1 Hyperbolic Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Einstein's Velocity Addition Law and Its Hyperbolic Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 53 (2007) 1228–1250 www.elsevier.com/locate/camwa Einstein’s velocity addition law and its hyperbolic geometry Abraham A. Ungar∗ Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA Received 6 March 2006; accepted 19 May 2006 Abstract Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the introduction of these concepts into hyperbolic geometry leads to hyperbolic vector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. c 2007 Elsevier Ltd. All rights reserved. Keywords: Special relativity; Einstein’s velocity addition law; Thomas precession; Hyperbolic geometry; Hyperbolic trigonometry 1. Introduction The hyperbolic law of cosines is nearly a century old result that has sprung from the soil of Einstein’s velocity addition law that Einstein introduced in his 1905 paper [1,2] that founded the special theory of relativity. It was established by Sommerfeld (1868–1951) in 1909 [3] in terms of hyperbolic trigonometric functions as a consequence of Einstein’s velocity addition of relativistically admissible velocities. Soon after, Varicakˇ (1865–1942) established in 1912 [4] the interpretation of Sommerfeld’s consequence in the hyperbolic geometry of Bolyai and Lobachevski. -
Hyperbolic Geometry
Hyperbolic Geometry David Gu Yau Mathematics Science Center Tsinghua University Computer Science Department Stony Brook University [email protected] September 12, 2020 David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 1 / 65 Uniformization Figure: Closed surface uniformization. David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 2 / 65 Hyperbolic Structure Fundamental Group Suppose (S; g) is a closed high genus surface g > 1. The fundamental group is π1(S; q), represented as −1 −1 −1 −1 π1(S; q) = a1; b1; a2; b2; ; ag ; bg a1b1a b ag bg a b : h ··· j 1 1 ··· g g i Universal Covering Space universal covering space of S is S~, the projection map is p : S~ S.A ! deck transformation is an automorphism of S~, ' : S~ S~, p ' = '. All the deck transformations form the Deck transformation! group◦ DeckS~. ' Deck(S~), choose a pointq ~ S~, andγ ~ S~ connectsq ~ and '(~q). The 2 2 ⊂ projection γ = p(~γ) is a loop on S, then we obtain an isomorphism: Deck(S~) π1(S; q);' [γ] ! 7! David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 3 / 65 Hyperbolic Structure Uniformization The uniformization metric is ¯g = e2ug, such that the K¯ 1 everywhere. ≡ − 2 Then (S~; ¯g) can be isometrically embedded on the hyperbolic plane H . The On the hyperbolic plane, all the Deck transformations are isometric transformations, Deck(S~) becomes the so-called Fuchsian group, −1 −1 −1 −1 Fuchs(S) = α1; β1; α2; β2; ; αg ; βg α1β1α β αg βg α β : h ··· j 1 1 ··· g g i The Fuchsian group generators are global conformal invariants, and form the coordinates in Teichm¨ullerspace. -
Old-Fashioned Relativity & Relativistic Space-Time Coordinates
Relativistic Coordinates-Classic Approach 4.A.1 Appendix 4.A Relativistic Space-time Coordinates The nature of space-time coordinate transformation will be described here using a fictional spaceship traveling at half the speed of light past two lighthouses. In Fig. 4.A.1 the ship is just passing the Main Lighthouse as it blinks in response to a signal from the North lighthouse located at one light second (about 186,000 miles or EXACTLY 299,792,458 meters) above Main. (Such exactitude is the result of 1970-80 work by Ken Evenson's lab at NIST (National Institute of Standards and Technology in Boulder) and adopted by International Standards Committee in 1984.) Now the speed of light c is a constant by civil law as well as physical law! This came about because time and frequency measurement became so much more precise than distance measurement that it was decided to define the meter in terms of c. Fig. 4.A.1 Ship passing Main Lighthouse as it blinks at t=0. This arrangement is a simplified model for a 1Hz laser resonator. The two lighthouses use each other to maintain a strict one-second time period between blinks. And, strict it must be to do relativistic timing. (Even stricter than NIST is the universal agency BIGANN or Bureau of Intergalactic Aids to Navigation at Night.) The simulations shown here are done using RelativIt. Relativistic Coordinates-Classic Approach 4.A.2 Fig. 4.A.2 Main and North Lighthouses blink each other at precisely t=1. At p recisel y t=1 sec. -
Cross-Ratio Dynamics on Ideal Polygons
Cross-ratio dynamics on ideal polygons Maxim Arnold∗ Dmitry Fuchsy Ivan Izmestievz Serge Tabachnikovx Abstract Two ideal polygons, (p1; : : : ; pn) and (q1; : : : ; qn), in the hyperbolic plane or in hyperbolic space are said to be α-related if the cross-ratio [pi; pi+1; qi; qi+1] = α for all i (the vertices lie on the projective line, real or complex, respectively). For example, if α = 1, the respec- − tive sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of M¨obius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invari- ant Poisson structures, and show that these relations, with different values of the constants α, commute, in an appropriate sense. We inves- tigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many α-related polygons, and study the ideal polygons that are α-related to themselves (with a cyclic shift of the indices). Contents 1 Introduction 3 ∗Department of Mathematics, University of Texas, 800 West Campbell Road, Richard- son, TX 75080; [email protected] yDepartment of Mathematics, University of California, Davis, CA 95616; [email protected] zDepartment of Mathematics, University of Fribourg, Chemin du Mus´ee 23, CH-1700 Fribourg; [email protected] xDepartment of Mathematics, Pennsylvania State University, University Park, PA 16802; [email protected] 1 1.1 Motivation: iterations of evolutes . .3 1.2 Plan of the paper and main results . -
Hyperbolic Trigonometry in Two-Dimensional Space-Time Geometry
Hyperbolic trigonometry in two-dimensional space-time geometry F. Catoni, R. Cannata, V. Catoni, P. Zampetti ENEA; Centro Ricerche Casaccia; Via Anguillarese, 301; 00060 S.Maria di Galeria; Roma; Italy January 22, 2003 Summary.- By analogy with complex numbers, a system of hyperbolic numbers can be intro- duced in the same way: z = x + hy; h2 = 1 x, y R . As complex numbers are linked to the { ∈ } Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representa- tion, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalise the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry. PACS 03 30 - Special Relativity PACS 02.20. Hj - Classical groups and Geometries 1 Introduction Complex numbers are strictly related to the Euclidean geometry: indeed their invariant (the module) arXiv:math-ph/0508011v1 3 Aug 2005 is the same as the Pythagoric distance (Euclidean invariant) and their unimodular multiplicative group is the Euclidean rotation group. It is well known that these properties allow to use complex numbers for representing plane vectors. In the same way hyperbolic numbers, an extension of complex numbers [1, 2] defined as z = x + hy; h2 =1 x, y R , { ∈ } are strictly related to space-time geometry [2, 3, 4]. Indeed their square module given by1 z 2 = 2 2 | | zz˜ x y is the Lorentz invariant of two dimensional special relativity, and their unimodular multiplicative≡ − group is the special relativity Lorentz group [2]. -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. -
Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray
Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray Tevian Dray ([email protected]) received his B.S. in mathematics from MIT in 1976, his Ph.D. in mathematics from Berkeley in 1981, spent several years as a physics postdoc, and is now a professor of mathematics at Oregon State University. A Fellow of the American Physical Society for his early work in general relativity, his current research interests include the octonions as well as science education. He directs the Vector Calculus Bridge Project. (http://www.math.oregonstate.edu/bridge) Differentiating a polynomial is easy. To differentiate u2 with respect to u, start by computing d.u2/ D .u C du/2 − u2 D 2u du C du2; and then dropping the last term, an operation that can be justified in terms of limits. Differential notation, in general, can be regarded as a shorthand for a formal limit argument. Still more informally, one can argue that du is small compared to u, so that the last term can be ignored at the level of approximation needed. After dropping du2 and dividing by du, one obtains the derivative, namely d.u2/=du D 2u. Even if one regards this process as merely a heuristic procedure, it is a good one, as it always gives the correct answer for a polynomial. (Physicists are particularly good at knowing what approximations are appropriate in a given physical context. A physicist might describe du as being much smaller than the scale imposed by the physical situation, but not so small that quantum mechanics matters.) However, this procedure does not suffice for trigonometric functions. -
Cheryl Jaeger Balm Hyperbolic Function Project
Math 43 Fall 2016 Instructor: Cheryl Jaeger Balm Hyperbolic Function Project Circles are part of a family of curves called conics. The various conic sections can be derived by slicing a plane through a double cone. A hyperbola is a conic with two basic forms: y y 4 4 2 2 • (0; 1) (−1; 0) (1; 0) • • x x -4 -2 2 4 -4 -2 2 4 (0; −1) • -2 -2 -4 -4 x2 − y2 = 1 y2 − x2 = 1 x2 − y2 = 1 is the unit hyperbola. Hyperbolic Functions: Similar to how the trigonometric functions, cosine and sine, correspond to the x and y values of the unit circle (x2 + y2 = 1), there are hyperbolic functions, hyperbolic cosine (cosh) and hyperbolic sine (sinh), which correspond to the x and y values of the right side of the unit hyperbola (x2 − y2 = 1). y y -1 1 x x π π 3π 2π π π 3π 2π 2 2 2 2 -1 -1 cos x sin x y y 6 2 4 x 2 -2 2 • (0; 1) -2 x -2 2 cosh x sinh x Hyperbolic Angle: Just like how the argument for the trigonometric functions is an angle, the argument for the hyperbolic functions is something called a hyperbolic angle. Instead of being defined by arc length, the hyperbolic angle is defined by area. If that seems confusing, consider the area of the circular sector of the unit circle. The r2θ equation for the area of a circular sector is Area = 2 , so because the radius of the unit circle is 1, any sector of the unit circle will have an area equal to half of the sector's central θ angle, A = 2 . -
MATH32052 Hyperbolic Geometry
MATH32052 Hyperbolic Geometry Charles Walkden 12th January, 2019 MATH32052 Contents Contents 0 Preliminaries 3 1 Where we are going 6 2 Length and distance in hyperbolic geometry 13 3 Circles and lines, M¨obius transformations 18 4 M¨obius transformations and geodesics in H 23 5 More on the geodesics in H 26 6 The Poincar´edisc model 39 7 The Gauss-Bonnet Theorem 44 8 Hyperbolic triangles 52 9 Fixed points of M¨obius transformations 56 10 Classifying M¨obius transformations: conjugacy, trace, and applications to parabolic transformations 59 11 Classifying M¨obius transformations: hyperbolic and elliptic transforma- tions 62 12 Fuchsian groups 66 13 Fundamental domains 71 14 Dirichlet polygons: the construction 75 15 Dirichlet polygons: examples 79 16 Side-pairing transformations 84 17 Elliptic cycles 87 18 Generators and relations 92 19 Poincar´e’s Theorem: the case of no boundary vertices 97 20 Poincar´e’s Theorem: the case of boundary vertices 102 c The University of Manchester 1 MATH32052 Contents 21 The signature of a Fuchsian group 109 22 Existence of a Fuchsian group with a given signature 117 23 Where we could go next 123 24 All of the exercises 126 25 Solutions 138 c The University of Manchester 2 MATH32052 0. Preliminaries 0. Preliminaries 0.1 Contact details § The lecturer is Dr Charles Walkden, Room 2.241, Tel: 0161 27 55805, Email: [email protected]. My office hour is: WHEN?. If you want to see me at another time then please email me first to arrange a mutually convenient time. 0.2 Course structure § 0.2.1 MATH32052 § MATH32052 Hyperbolic Geoemtry is a 10 credit course. -
University Microfilms 300 North Zmb Road Ann Arbor, Michigan 40106 a Xerox Education Company 73-11,544
INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. -
Arxiv:1808.05573V2 [Math.GT] 13 May 2020 A.K.A
THE MAXIMAL INJECTIVITY RADIUS OF HYPERBOLIC SURFACES WITH GEODESIC BOUNDARY JASON DEBLOIS AND KIM ROMANELLI Abstract. We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function. The main results of this paper relate to maximal injectivity radius among hyperbolic surfaces with geodesic boundary. For a point p in the interior of a hyperbolic surface F , by the injectivity radius of F at p we mean the supremum injrad p(F ) of all r > 0 such that there is a locally isometric embedding of an open metric neighborhood { a disk { of radius r into F that takes 1 the disk's center to p. If F is complete and without boundary then injrad p(F ) = 2 sysp(F ) at p, where sysp(F ) is the systole of loops at p, the minimal length of a non-constant geodesic arc in F with both endpoints at p. But if F has boundary then injrad p(F ) is bounded above by the distance from p to the boundary, so it approaches 0 as p approaches @F (and we extend it continuously to @F as 0). On the other hand, sysp(F ) does not approach 0 as p @F . -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31, 1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a ThisworkwassupportedinpartbyTheGeometryCenter,UniversityofMinnesota,anSTC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. This has not been true of the mathematicians and physicists of our generation.