Poincaré half-plane model | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-06-02

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model


00:01:51 1 Metric
00:03:19 1.1 Distance calculation
00:12:43 2 Special points and curves
00:15:15 2.1 Euclidean synopsis
00:17:42 3 Compass and straightedge constructions
00:18:14 3.1 Creating the line through two existing points
00:18:58 3.2 Creating the circle through one point with center another point
00:21:04 3.3 Given a circle find its (hyperbolic) center
00:21:44 3.4 Other constructions
00:22:27 4 Symmetry groups
00:26:07 5 Isometric symmetry
00:31:09 6 Geodesics
00:33:00 7 The model in three dimensions
00:34:29 8 The model in in/i dimensions
00:37:57 9 See also



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SUMMARY
=======
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H 
  
    
      
        {
        (
        x
        ,
        y
        )
        
          |
        
        y
        
        0
        ;
        x
        ,
        y
        ∈
        
          R
        
        }
      
    
    {\displaystyle \{(x,y)|y0;x,y\in \mathbb {R} \}}
   , 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 (due to Bernhard Riemann), 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 
  
    
      
        n
        +
        1
      
    
    {\displaystyle n+1}
   dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.