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Concept
Poincaré half-plane model
Summary
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 (the y coordinate mentioned above) 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.
The metric of the model on the half-plane, is:
where s measures the 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 centers are on the x-axis) and straight vertical rays perpendicular to the x-axis.
If and are two points in the half-plane and is the reflection of across the x-axis into the lower half plane, the distance between the two points under the hyperbolic-plane metric is:
where is the Euclidean distance between points and is the inverse hyperbolic sine, and is the inverse hyperbolic tangent. This 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.
Official source
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