Geometric Inversion, Cross Ratio, Projective Geometry and Poincaré Disk Model

2020 ◽  
pp. 39-64
Author(s):  
Wladimir-Georges Boskoff ◽  
Salvatore Capozziello
Keyword(s):  
Projective Geometry ◽  
Cross Ratio ◽  
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 Non-Euclidean Biosymmetries and Algebraic Harmony in Genomes of Higher and Lower Organisms

2021 ◽  
Vol 248 ◽  
pp. 01010
Author(s):  
Sergey Petoukhov ◽  
Elena Petukhova ◽  
Vitaly Svirin
Keyword(s):  
Projective Geometry ◽  
Double Helix ◽  
Nucleotide Sequences ◽  
Cross Ratio ◽  
Harmonic Progression ◽  
The Cross ◽  
Dna Double Helix ◽  
Cooperative Organization ◽  
Relationship Of ◽  
The Relationship

The article is devoted to the study of the relationship of non-Euclidean symmetries in inherited biostructures with algebraic features of information nucleotide sequences in DNA molecules in the genomes of eukaryotes and prokaryotes. These genomic sequences obey the universal hyperbolic rules of the oligomer cooperative organization, which are associated with the harmonic progression 1/1, 1/2, 1/3,.., 1/n. The progression has long been known and studied in various branches of mathematics and physics. Now it has manifested itself in genetic informatics. The performed analysis of the harmonic progression revealed its connection with the cross-ratio, which is the main invariant of projective geometry. This connection consists in the fact that the magnitude of the cross-ratio is the same and is equal to 4/3 for any four adjacent members of this progression. The long DNA nucleotide sequences have fractal-like structure with so called epi-chains, whose structures are also related to the harmonic progression and the projective-geometrical symmetries. The received results are related additionally to a consideration of DNA double helix as helical antenna. This fact of the connection of genetic informatics with the main invariant of projective geometry can be used to explain the implementation of some non-Euclidean symmetries in genetically inherited structures of living bodies.

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 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.

RECALCULATION OF THE LOADS CURRENT OF ACTIVE MULTI-PORT NETWORKS ON THE BASIS OF PROJECTIVE GEOMETRY

2013 ◽  
Vol 22 (05) ◽  
pp. 1350031 ◽  
Author(s):  
A. A. PENIN
Keyword(s):  
Projective Geometry ◽  
Direct Current ◽  
Power Supply ◽  
Power Supply System ◽  
Traditional Approach ◽  
Cross Ratio ◽  
Supply System ◽  
The Cross ◽  
The Given

For an active multi-port network of a direct current, as a model of a power supply system, the problem of recalculation of the changeable loads currents is considered. The approach on the basis of projective geometry is used for interpretation of changes or "kinematics" of regime parameters of a circuit. The changes of the regime parameters are introduced otherwise, through the cross ratio of four points with use of projective coordinates. Easy-to-use formulas of the recalculation of the currents, which possess the group properties at change of conductivity of the loads, are obtained. It allows expressing the final values of the currents through the intermediate changes of the currents and conductivities. Disadvantages of the traditional approach, which uses the changes of resistance in the form of increments, are shown. The given approach is applicable to the analysis of "flowed" form processes of the various physical natures.

Sign in / Sign up

Export Citation Format

Share Document