Poincaré half-plane model

⁠ coordinate is greater than zero, the upper half-plane, and a metric tensor (definition of distance) called the Poincaré metric is adopted, in which the local scale is inversely proportional to the ⁠

Sometimes the points of the half-plane model are considered to lie in the complex plane with positive imaginary part.

From the hyperboloid model (a representation of the hyperbolic plane on a hyperboloid of two sheets embedded in 3-dimensional Minkowski space, analogous to the sphere embedded in 3-dimensional Euclidean space), the half-plane model is obtained by orthographic projection in a direction parallel to a null vector, which can also be thought of as a kind of stereographic projection centered on an ideal point.

In particular, geodesics (analogous to straight lines), project to either half-circles whose center has ⁠

When points in the plane are taken to be complex numbers, any Möbius transformation is represented by a linear fractional transformation of complex numbers, and the hyperbolic motions are represented by elements of the projective special linear group ⁠

The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model, 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.

across the x-axis into the lower half plane, the distance between the two points under the hyperbolic-plane metric is:

formula can be thought of as coming from the chord length in the Minkowski metric between points in the hyperboloid model,

, analogous to finding arclength on a sphere in terms of chord length.

formula can be thought of as coming from Euclidean distance in the Poincaré disk model 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.

are on a hyperbolic line (Euclidean half-circle) which intersects the x-axis at the ideal points

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 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, 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.

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 ⁠

[2] 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.

Draw the model circle around that new center and passing through the given non-central point.

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 line tangent to the circle which passes through the given non-central point.

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.

The projective linear group 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.

There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.

In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area.

The upper half-plane is tessellated into free regular sets by the modular group

The straight lines in the hyperbolic space (geodesics 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.