DOCSLIB.ORG

MATH32052 Hyperbolic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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. There will be about 22 lectures and a weekly examples class. The examples classes will start in Week 2. 0.2.2 Learning outcomes § On successfully completing the course you will be able to: ILO1 calculate the hyperbolic distance between and the geodesic through points in the hyperbolic plane, ILO2 compare different models (the upper half-plane model and the Poincar´e disc model) of hyperbolic geometry, ILO3 prove results (Gauss-Bonnet Theorem, angle formulæ for triangles, etc as listed in the syllabus) in hyperbolic trigonometry and use them to calcu- late angles, side lengths, hyperbolic areas, etc, of hyperbolic triangles and polygons, ILO4 classify M¨obius transformations in terms of their actions on the hyperbolic plane, ILO5 calculate a fundamental domain and a set of side-pairing transformations for a given Fuchsian group, ILO6 define a finitely presented group in terms of generators and relations, ILO7 use Poincar´e’s Theorem to construct examples of Fuchsian groups and calculate presentations in terms of generators and relations, ILO8 relate the signature of a Fuchsian group to the algebraic and geometric properties of the Fuchsian group and to the geometry of the corresponding hyperbolic surface. c The University of Manchester 2019 3 MATH32052 0. Preliminaries 0.2.3 Lecture notes § This file contains a complete set of lecture notes. The lecture notes contain more material than I present in the lectures. This allows me to expand on minor points for the interested student, present alternative explanations, etc. Only the material I cover in the lectures is examinable. The lecture notes are available on the course webpage. The course webpage is available via Blackboard or directly at www.maths.manchester.ac.uk/~cwalkden/complex-analysis. Please let me know of any mistakes or typos that you find in the notes. I will use the visualiser for the majority of the lectures. I will upload scanned copies of what I write on the visualiser onto the course webpage. I will normally upload these onto the course webpage within 3 working days of the lectures. 0.2.4 Exercises § The lectures also contain the exercises. For your convenience I’ve collated all the exercises into a single section at the end of the notes; here, I’ve indicated which exercises are partic- ularly important and which are there for completeness only. The exercises are a key part of the course. 0.2.5 Solutions to the exercises § This file contains the solutions to all of the exercises. I will trust you to have a serious attempt at the exercises before you refer to the solutions. 0.2.6 Support classes § The support classes are a key part of the course. I will try to make them as interactive as possible by getting you to revise material that will be useful in the course or getting you to work through some of the exercises, perhaps with additional hints. Towards the end of the course we will spend time doing past exam papers. You should consider attendance at the examples classes to be compulsory. The handouts in the support classes comprise of questions from the exercises and past exams. These handouts do not contain any material that is not already available within these notes or within the past exam papers on the course webpage; as such I will not be putting these handouts on the course webpage. 0.2.7 Lecture capture § The lectures will be recorded using the University’s lecture capture (‘podcast’) system. I will mostly use the visualiser. Each week, I will scan copies of the visualiser slides and put them on the course webpage. 0.3 Coursework and the exam § The coursework for this year will be a 40 minute closed-book test taking place during WHEN???? All questions on the test are compulsory and it will be in the format of an exam question. Thus, looking at past exam papers will provide excellent preparation for the test. You will need to know the material from sections 1–1 in the lecture notes for the test (this is the material that we will cover in weeks 1–5). c The University of Manchester 2019 4 MATH32052 0. Preliminaries Your coursework script, with feedback, will be returned to you within 15 working days of the test. You will be able to collect your script from the Teaching & Learning Office reception on the ground floor of the Alan Turing Building. The course is examined by a 2 hour written examination in May/June. The format of the exam will change this year: there will be 4 questions of which you have to answer 3. (If you attempt all 4 questions then only your best 3 answers will count.) Past exam papers still give a good guide as to what you could be asked to do. 0.4 Recommended texts § J. Anderson, Hyperbolic Geometry, 1st ed., Springer Undergraduate Mathematics Series, Springer-Verlag, Berlin, New York, 1999. S. Katok, Fuchsian Groups, Chicago Lecture Notes in Mathematics, Chicago University Press, 1992. A. Beardon, The Geometry of Discrete Groups, Springer-Verlag, Berlin, New York, 1983. The book by Anderson is the most suitable for the first half of the course. Katok’s book is probably the best source for the second half of the course. Beardon’s book contains everything in the course, and much more. You probably do not need to buy any book and can rely solely on the lecture notes. c The University of Manchester 2019 5 MATH32052 1. Where we are going 1. Where we are going 1.1 Introduction § One purpose of this course is to provide an introduction to some aspects of hyperbolic ge- ometry. Hyperbolic geometry is one of the richest areas of mathematics, with connections not only to geometry but to dynamical systems, chaos theory, number theory, relativity, and many other areas of mathematics and physics. Unfortunately, it would be impossible to discuss all of these aspects of hyperbolic geometry within the confines of a single lecture course. Instead, we will develop hyperbolic geometry in a way that emphasises the similar- ities and (more interestingly!) the many differences with Euclidean geometry (that is, the ‘real-world’ geometry that we are all familiar with). 1.2 Euclidean geometry § Euclidean geometry is the study of geometry in the Euclidean plane R2, or more generally in n-dimensional Euclidean space Rn. This is the geometry that we are familiar with from the real world. For example, in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides; this is Pythagoras’ Theorem. But what makes Euclidean geometry ‘Euclidean’? And what is ‘geometry’ anyway? One convenient meta-definition is due to Felix Klein (1849-1929) in his Erlangen programme (1872), which we paraphrase here: given a set with some structure and a group of trans- formations that preserve that structure, geometry is the study of objects that are invariant under these transformations. For 2-dimensional Euclidean geometry, the set is the plane R2 equipped with the Euclidean distance function (the normal way of defining the distance be- tween two points) together with a group of transformations (such as rotations, translations) that preserve the distance between points. We will define hyperbolic geometry in a similar way: we take a set, define a notion of distance on it, and study the transformations which preserve this distance. 1.3 Distance in the Euclidean plane § Consider the Euclidean plane R2. Take two points x,y R2. What do we mean by the ∈ distance between x and y? If x = (x1,x2) and y = (y1,y2) then one way of calculating the distance between x and y is by using Pythagoras’ Theorem: distance(x,y)= x y = (y x )2 + (y x )2; (1.3.1) k − k 1 − 1 2 − 2 this is the length of the straight line drawn in Figurep 1.3.1. Writing d(x,y) for distance(x,y) we can see that there are some natural properties satisfied by this formula for distance: (i) d(x,y) 0 for all x,y with equality if and only if x = y, ≥ (ii) d(x,y)= d(y,x) for all x,y, c The University of Manchester 2019 6 MATH32052 1.
Recommended publications
  • Analytic Geometry

    Analytic Geometry

    Guide Study Georgia End-Of-Course Tests Georgia ANALYTIC GEOMETRY TABLE OF CONTENTS INTRODUCTION ...........................................................................................................5 HOW TO USE THE STUDY GUIDE ................................................................................6 OVERVIEW OF THE EOCT .........................................................................................8 PREPARING FOR THE EOCT ......................................................................................9 Study Skills ........................................................................................................9 Time Management .....................................................................................10 Organization ...............................................................................................10 Active Participation ...................................................................................11 Test-Taking Strategies .....................................................................................11 Suggested Strategies to Prepare for the EOCT ..........................................12 Suggested Strategies the Day before the EOCT ........................................13 Suggested Strategies the Morning of the EOCT ........................................13 Top 10 Suggested Strategies during the EOCT .........................................14 TEST CONTENT ........................................................................................................15
  • Geometric Manifolds

    Geometric Manifolds

    Wintersemester 2015/2016 University of Heidelberg Geometric Structures on Manifolds Geometric Manifolds by Stephan Schmitt Contents Introduction, first Definitions and Results 1 Manifolds - The Group way .................................... 1 Geometric Structures ........................................ 2 The Developing Map and Completeness 4 An introductory discussion of the torus ............................. 4 Definition of the Developing map ................................. 6 Developing map and Manifolds, Completeness 10 Developing Manifolds ....................................... 10 some completeness results ..................................... 10 Some selected results 11 Discrete Groups .......................................... 11 Stephan Schmitt INTRODUCTION, FIRST DEFINITIONS AND RESULTS Introduction, first Definitions and Results Manifolds - The Group way The keystone of working mathematically in Differential Geometry, is the basic notion of a Manifold, when we usually talk about Manifolds we mean a Topological Space that, at least locally, looks just like Euclidean Space. The usual formalization of that Concept is well known, we take charts to ’map out’ the Manifold, in this paper, for sake of Convenience we will take a slightly different approach to formalize the Concept of ’locally euclidean’, to formulate it, we need some tools, let us introduce them now: Definition 1.1. Pseudogroups A pseudogroup on a topological space X is a set G of homeomorphisms between open sets of X satisfying the following conditions: • The Domains of the elements g 2 G cover X • The restriction of an element g 2 G to any open set contained in its Domain is also in G. • The Composition g1 ◦ g2 of two elements of G, when defined, is in G • The inverse of an Element of G is in G. • The property of being in G is local, that is, if g : U ! V is a homeomorphism between open sets of X and U is covered by open sets Uα such that each restriction gjUα is in G, then g 2 G Definition 1.2.
  • Black Hole Physics in Globally Hyperbolic Space-Times

    Black Hole Physics in Globally Hyperbolic Space-Times

    Pram[n,a, Vol. 18, No. $, May 1982, pp. 385-396. O Printed in India. Black hole physics in globally hyperbolic space-times P S JOSHI and J V NARLIKAR Tata Institute of Fundamental Research, Bombay 400 005, India MS received 13 July 1981; revised 16 February 1982 Abstract. The usual definition of a black hole is modified to make it applicable in a globallyhyperbolic space-time. It is shown that in a closed globallyhyperbolic universe the surface area of a black hole must eventuallydecrease. The implications of this breakdown of the black hole area theorem are discussed in the context of thermodynamics and cosmology. A modifieddefinition of surface gravity is also given for non-stationaryuniverses. The limitations of these concepts are illustrated by the explicit example of the Kerr-Vaidya metric. Keywocds. Black holes; general relativity; cosmology, 1. Introduction The basic laws of black hole physics are formulated in asymptotically flat space- times. The cosmological considerations on the other hand lead one to believe that the universe may not be asymptotically fiat. A realistic discussion of black hole physics must not therefore depend critically on the assumption of an asymptotically flat space-time. Rather it should take account of the global properties found in most of the widely discussed cosmological models like the Friedmann models or the more general Robertson-Walker space times. Global hyperbolicity is one such important property shared by the above cosmo- logical models. This property is essentially a precise formulation of classical deter- minism in a space-time and it removes several physically unreasonable pathological space-times from a discussion of what the large scale structure of the universe should be like (Penrose 1972).
  • Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances

    Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances

    Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent.
  • Möbius Transformations

    Möbius Transformations

    Möbius transformations Möbius transformations are simply the degree one rational maps of C: az + b σ : z 7! : ! A cz + d C C where ad − bc 6= 0 and a b A = c d Then A 7! σA : GL(2C) ! fMobius transformations g is a homomorphism whose kernel is ∗ fλI : λ 2 C g: The homomorphism is an isomorphism restricted to SL(2; C), the subgroup of matri- ces of determinant 1. We have an action of GL(2; C) on C by A:(B:z) = (AB):z; A; B 2 GL(2; C); z 2 C: The action of SL(2; R) preserves the upper half-plane fz 2 C : Im(z) > 0g and also R [ f1g and the lower half-plane. The action of the subgroup a b SU(1; 1) = : jaj2 − jbj2 = 1 b a preserves the open unit disc, the closed unit disc, and its exterior. All of these actions are transitive that is, for all z and w in the domain there is A in the group with A:z = w. Kleinian groups Definition 1 A Kleinian group is a subgroup Γ of P SL(2; C) which is discrete, that is, there is an open neighbourhood U ⊂ P SL(2; C) of the identity element I such that U \ Γ = fIg: Definition 2 A Fuchsian group is a discrete subgroup of P SL(2; R) Equivalently (as usual with topological groups) there is an open neighbourhood V of I such that γV \ γ0V = ; for all γ, γ0 2 Γ, γ 6= γ0. To get this, choose V with V = V −1 and V:V ⊂ U.
  • Lesson 3: Rectangles Inscribed in Circles

    Lesson 3: Rectangles Inscribed in Circles

    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection .
  • Squaring the Circle a Case Study in the History of Mathematics the Problem

    Squaring the Circle a Case Study in the History of Mathematics the Problem

    Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A.
  • Holley GM LS Race Single-Plane Intake Manifold Kits

    Holley GM LS Race Single-Plane Intake Manifold Kits

    Holley GM LS Race Single-Plane Intake Manifold Kits 300-255 / 300-255BK LS1/2/6 Port-EFI - w/ Fuel Rails 4150 Flange 300-256 / 300-256BK LS1/2/6 Carbureted/TB EFI 4150 Flange 300-290 / 300-290BK LS3/L92 Port-EFI - w/ Fuel Rails 4150 Flange 300-291 / 300-291BK LS3/L92 Carbureted/TB EFI 4150 Flange 300-294 / 300-294BK LS1/2/6 Port-EFI - w/ Fuel Rails 4500 Flange 300-295 / 300-295BK LS1/2/6 Carbureted/TB EFI 4500 Flange IMPORTANT: Before installation, please read these instructions completely. APPLICATIONS: The Holley LS Race single-plane intake manifolds are designed for GM LS Gen III and IV engines, utilized in numerous performance applications, and are intended for carbureted, throttle body EFI, or direct-port EFI setups. The LS Race single-plane intake manifolds are designed for hi-performance/racing engine applications, 5.3 to 6.2+ liter displacement, and maximum engine speeds of 6000-7000 rpm, depending on the engine combination. This single-plane design provides optimal performance across the RPM spectrum while providing maximum performance up to 7000 rpm. These intake manifolds are for use on non-emissions controlled applications only, and will not accept stock components and hardware. Port EFI versions may not be compatible with all throttle body linkages. When installing the throttle body, make certain there is a minimum of ¼” clearance between all linkage and the fuel rail. SPLIT DESIGN: The Holley LS Race manifold incorporates a split feature, which allows disassembly of the intake for direct access to internal plenum and port surfaces, making custom porting and matching a snap.
  • Hyperbolic Geometry

    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.
  • Handlebody Orbifolds and Schottky Uniformizations of Hyperbolic 2-Orbifolds

    Handlebody Orbifolds and Schottky Uniformizations of Hyperbolic 2-Orbifolds

    proceedings of the american mathematical society Volume 123, Number 12, December 1995 HANDLEBODY ORBIFOLDS AND SCHOTTKY UNIFORMIZATIONS OF HYPERBOLIC 2-ORBIFOLDS MARCO RENI AND BRUNO ZIMMERMANN (Communicated by Ronald J. Stern) Abstract. The retrosection theorem says that any hyperbolic or Riemann sur- face can be uniformized by a Schottky group. We generalize this theorem to the case of hyperbolic 2-orbifolds by giving necessary and sufficient conditions for a hyperbolic 2-orbifold, in terms of its signature, to admit a uniformization by a Kleinian group which is a finite extension of a Schottky group. Equiva- lent^, the conditions characterize those hyperbolic 2-orbifolds which occur as the boundary of a handlebody orbifold, that is, the quotient of a handlebody by a finite group action. 1. Introduction Let F be a Fuchsian group, that is a discrete group of Möbius transforma- tions—or equivalently, hyperbolic isometries—acting on the upper half space or the interior of the unit disk H2 (Poincaré model of the hyperbolic plane). We shall always assume that F has compact quotient tf = H2/F which is a hyperbolic 2-orbifold with signature (g;nx,...,nr); here g denotes the genus of the quotient which is a closed orientable surface, and nx, ... , nr are the orders of the branch points (the singular set of the orbifold) of the branched covering H2 -» H2/F. Then also F has signature (g;nx,... , nr), and (2 - 2g - £-=1(l - 1/«/)) < 0 (see [12] resp. [15] for information about orbifolds resp. Fuchsian groups). In case r — 0, or equivalently, if the Fuch- sian group F is without torsion, the quotient H2/F is a hyperbolic or Riemann surface without branch points.
  • And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent

    And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent

    9-3 Arcs and Chords ALGEBRA Find the value of x. 3. SOLUTION: 1. In the same circle or in congruent circles, two minor SOLUTION: arcs are congruent if and only if their corresponding Arc ST is a minor arc, so m(arc ST) is equal to the chords are congruent. Since m(arc AB) = m(arc CD) measure of its related central angle or 93. = 127, arc AB arc CD and . and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = 93. ANSWER: 93 ANSWER: 3 In , JK = 10 and . Find each measure. Round to the nearest hundredth. 2. SOLUTION: Since HG = 4 and FG = 4, and are 4. congruent chords and the corresponding arcs HG and FG are congruent. SOLUTION: m(arc HG) = m(arc FG) = x Radius is perpendicular to chord . So, by Arc HG, arc GF, and arc FH are adjacent arcs that Theorem 10.3, bisects arc JKL. Therefore, m(arc form the circle, so the sum of their measures is 360. JL) = m(arc LK). By substitution, m(arc JL) = or 67. ANSWER: 67 ANSWER: 70 eSolutions Manual - Powered by Cognero Page 1 9-3 Arcs and Chords 5. PQ ALGEBRA Find the value of x. SOLUTION: Draw radius and create right triangle PJQ. PM = 6 and since all radii of a circle are congruent, PJ = 6. Since the radius is perpendicular to , bisects by Theorem 10.3. So, JQ = (10) or 5. 7. Use the Pythagorean Theorem to find PQ.
  • Cross-Ratio Dynamics on Ideal Polygons

    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 .