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Unit disk - Wikipedia
Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term unit disk is used for the open unit disk about the origin , D 1 ( 0 ) {\displaystyle D_{1}(0)} , with respect to the standard ... See more
In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1:
$${\displaystyle D_{1}(P)=\{Q:\vert P-Q\vert <1\}.\,}$$ See moreOne also considers unit disks with respect to other metrics. For instance, with the taxicab metric and the Chebyshev metric disks look like squares (even though the underlying See more
• Weisstein, Eric W. "Unit disk". MathWorld.
• On the Perimeter and Area of the Unit Disc, by J.C. Álvarez Pavia and A.C. Thompson See moreThe open unit disk forms the set of points for the Poincaré disk model of the hyperbolic plane. Circular arcs perpendicular to … See more
Wikipedia text under CC-BY-SA license Poincaré disk model - Wikipedia
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.
The group of orientation preserving isometries of the disk model is given by th…Wikipedia · Text under CC-BY-SA license- Estimated Reading Time: 10 mins
Unit disk graph - Wikipedia
In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with …
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7. The Poincaré disk model for the hyperbolic plane
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 …
5.1: The Poincaré Disk Model - Mathematics LibreTexts
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.…
In order to give a model for H 2, we need to decide on a set of points, then determine what lines are and how to measure distance. For Poincar ́e’s Disk Model we take the set of points that lie …
Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in …
Unit Disk -- from Wolfram MathWorld
4 days ago · Unit Disk. A disk with radius 1. The (open) unit disk can also be considered to be the region in the complex plane defined by , where denotes the complex modulus. (The closed …
unit disk - PlanetMath.org
Mar 22, 2013 · unit disk. The unit disk in the complex plane, denoted Δ Δ, is defined as {z ∈C:|z| <1} { z ∈ ℂ: | z | < 1 }. The unit circle, denoted ∂Δ ∂. Δ or S1 S 1 is the boundary {z∈ C:|z|= 1} …
a disk of unit radius r = 1 in the plane, and is connected to all nodes within (or on the edge of) its corresponding disk. Unit disk graphs have proven to be useful in mod-eling various physical …
Automorphisms of the Unit Disk. Let D = fz : jzj < 1. We want to describe all conformal maps from D onto D. We will postpone doing this and instead describe all linear fractional transformations …
GCT The Poincaré Disk Model - mphitchman.com
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 …
Poincaré Hyperbolic Disk -- from Wolfram MathWorld
4 days ago · 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 …
Poincaré metric - Wikipedia
The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius …
Unit disk graph and physical interference model: Putting pieces ...
As a tractable mathematical object, the unit disk graph (UDG) is a popular model that has enabled the development of efficient algorithms for crucial networking problems. In a rho-UDG, two …
hyperbolic geometry - Measuring distance on the Poincare disk ...
Start with some basic definitions for the Poincaré disk model. A point of the hyperbolic plane is a point inside the unit disk. A hyperbolic line is a circle arc which is perpendicular to the unit disk.
5.5: The Upper Half-Plane Model - Mathematics LibreTexts
The upper half-plane model of hyperbolic geometry has space U consisting of all complex numbers z such that Im ( z) > 0, and transformation group U consisting of all Möbius …
Unit disk graphs - ScienceDirect
Dec 14, 1990 · Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some …
Distance between points in hyperbolic disk models
The correct distance formula in the Poincare disc is obtained by integrating the length element, also provided by @Post No Bulls, along Euclidean circles that hit the boundary of the Poincare …
2: Unit Disk Graph model for wireless communication networks.
A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks.
Isometry from Half-Plane metric to Poincare Disk metric
Jun 16, 2022 · If I take the metric of the upper half-plane model $$ (ds)^2=\frac{(dx)^2+(dy)^2}{y^2} $$ there is obviously an isometry which should provide the …
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