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In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
Sep 5, 2021 · The Poincaré disk model for hyperbolic geometry is the pair (D, H) where D consists of all points z in C such that | z | < 1, and H consists of all Möbius transformations T for which T(D) = D. The set D is called the hyperbolic plane, and H is called the transformation group in hyperbolic geometry.
They are known as the Klein model, the Poincar ́e Disk model, and the Poincar ́e Half-Plane model. We will start with the Disk model and move to the Half-Plane model later. There are geometric “isomorphisms” between these models, it is just that some properties are easier to see in one model than the other.
May 11, 2024 · The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the ...
The Poincaré disk model for the hyperbolic plane. The second model that we use to represent the hyperbolic plane is called the Poincaré disk model, named after the great French mathematician, Henri Poincaré (1854 - 1912). This model is constructed starting from the previous one.
There are several standard ones but the only one we will work with is the Poincaré disk model for hyperbolic geometry: \(\mathcal{l}=(\rho, L, \varrho)\) where: \(\mathcal{P}\) , the set of points, is the set of interior points of a fixed Euclidean circle \(\gamma\) centered at \(\mathrm{O}\)
The Poincaré disk model for hyperbolic geometry is the pair \((\mathbb{D},{\cal H})\) where \(\mathbb{D}\) consists of all points \(z\) in \(\mathbb{C}\) such that \(|z| \lt 1\text{,}\) and \({\cal H}\) consists of all Möbius transformations \(T\) for which \(T(\mathbb{D}) = \mathbb{D}\text{.}\)
