What is the Poincare half-plane model in hyperbolic geometry?

What is the Poincaré half-plane model in hyperbolic geometry?

, 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.

What is a model of hyperbolic geometry?

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.

Are the Poincaré disk model and upper half-plane models of hyperbolic geometry isomorphic?

The fact that the disk model and the upper half-plane model of Hyperbolic Geometry are isomorphic, is usually proved through a formula using the Möbius transformation [1, p.

Which type of geometry uses a flat plane called a Poincaré disk?

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus …

What are examples of hyperbolic geometry?

The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood in two dimensions, but not in three dimensions. Hilbert extended the definition to general bounded sets in a Euclidean space.

Why is hyperbolic geometry called hyperbolic?

Why Call it Hyperbolic Geometry? The non-Euclidean geometry of Gauss, Lobachevski˘ı, and Bolyai is usually called hyperbolic geometry because of one of its very natural analytic models.

What is Poincare line?

How do you build a hyperbolic tile?

Make Hyperbolic Tilings of Images

  1. Choose an image-file from your computer.
  2. Choose a hyperbolic polygon with p vertices, click on a p.
  3. Choose the number of hyperbolic polygons adjacent to each vertex, click on a q.
  4. Move the image by dragging if you wish.
  5. Click on “generate tiling”.