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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 hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...
There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models ...
The order 7-3 rhombic tiling shown in a portion of the band model. The band model is a conformal model of the hyperbolic plane. The band model employs a portion of the Euclidean plane between two parallel lines. [1] Distance is preserved along one line through the middle of the band.
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 ...
Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic ...
In the Poincaré half-plane model one convenient choice is the portion of the half-plane with y ≥ 1. [7] Then the covering map is periodic in the x direction of period 2 π , and takes the horocycles y = c to the meridians of the pseudosphere and the vertical geodesics x = c to the tractrices that generate the pseudosphere.
Considering the action of SL(2, R) on the upper half-plane model H 2 of the hyperbolic plane by Möbius transformations, the affine Klein quartic can be realized as the quotient Γ(7)\H 2. (Here Γ(7) is the congruence subgroup of SL(2, Z) consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)
The proof is very simple: choose an homeomorphism () and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since is compact.. This result can be seen as the equivalence between two models for Teichmüller space of : the set of discrete faithful representations of the fundamental group () into () modulo conjugacy and the set of marked Riemann surfaces (,) where ...