circles whose origin is on the x-axis, or straight vertical rays orthogonal to the x-axis. A circle (curve equidistant from a central point) with center \langle x, y \rangle and radius r is modeled by a circle with center and radius . A hypercycle (a curve equidistant from a straight line, its axis) is modeled by either a circular arc which intersects the -axis at the same two ideal points as the half-circle which models its axis but at an acute or obtuse
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
, or a straight line which intersects the -axis at the same point as the vertical line which models its axis, but at an acute or obtuse
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
. A horocycle (a curve whose normals all converge asymptotically in the same direction, its center) is modeled by either a circle tangent to the -axis (but excluding the ideal point of intersection, which is its center), or a line parallel to the -axis, in which case the center is the ideal point at .


Euclidean synopsis

A Euclidean circle with center \langle x_e, y_e \rangle and radius r_e represents: * when the circle is completely inside the halfplane a hyperbolic circle with center \textstyle \big\langle x_e, \sqrt\, \big\rangle and radius \tfrac12 \ln \left( \frac \right). * when the circle is completely inside the halfplane and touches the boundary a horocycle centered around the ideal point (x_e, 0 ) * when the circle intersects the boundary
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
( y_e = 0 ) a hyperbolic line * when the circle intersects the boundary non- orthogonal a hypercycle.


Compass and straightedge constructions

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. 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

If the two points are not on a vertical line: 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. If the two given points lie on a vertical line and the given center is above the other given point: Draw a circle around the intersection of the vertical line and the ''x''-axis which passes through the given central point. Draw a horizontal line through the non-central point. Construct the tangent to the circle at its intersection with that 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. If the two given points lie on a vertical line and the given center is below the other given point: 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.


Given a circle find its (hyperbolic) center

Drop a perpendicular ''p'' from the Euclidean center of the circle to the ''x''-axis. Let point ''q'' be the intersection of this line and the ''x''- axis. Draw a line tangent to the circle going through ''q''. Draw the half circle ''h'' with center ''q'' going through the point where the tangent and the circle meet. The (hyperbolic) center is the point where ''h'' and ''p'' intersect.Flavors of Geometry, MSRI Publications, Volume 31, 1997, Hyperbolic Geometry, J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, page 87, Figure 19. Constructing the hyperbolic center of a circle


Other constructions

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 In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
PGL(2,C) acts on the Riemann sphere by the
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s. 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 In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
. There are four closely related
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance. * The
special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...
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 In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
PSL(2,R) = SL(2,R)/, 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)/=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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
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 z\rightarrow -\overline. * 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 groups. One also frequently sees the
modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
SL(2,Z). This group is important in two ways. First, it is a symmetry group of the square 2x2 lattice of points. Thus, functions that are periodic on a square grid, such as
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
s and
elliptic functions In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
, 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 In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
of the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
(2,\R) on \mathbb is defined by :\begina&b\\ c&d\\ \end \cdot z = \frac = \frac. Note that the action is transitive: for any z_1,z_2\in\mathbb, there exists a g\in (2,\R) such that gz_1=z_2. It is also faithful, in that if gz=z for all z\in\mathbb, then ''g'' = ''e''. The stabilizer or ''isotropy subgroup'' of an element z\in\mathbb is the set of g\in(2,\R) which leave ''z'' unchanged: ''gz'' = ''z''. The stabilizer of ''i'' is the rotation group :(2) = \left.\left\. Since any element z\in\mathbb is mapped to ''i'' by some element of (2, \R), this means that the isotropy subgroup of any ''z'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to SO(2). Thus, \mathbb=(2,\R)/(2). Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to (2, \R). The upper half-plane is tessellated into free regular sets by the
modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
(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 :\gamma(t) = \begine^&0\\ 0&e^\\ \end\cdot i = ie^t. 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 :\gamma(t)=\begina&b\\ c&d\\ \end \begine^&0\\0&e^\\ \end\cdot i = \frac . This provides a basic description of the geodesic flow on the unit-length tangent bundle (complex
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 organis ...
) on the upper half-plane. Starting with this model, one can obtain the flow on arbitrary
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, as described in the article on the
Anosov flow In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
.


The model in three dimensions

The
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
of the model on the half- space \ is given by (d s)^2 = \frac \, where ''s'' measures length along a possibly curved line. The ''straight lines'' in the hyperbolic space (
geodesics In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
for this metric tensor, i.e. curves which minimize the distance) are represented in this model by circular arcs normal to the ''z = 0''-plane (half-circles whose origin is on the ''z = 0''-plane) and straight vertical rays normal to the ''z = 0''-plane. The ''distance'' between two points p_1 = \langle x_1, y_1,z_1 \rangle and p_2 = \langle x_2, y_2, z_2 \rangle measured in this metric along such a geodesic is: \operatorname (p_1, p_2) = 2 \operatorname \frac.


The model in ''n'' dimensions

This model can be generalized to model an n+1 dimensional
hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...
by replacing the real number ''x'' by a vector in an ''n'' dimensional Euclidean vector space.


See also

*
Angle of parallelism In hyperbolic geometry, angle of parallelism \Pi(a) is the angle at the non-right angle vertex of a right hyperbolic triangle having two limiting parallel, asymptotic parallel sides. The angle depends on the segment length ''a'' between the ri ...
*
Anosov flow In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
* Fuchsian group * Fuchsian model * Hyperbolic motion * Kleinian model * Models of the hyperbolic plane *
Pseudosphere In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius is a surface in \mathbb^3 having Gaussian curvature, curvature −1/''R''2 at each point. Its name comes from the analogy with the sphere ...
* Schwarz–Ahlfors–Pick theorem * Ultraparallel theorem


References

;Notes ;Sources * Eugenio Beltrami, ''Teoria fondamentale degli spazi di curvatura constante'', Annali di Matematica Pura ed Applicata, ser II 2 (1868), 232–255 *
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
(1882) "Théorie des Groupes Fuchsiens", '' Acta Mathematica'' v.1, p. 1. First article in a series exploiting the half-plane model. A
archived copy
is freely available. On page 52 one can see an example of the semicircle diagrams so characteristic of the model. * Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'' (1980), Springer-Verlag, New York. . * Jürgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ''(See Section 2.3)''. * Saul Stahl, ''The Poincaré Half-Plane'', Jones and Bartlett, 1993, . * John Stillwell (1998) ''Numbers and Geometry'', pp. 100–104, Springer-Verlag, NY . An elementary introduction to the Poincaré half-plane model of the hyperbolic plane. {{DEFAULTSORT:Poincare half-plane model Conformal geometry Hyperbolic geometry Half-plane model