Poincaré half-plane model


In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part is positive.
The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.
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.
The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model.
This model can be generalized to model an dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.

Metric

The metric of the model on the half-plane, is:
where s measures the length along a line.
The straight lines in the hyperbolic plane are represented in this model by circular arcs perpendicular to the x-axis and straight vertical rays perpendicular to the x-axis.

Distance calculation

In general, the distance between two points measured in this metric along such a geodesic is:
where arcosh and arsinh are inverse hyperbolic functions
Some special cases can be simplified:
Another way to calculate the distance between two points that are on a half circle is:
where are the points where the halfcircles meet the boundary line and is the euclidean length of the line segment connecting the points P and Q in the model.

Special points and curves

A Euclidean circle with center and radius represents:
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

Draw the radial line 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.
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.
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 center is the point where h and p intersect.

Other constructions

Find the intersection of the two given semicircles.
Find the intersection of the given semicircle with the given circle.
Find the intersection of the two given circles.

Symmetry groups

The projective linear group PGL acts on the Riemann sphere by the Möbius transformations. The subgroup that maps the upper half-plane, H, onto itself is PSL, the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space.
There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.
The relationship of these groups to the Poincaré model is as follows:
Important subgroups of the isometry group are the Fuchsian groups.
One also frequently sees the modular group SL. 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 forms and elliptic functions, will thus inherit an SL symmetry from the grid. Second, SL is of course a subgroup of SL, and thus has a hyperbolic behavior embedded in it. In particular, SL can be used to tessellate the hyperbolic plane into cells of equal area.

Isometric symmetry

The group action of the projective special linear group on is defined by
Note that the action is transitive: for any, there exists a such that. It is also faithful, in that if for all then g = e.
The stabilizer or isotropy subgroup of an element is the set of which leave z unchanged: gz = z. The stabilizer of i is the rotation group
Since any element is mapped to i by some element of, this means that the isotropy subgroup of any z is isomorphic to SO. Thus,. Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to.
The upper half-plane is tessellated into free regular sets by the modular group

Geodesics

The geodesics for this metric tensor are circular arcs perpendicular to 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 acts transitively by isometries of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL. Thus, the general unit-speed geodesic is given by
This provides a basic description of the geodesic flow on the unit-length tangent bundle on the upper half-plane. Starting with this model, one can obtain the flow on arbitrary Riemann surfaces, as described in the article on the Anosov flow.

The model in three dimensions

The metric of the model on the half- space
is given by
where s measures length along a possibly curved line.
The straight lines in the hyperbolic space are represented in this model by circular arcs normal to the z = 0-plane and straight vertical rays normal to the z = 0-plane.
The distance between two points measured in this metric along such a geodesic is:

The model in ''n'' dimensions

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