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Journal articles on the topic 'Hyperbolic Geometry'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid." American Mathematical Monthly 100, no. 5 (1993): 442. http://dx.doi.org/10.2307/2324297.

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2

Reynolds, William F. "Hyperbolic Geometry on a Hyperboloid." American Mathematical Monthly 100, no. 5 (1993): 442–55. http://dx.doi.org/10.1080/00029890.1993.11990430.

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3

Kostin, A. V., and I. Kh Sabitov. "Smarandache Theorem in Hyperbolic Geometry." Zurnal matematiceskoj fiziki, analiza, geometrii 10, no. 2 (2014): 221–32. http://dx.doi.org/10.15407/mag10.02.221.

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4

Li, Hongbo. "Hyperbolic geometry with geometric algebra." Chinese Science Bulletin 42, no. 3 (1997): 262–63. http://dx.doi.org/10.1007/bf02882454.

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5

Lord, Nick, and B. Iversen. "Hyperbolic Geometry." Mathematical Gazette 79, no. 486 (1995): 622. http://dx.doi.org/10.2307/3618121.

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6

Ungar, Abraham A. "Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry." Symmetry 15, no. 3 (2023): 649. http://dx.doi.org/10.3390/sym15030649.

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Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achiev

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7

Baseilhac, Stephane, and Riccardo Benedetti. "Quantum hyperbolic geometry." Algebraic & Geometric Topology 7, no. 2 (2007): 845–917. http://dx.doi.org/10.2140/agt.2007.7.845.

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8

Eppstein, David, and Michael T. Goodrich. "Succinct Greedy Geometric Routing Using Hyperbolic Geometry." IEEE Transactions on Computers 60, no. 11 (2011): 1571–80. http://dx.doi.org/10.1109/tc.2010.257.

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9

Eskew, Russell Clark. "Suspending the Principle of Relativity." Applied Physics Research 8, no. 2 (2016): 82. http://dx.doi.org/10.5539/apr.v8n2p82.

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A unique hyperbolic geometry paradigm requires suspending the Relativistic principle that absolute velocity is unmeasurable. The idea that of two observers each sees the same constant velocity, therefore there is no absolute velocity, is true only because Relativity uses a particular Lorentz geometry. Our mathematical geometry constructs circle and hyperbola vectors with hyperbolic terms in an original formulation of complex numbers. We use a point on a hyperbola as a frame of reference. A theory of time is given. The physical laws of motion by Galileo, Newton and Einstein are forged using the

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10

Eskew, Russell Clark. "Altering the Principle of Relativity." Applied Physics Research 10, no. 3 (2018): 21. http://dx.doi.org/10.5539/apr.v10n3p23.

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A unique hyperbolic geometry paradigm requires altering the Relativistic principle that absolute velocity is unmeasurable. There is no absolute velocity, but in the case where a constant velocity is made from a half-angle velocity, a variable velocity is the same as (absolute) acceleration. Relativity is based on local Lorentz geometry. Our mathematical geometry constructs circle and hyperbola vectors with hyperbolic terms in an original formulation of complex numbers. We use a point on a hyperbola as a frame of reference. A theory is given that time and our velocity are inversely related. The

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11

Lohkamp, Joachim. "Hyperbolic Unfoldings of Minimal Hypersurfaces." Analysis and Geometry in Metric Spaces 6, no. 1 (2018): 96–128. http://dx.doi.org/10.1515/agms-2018-0006.

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Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new conc

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12

Kemper, Matthias, and Joachim Lohkamp. "Potential Theory on Gromov Hyperbolic Spaces." Analysis and Geometry in Metric Spaces 10, no. 1 (2022): 394–431. http://dx.doi.org/10.1515/agms-2022-0147.

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Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geom

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13

Bao, Hui, and Xingdi Chen. "A New Hyperbolic Area Formula of a Hyperbolic Triangle and Its Applications." Journal of Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/838497.

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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.

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14

Zeitler, H. "Packings in hyperbolic geometry." Teaching Mathematics and Computer Science 2, no. 2 (2015): 209–29. http://dx.doi.org/10.5485/tmcs.2004.0040.

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15

Crilly, Tony, John Stillwell, and Michael Henle. "Sources of Hyperbolic Geometry." Mathematical Gazette 83, no. 497 (1999): 357. http://dx.doi.org/10.2307/3619103.

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16

Guillopé, Laurent. "Hyperfunctions in hyperbolic geometry." Complex Variables and Elliptic Equations 59, no. 11 (2013): 1559–71. http://dx.doi.org/10.1080/17476933.2013.805412.

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17

Ratcliffe, John G., Arlan Ramsay, and Robert D. Richtmyer. "Introduction to Hyperbolic Geometry." American Mathematical Monthly 103, no. 2 (1996): 185. http://dx.doi.org/10.2307/2975123.

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18

Gehring, F. W., and K. Hag. "Hyperbolic geometry and disks." Journal of Computational and Applied Mathematics 105, no. 1-2 (1999): 275–84. http://dx.doi.org/10.1016/s0377-0427(99)00016-3.

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19

Nash, C. "Geometry of hyperbolic monopoles." Journal of Mathematical Physics 27, no. 8 (1986): 2160–64. http://dx.doi.org/10.1063/1.526985.

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20

Horváth, Ákos G. "Hyperbolic plane geometry revisited." Journal of Geometry 106, no. 2 (2014): 341–62. http://dx.doi.org/10.1007/s00022-014-0252-0.

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21

Prishlyak, Oleksandr. "Regular Octagons in Hyperbolic Geometry." In the world of mathematics, no. 1 (2) (2024): 88–103. https://doi.org/10.17721/1029-4171.2024/2.10.

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When constructing hyperbolic structures on closed surfaces, one can use hyperbolic geometry (Lobachevsky geometry) on the plane. To do this, the surface must be represented as a 2n-gon on the hyperbolic plane, and a discrete group action, which is a subgroup of the movements of the hyperbolic plane, must be defined, for which the 2n-gon serves as a fundamental domain. If such a surface is a double torus (an oriented surface of genus 2), it can be obtained by gluing opposite sides of an octagon. In fact, the Lobachevsky plane is divided into octagons. The presence of symmetries simplifies calcu

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22

Knieper, Gerhard, Leonid Polterovich, and Leonid Potyagailo. "Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry." Oberwolfach Reports 9, no. 3 (2012): 2139–203. http://dx.doi.org/10.4171/owr/2012/35.

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23

Gallego, Eduardo, and Gil Solanes. "Integral geometry and geometric inequalities in hyperbolic space." Differential Geometry and its Applications 22, no. 3 (2005): 315–25. http://dx.doi.org/10.1016/j.difgeo.2005.01.006.

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24

Ungar, Abraham A. "The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry." Symmetry 15, no. 8 (2023): 1487. http://dx.doi.org/10.3390/sym15081487.

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Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

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25

Ungar, Abraham A. "When Four Cyclic Antipodal Pairs Are Ordered Counterclockwise in Euclidean and Hyperbolic Geometry." Symmetry 16, no. 6 (2024): 729. http://dx.doi.org/10.3390/sym16060729.

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A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) the relativistic model of hyperbolic geometry (also known as the Klein model)—motivates the study of four cyclic antipodal pairs of a circle, ordered arbitrarily counterclockwise. The translation of results from Euclidean geometry into hyperbolic geometry is obtained by means of hyperbolic trigonometry, called gyrotrigonometry, to which Einstein addition gives

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26

Zenginoğlu, Anıl. "Hyperbolic times in Minkowski space." American Journal of Physics 92, no. 12 (2024): 965–74. http://dx.doi.org/10.1119/5.0214271.

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Time functions with asymptotically hyperbolic geometry play an increasingly important role in many areas of relativity, from computing black hole perturbations to analyzing wave equations. Despite their significance, many of their properties remain underexplored. In this expository article, I discuss hyperbolic time functions by considering the hyperbola as the relativistic analog of a circle in two-dimensional Minkowski space and argue that suitably defined hyperboloidal coordinates are as natural in Lorentzian manifolds as spherical coordinates are in Riemannian manifolds.

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27

Mochizuki, Shinichi. "Galois Sections in Absolute Anabelian Geometry." Nagoya Mathematical Journal 179 (2005): 17–45. http://dx.doi.org/10.1017/s0027763000025599.

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AbstractWe show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structu

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28

IZUMIYA, SHYUICHI, DONGHE PEI, and TAKASI SANO. "SINGULARITIES OF HYPERBOLIC GAUSS MAPS." Proceedings of the London Mathematical Society 86, no. 2 (2003): 485–512. http://dx.doi.org/10.1112/s0024611502013850.

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In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the

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29

Long, D. D., and A. W. Reid. "Virtually spinning hyperbolic manifolds." Proceedings of the Edinburgh Mathematical Society 63, no. 2 (2019): 305–13. http://dx.doi.org/10.1017/s0013091519000324.

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AbstractWe give a new proof of a result of Sullivan [Hyperbolic geometry and homeomorphisms, in Geometric topology (ed. J. C. Cantrell), pp. 543–555 (Academic Press, New York, 1979)] establishing that all finite volume hyperbolic n-manifolds have a finite cover admitting a spin structure. In addition, in all dimensions greater than or equal to 5, we give the first examples of finite-volume hyperbolic n-manifolds that do not admit a spin structure.

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30

Hongnan Chen. "A Steward inequality in the Hyperbolic Geometry." Journal of Electrical Systems 20, no. 10s (2024): 2326–33. http://dx.doi.org/10.52783/jes.5579.

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The main aim of this study is to show that the steward inequality holds for any hyperbolic domain and distinct points , where denote the hyperbolic metric in and be a hyperbolic geodesic triangle with vertices and sides and be a point on .

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31

Manin, Yu I. "Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry." Inventiones mathematicae 104, no. 1 (1991): 223–43. http://dx.doi.org/10.1007/bf01245074.

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32

Guo, Ren. "A Characterization of Hyperbolic Geometry among Hilbert Geometry." Journal of Geometry 89, no. 1-2 (2008): 48–52. http://dx.doi.org/10.1007/s00022-008-1989-0.

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33

Pratama, Febriyana Putra, and Julan Hernadi. "KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE." EDUPEDIA 2, no. 2 (2018): 161. http://dx.doi.org/10.24269/ed.v2i2.148.

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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 refer

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34

PLATIS, IOANNIS D. "The geometry of complex hyperbolic packs." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 1 (2009): 205–34. http://dx.doi.org/10.1017/s0305004109002333.

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AbstractComplex hyperbolic packs are hypersurfaces of complex hyperbolic planeH2ℂwhich may be considered as dual to the well known bisectors. In this paper we study the geometric aspects associated to packs.

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35

Girsh, A. "New Problems of Descriptive Geometry. Continuation." Geometry & Graphics 9, no. 4 (2022): 3–10. http://dx.doi.org/10.12737/2308-4898-2022-9-4-3-10.

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Complex geometry is a synthesis of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually defines a non-closed system in which a correctly posed problem may not give a solution. Analytical geometry represents a closed system. In it, a correctly posed problem always gives solutions in the form of complex numbers, for each of which, one of the parts may be equal to zero. Finding imaginary solutions and imaginary figures formed by a set of such solutions is a new problem in descriptive geometry. Degenerated conics and quadrics, or cu

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36

Zhou, Yuansheng, and Tatyana O. Sharpee. "Hyperbolic geometry of gene expression." iScience 24, no. 3 (2021): 102225. http://dx.doi.org/10.1016/j.isci.2021.102225.

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37

Fei, Bao Jun, and Zhi Gang Li. "Relativistic Velocity and Hyperbolic Geometry." Physics Essays 10, no. 2 (1997): 248–55. http://dx.doi.org/10.4006/1.3028713.

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38

Mejia, Diego, and C. David Minda. "Hyperbolic geometry ink-convex regions." Pacific Journal of Mathematics 141, no. 2 (1990): 333–54. http://dx.doi.org/10.2140/pjm.1990.141.333.

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39

Shenitzer, Abe. "How Hyperbolic Geometry Became Respectable." American Mathematical Monthly 101, no. 5 (1994): 464. http://dx.doi.org/10.2307/2974912.

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40

Farup, Ivar. "Hyperbolic geometry for colour metrics." Optics Express 22, no. 10 (2014): 12369. http://dx.doi.org/10.1364/oe.22.012369.

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41

KIM, INKANG, and JOHN R. PARKER. "Geometry of quaternionic hyperbolic manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 135, no. 2 (2003): 291–320. http://dx.doi.org/10.1017/s030500410300687x.

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42

Bal, Guillaume. "Ray transforms in hyperbolic geometry." Journal de Mathématiques Pures et Appliquées 84, no. 10 (2005): 1362–92. http://dx.doi.org/10.1016/j.matpur.2005.02.001.

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43

Marshall, T. H. "Hyperbolic geometry and reflection groups." Bulletin of the Australian Mathematical Society 53, no. 1 (1996): 173–74. http://dx.doi.org/10.1017/s0004972700016853.

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44

Gramlich, Ralf. "On the hyperbolic symplectic geometry." Journal of Combinatorial Theory, Series A 105, no. 1 (2004): 97–110. http://dx.doi.org/10.1016/j.jcta.2003.10.005.

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45

Wildberger, N. J. "Universal hyperbolic geometry I: trigonometry." Geometriae Dedicata 163, no. 1 (2012): 215–74. http://dx.doi.org/10.1007/s10711-012-9746-9.

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46

Ruggiero, Rafael O. "Expansive dynamics and hyperbolic geometry." Boletim da Sociedade Brasileira de Matem�tica 25, no. 2 (1994): 139–72. http://dx.doi.org/10.1007/bf01321305.

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47

Bielawski, Roger, and Lorenz Schwachhöfer. "Pluricomplex Geometry and Hyperbolic Monopoles." Communications in Mathematical Physics 323, no. 1 (2013): 1–34. http://dx.doi.org/10.1007/s00220-013-1761-7.

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48

Shenitzer, Abe. "How Hyperbolic Geometry Became Respectable." American Mathematical Monthly 101, no. 5 (1994): 464–70. http://dx.doi.org/10.1080/00029890.1994.11996976.

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49

Corlette, Kevin. "Archimedean Superrigidity and Hyperbolic Geometry." Annals of Mathematics 135, no. 1 (1992): 165. http://dx.doi.org/10.2307/2946567.

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50

Popescu, Gelu. "Hyperbolic geometry on noncommutative balls." Documenta Mathematica 14 (2009): 595–651. http://dx.doi.org/10.4171/dm/283.

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