The Poincaré disk model of hyperbolic geometry

2012 ◽  
pp. 125-142
Author(s):  
A. Sossinsky
Keyword(s):  
Hyperbolic Geometry ◽  
Disk Model ◽  
Poincaré Disk

scholarly journals KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE

EDUPEDIA ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 161
Author(s):  
Febriyana Putra Pratama ◽  
Julan Hernadi
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Cross Ratio ◽  
Literature Study ◽  
Disk Model ◽  
Non Euclidean Geometry ◽  
The Cross ◽  
Definition Of ◽  
Poincaré Disk

This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History)  by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of   . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from  to  is , where  is cross-ratio defined by  . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.

The Nature of Arguments Provided by College Geometry Students With Access to Technology While Solving Problems

2010 ◽  
Vol 41 (4) ◽  
pp. 324-350 ◽  
Author(s):  
Karen F. Hollebrands ◽  
AnnaMarie Conner ◽  
Ryan C. Smith
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Dynamic Geometry ◽  
Disk Model ◽  
Geometric Objects ◽  
Access To Technology ◽  
Poincaré Disk ◽  
Support Students

Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.

scholarly journals A hyperbolic variant of the Nelder–Mead simplex method in low dimensions

2013 ◽  
Vol 5 (2) ◽  
pp. 169-183 ◽  
Author(s):  
Levente Lócsi
Keyword(s):  
Simplex Method ◽  
Numerical Optimization ◽  
Optimization Method ◽  
Convergence Properties ◽  
Disk Model ◽  
Vast Number ◽  
Low Dimensions ◽  
Practical Applications ◽  
Matlab Implementation ◽  
Poincaré Disk

Abstract The Nelder-Mead simplex method is a widespread applied numerical optimization method with a vast number of practical applications, but very few mathematically proven convergence properties. The original formulation of the algorithm is stated in Rn using terms of Euclidean geometry. In this paper we introduce the idea of a hyperbolic variant of this algorithm using the Poincaré disk model of the Bolyai- Lobachevsky geometry. We present a few basic properties of this method and we also give a Matlab implementation in 2 and 3 dimensions

scholarly journals A novel way of constraining the α-attractor chaotic inflation through Planck data

2021 ◽  
Vol 2021 (11) ◽  
pp. 029
Author(s):  
Arunoday Sarkar ◽  
Chitrak Sarkar ◽  
Buddhadeb Ghosh
Keyword(s):  
Supergravity Models ◽  
Hyperbolic Geometry ◽  
Quantum Fluctuations ◽  
Cosmological Models ◽  
Planck Data ◽  
Model Potentials ◽  
First Order ◽  
Maximal Supergravity ◽  
Minimal Supergravity ◽  
Poincaré Disk

Abstract Defining a scale of k-modes of the quantum fluctuations during inflation through the dynamical horizon crossing condition k = aH we go from the physical t variable to k variable and solve the equations of cosmological first-order perturbations self consistently, with the chaotic α-attractor type potentials. This enables us to study the behaviour of ns , r, nt and N in the k-space. Comparison of our results in the low-k regime with the Planck data puts constraints on the values of the α parameter through microscopic calculations. Recent studies had already put model-dependent constraints on the values of α through the hyperbolic geometry of a Poincaré disk: consistent with both the maximal supergravity model 𝒩 = 8 and the minimal supergravity model 𝒩 = 1, the constraints on the values of α are 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3. The minimal 𝒩 = 1 supersymmetric cosmological models with B-mode targets, derived from these supergravity models, predicted the values of r between 10-2 and 10-3. Both in the E-model and the T-model potentials, we have obtained, in our calculations, the values of r in this range for all the constrained values of α stated above, within 68% CL. Moreover, we have calculated r for some other possible values of α both in low-α limit, using the formula r = 12α/N 2, and in the high-α limit, using the formula r = 4n/N, for n = 2 and 4. With all such values of α, our calculated results match with the Planck-2018 data with 68% or near 95% CL.

scholarly journals The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1019
Author(s):  
Frank Nielsen
Keyword(s):  
Convex Set ◽  
Geometric Algorithms ◽  
Siegel Disk ◽  
Algorithms And Data Structures ◽  
Lower And Upper Bounds ◽  
Disk Model ◽  
Hilbert Geometry ◽  
Core Set ◽  
Poincaré Disk ◽  
Bounded Convex

We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

scholarly journals A New Hyperbolic Area Formula of a Hyperbolic Triangle and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hui Bao ◽  
Xingdi Chen
Keyword(s):  
Hyperbolic Geometry ◽  
Area Formula ◽  
Poincaré Disk

We study some characterizations of hyperbolic geometry in the Poincaré disk. We first obtain the hyperbolic area and length formula of Euclidean disk and a circle represented by its Euclidean center and radius. Replacing interior angles with vertices coordinates, we also obtain a new hyperbolic area formula of a hyperbolic triangle. As its application, we give the hyperbolic area of a Lambert quadrilateral and some geometric characterizations of Lambert quadrilaterals and Saccheri quadrilaterals.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

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