The Hyperbolic Pythagorean Theorem in the Poincare Disc Model of Hyperbolic Geometry

1999 ◽  
Vol 106 (8) ◽  
pp. 759 ◽  
Author(s):  
Abraham A. Ungar
Keyword(s):  
Hyperbolic Geometry ◽  
Pythagorean Theorem ◽  
Disc Model

scholarly journals The Formulas of Möbius-Bretschneider and Möbius-Cagnoli in the Poincaré Disc Model of Hyperbolic Geometry

2021 ◽  
Vol 32 (1) ◽  
pp. 31
Author(s):  
Gülcan Balakan ◽  
Oğuzhan Demirel
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this paper, we present two gyroarea formulas (Möbius-Bretschneider’s formula and Möbius-Cagnoli’s formula) for Möbius gyroquadrilaterals in the Poincaré disc model of hyperbolic geometry.

scholarly journals The hyperbolic geometry of financial networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Martin Keller-Ressel ◽  
Stephanie Nargang
Keyword(s):  
Hyperbolic Geometry ◽  
Structural Changes ◽  
Banking System ◽  
Financial Networks ◽  
Stress Tests ◽  
Disc Model ◽  
European Banking ◽  
European Banks ◽  
Hyperbolic Embedding ◽  
Embedding Methods

AbstractBased on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we construct networks of European banks and demonstrate that the latent geometry of these financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. Using two different hyperbolic embedding methods, hydra+ and Mercator, this allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincaré disc model of hyperbolic geometry. We show that the latent dimensions of ‘popularity’ and ‘similarity’ in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system, independent of the embedding method that is used. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability. Based on our analysis we argue that embeddings into hyperbolic geometry can be used to monitor structural change in financial networks and are able to distinguish between changes in systemic relevance and other (peripheral) structural changes.

7. Spheres and more

2020 ◽  
pp. 128-152
Author(s):  
Glen Van Brummelen
Keyword(s):  
Hyperbolic Geometry ◽  
Spherical Geometry ◽  
Pythagorean Theorem ◽  
Spherical Trigonometry ◽  
Spherical Triangles ◽  
The Times ◽  
The Impact

‘Spheres and more’ considers the ten formulas for right-angled spherical triangles (and how they can be generated), the spherical Pythagorean theorem, and Napier’s rules. Spherical trigonometry was intended originally for astronomers, but medieval Islamic scholars used it to predict the beginning of the sacred month of Ramadan and the times of the five daily prayers. The impact of spherical geometry on Euclid’s axioms resulting in two types of non-Euclidean geometries—elliptical geometry and hyperbolic geometry—is also considered.

scholarly journals The Beckman–Quarles Theorem in Hyperbolic Geometry

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Oğuzhan Demirel ◽  
Damla Topal ◽  
Leyla Aslan
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this paper, we present the counterpart of the Beckman–Quarles theorem in the Poincaré disc model of hyperbolic geometry to characterize the gyroisometries (hyperbolic isometries) with a single nonzero distance a ∈ 0,1 satisfying a 2 ∈ ℚ .

scholarly journals Mathieu’s theorem in the Poincar´e disc model of hyperbolic geometry

2011 ◽  
Vol 20 (1) ◽  
pp. 16-19
Author(s):  
CATALIN BARBU ◽  
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this note, we present a proof of Mathieu’s hyperbolic theorem in the Poincare disc model of hyperbolic geometry.

Testing the DISC-Model in Service Work: Predicting Emotional Exhaustion

2013 ◽  
Author(s):  
Natasja van Vegchel ◽  
Jan de Jonge ◽  
Christian Dormann ◽  
Wilmar Schaufeli
Keyword(s):  
Emotional Exhaustion ◽  
Service Work ◽  
Disc Model
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