H.S.M. Coxeter, Angles and Arcs in the Hyperbolic Plane, P 17-34Mathschron009-004.Pdf
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Notes on Euclidean Geometry Kiran Kedlaya Based
Notes on Euclidean Geometry Kiran Kedlaya based on notes for the Math Olympiad Program (MOP) Version 1.0, last revised August 3, 1999 c Kiran S. Kedlaya. This is an unfinished manuscript distributed for personal use only. In particular, any publication of all or part of this manuscript without prior consent of the author is strictly prohibited. Please send all comments and corrections to the author at [email protected]. Thank you! Contents 1 Tricks of the trade 1 1.1 Slicing and dicing . 1 1.2 Angle chasing . 2 1.3 Sign conventions . 3 1.4 Working backward . 6 2 Concurrence and Collinearity 8 2.1 Concurrent lines: Ceva’s theorem . 8 2.2 Collinear points: Menelaos’ theorem . 10 2.3 Concurrent perpendiculars . 12 2.4 Additional problems . 13 3 Transformations 14 3.1 Rigid motions . 14 3.2 Homothety . 16 3.3 Spiral similarity . 17 3.4 Affine transformations . 19 4 Circular reasoning 21 4.1 Power of a point . 21 4.2 Radical axis . 22 4.3 The Pascal-Brianchon theorems . 24 4.4 Simson line . 25 4.5 Circle of Apollonius . 26 4.6 Additional problems . 27 5 Triangle trivia 28 5.1 Centroid . 28 5.2 Incenter and excenters . 28 5.3 Circumcenter and orthocenter . 30 i 5.4 Gergonne and Nagel points . 32 5.5 Isogonal conjugates . 32 5.6 Brocard points . 33 5.7 Miscellaneous . 34 6 Quadrilaterals 36 6.1 General quadrilaterals . 36 6.2 Cyclic quadrilaterals . 36 6.3 Circumscribed quadrilaterals . 38 6.4 Complete quadrilaterals . 39 7 Inversive Geometry 40 7.1 Inversion . -
On the Deformation of Inversive Distance Circle Packings, Ii
ON THE DEFORMATION OF INVERSIVE DISTANCE CIRCLE PACKINGS, II HUABIN GE, WENSHUAI JIANG ABSTRACT. We show that the results in [GJ16] are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo’s combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston’s theorem to some extent. 1. INTRODUCTION 1.1. Background. We continue our study about the deformation of inversive distance circle patterns on a surface M with triangulation T. If the triangulated surface is obtained by taking a finite collection of Eu- clidean triangles in E2 and identifying their edges in pairs by isometries, we call it in Euclidean background geometry. While if the triangulated surface is obtained by taking a finite collection of hyperbolic triangles in H2 and identifying their edges in pairs by isometries, we shall call it in hyperbolic background geometry. Consider a closed surface M with a triangulation T = fV; E; Fg, where V; E; F represent the sets of ver- tices, edges and faces respectively. Thurston [Th76] once introduced a metric structure on (M; T) by circle patterns. The idea is taking the triangulation as the nerve of a circle pattern, while the length structure on (M; T) can be constructed from radii of circles and intersection angles between circles in the pattern. More π precisely, let Φi j 2 [0; 2 ] be intersection angles between two circles ci, c j for each nerve fi jg 2 E, and let r 2 (0; +1) be the radius of circle c for each vertex i 2 V, see Figure 1.1. -
On a Combinatorial Curvature for Surfaces with Inversive Distance Circle Packing Metrics
On a combinatorial curvature for surfaces with inversive distance circle packing metrics Huabin Ge, Xu Xu May 30, 2018 Abstract In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton [21]. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to α-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture [6]. We also use the combinatorial Ricci flow to study the corresponding α-Yamabe problem. Mathematics Subject Classification (2010). 52C25, 52C26, 53C44. 1 Introduction This is a continuation of our work on combinatorial curvature in [17]. This paper gen- eralizes our results in [17] to triangulated surfaces with inversive distance circle packing arXiv:1701.01795v2 [math.GT] 29 May 2018 metrics. Circle packing is a powerful tool in the study of differential geometry and geo- metric topology and there are lots of research on this topic. In his work on constructing hyperbolic structure on 3-manifolds, Thurston ([28], Chapter 13) introduced the notion of Euclidean and hyperbolic circle packing metrics on triangulated surfaces with prescribed intersection angles. -
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. -
The Euler Line in Hyperbolic Geometry
The Euler Line in Hyperbolic Geometry Jeffrey R. Klus Abstract- In Euclidean geometry, the most commonly known system of geometry, a very interesting property has been proven to be common among all triangles. For every triangle, there exists a line that contains three major points of concurrence for that triangle: the centroid, orthocenter, and circumcenter. The centroid is the concurrence point for the three medians of the triangle. The orthocenter is the concurrence point of the altitudes. The circumcenter is the point of concurrence of the perpendicular bisectors of each side of the triangle. This line that contains these three points is called the Euler line. This paper discusses whether or not the Euler line exists in Hyperbolic geometry, an alternate system of geometry. The Poincare Disc Model for Hyperbolic geometry, together with Geometer’s Sketchpad software package was used extensively to answer this question. Introduction In Euclidean geometry, the most Second, for any circle that is orthogonal commonly known system of geometry, a to the unit circle, the points on that circle very interesting property has been lying in the interior of the unit circle will proven to be common among all be considered a hyperbolic line triangles. For every triangle, there exists (Wallace, 1998, p. 336). a line that contains three major points of concurrence for that triangle: the Methods centroid, orthocenter, and circumcenter. Locating the Centroid The centroid is the concurrence point for In attempting to locate the Euler line, the the three medians of the triangle. The first thing that needs to be found is the orthocenter is the concurrence point of midpoint of each side of the hyperbolic the altitudes. -
DIY Hyperbolic Geometry
DIY hyperbolic geometry Kathryn Mann written for Mathcamp 2015 Abstract and guide to the reader: This is a set of notes from a 5-day Do-It-Yourself (or perhaps Discover-It-Yourself) intro- duction to hyperbolic geometry. Everything from geodesics to Gauss-Bonnet, starting with a combinatorial/polyhedral approach that assumes no knowledge of differential geometry. Al- though these are set up as \Day 1" through \Day 5", there is certainly enough material hinted at here to make a five week course. In any case, one should be sure to leave ample room for play and discovery before moving from one section to the next. Most importantly, these notes were meant to be acted upon: if you're reading this without building, drawing, and exploring, you're doing it wrong! Feedback: As often happens, these notes were typed out rather more hastily than I intended! I would be quite happy to hear suggestions/comments and know about typos or omissions. For the moment, you can reach me at [email protected] As some of the figures in this work are pulled from published work (remember, this is just a class handout!), please only use and distribute as you would do so with your own class mate- rials. Day 1: Wrinkly paper If we glue equilateral triangles together, 6 around a vertex, and keep going forever, we build a flat (Euclidean) plane. This space is called E2. Gluing 5 around a vertex eventually closes up and gives an icosahedron, which I would like you to think of as a polyhedral approximation of a round sphere. -
The Hyperbolic Smarandache Theorem in the Poincaré Upper Half Plane
The hyperbolic Smarandache theorem in the Poincaré upper half-plane model of hyperbolic geometry Nilgün Sönmez and CA¼tA¼lin Barbu Abstract. In this study, we give a hyperbolic version of the Smarandache’s theorem in the Poincaré upper half-plane model. 2000 Mathematical Subject Classi…cation: 30F45, 51M10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, Smarandache’s theorem, Poincaré upper half-plane model 1. Introduction Non-Euclidean geometry in general, and hyperbolic geometry in particular, is an area of mathematics which has an interesting history and which is still being actively studied by researchers around the world. One reason for the continuing interest in hyperbolic geometry is that it touches on a number of di¤erent …elds, including but not limited to Complex Analysis, Abstract Algebra and Group Theory, Number Theory, Diferential Geometry, and Low-dimensional Topology [1]. In hyperbolic geometry, a type of non-Eucliean geometry that retains the …rst four postulates of Euclid, the parallel postulate is replaced by the following: Through a given point not on a given line there can be drawn at least two lines parallel to the given line. The development of hyperbolic geometry is attributed to Lobachevsky, Bolyai, and Gauss, but there is certainly some evidence that Euclid had an inkling of its existence [2]. There are known many models for hyperbolic geometry, such as: Poincaré upper half- plane, Poincaré disc model, Klein model, etc. Here, in this study, we give a hyperbolic version of the Smarandache theorem in the Poincaré upper half-plane model. The Euclidean version of this well-known theorem states that if the points Mi; i = 1; n are the projections of a point M on the sides AiAi+1; i = 1; n; where An+1 = A1, of the polygon A1A2:::An; then 2 2 2 2 2 2 2 M1A1 + M2A2 + ::: + MnAn = M1A2 + M2A3 + ::: + Mn 1An + MnA1 (1) (see [5] or [8]). -
Non-Euclidean Shadows of Classical Projective Theorems Arxiv
Non-euclidean shadows of classical projective theorems Ruben Vigara Centro Universitario de la Defensa - Zaragoza I.U.M.A. - Universidad de Zaragoza January 23, 2015 arXiv:1412.7589v2 [math.MG] 22 Jan 2015 Contents 1 Introduction 3 2 Basics of planar projective geometry. 12 2.1 Cross ratios . 13 2.2 Segments . 15 2.3 Triangles . 15 2.4 Quadrangles . 15 2.5 Poles and polars . 16 2.6 Conjugate points and lines . 17 2.7 Harmonic sets and harmonic conjugacy . 18 2.8 Conics whose polarity preserves real elements . 20 3 Cayley-Klein models for hyperbolic and elliptic planar ge- ometries 21 3.1 Distances and angles . 22 3.2 Midpoints of a segment . 23 3.3 Proyective trigonometric ratios . 27 4 Pascal's and Chasles' theorems and classical triangle centers 31 4.1 Quad theorems . 31 4.2 Triangle notation . 32 4.3 Chasles' Theorem and the orthocenter . 35 4.4 Pascal's Theorem and classical centers . 36 5 Desargues' Theorem, alternative triangle centers and the Euler-Wildberger line 43 5.1 Pseudobarycenter . 43 5.2 Pseudocircumcenter . 45 5.3 The Euler-Wildberger line . 47 5.4 The nine-point conic . 50 1 Contents 6 Menelaus' Theorem and non-euclidean trigonometry 55 6.1 Trigonometry of generalized right-angled triangles . 56 6.1.1 Non-euclidean Pythagorean Theorems . 59 6.1.2 More trigonometric relations . 61 6.2 Trigonometry of generalized, non right-angled, triangles . 64 6.2.1 The general (squared) law of sines. 64 6.2.2 The general (squared) law of cosines . 65 7 Carnot's Theorem and.. -
C 2009 Karan Mohan Puri ALL RIGHTS RESERVED FACTORIZATION of ISOMETRIES of HYPERBOLIC
c 2009 Karan Mohan Puri ALL RIGHTS RESERVED FACTORIZATION OF ISOMETRIES OF HYPERBOLIC 4-SPACE AND A DISCRETENESS CONDITION By Karan Mohan Puri A dissertation submitted to the Graduate School-Newark Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematical Sciences written under the direction of Professor Jane P. Gilman and approved by Newark, New Jersey May, 2009 Abstract Factorization of isometries of Hyperbolic 4-space and a Discreteness Condition By Karan Mohan Puri Dissertation Director: Professor Jane P. Gilman Gilman's NSDC condition is a sufficient condition for the discrete- ness of a two generator subgroup of P SL(2; C). We address the question of the extension of this condition to subgroups of isometries of hyper- bolic 4-space. While making this new construction, namely the NSDS condition, we are led to ask whether every orientation preserving isom- etry of hyperbolic 4-space can be factored into the product of two half- turns. We use some techniques developed by Wilker to first, define a half-turn suitably in dimension 4 and then answer the former question. It turns out that defining a half-turn in this way in any dimension n enables us to generalize some of Gilman's theorems to dimension n ≥ 4. We also give an exposition on part of Wilker's work and give new proofs for some of his results. ii To my parents, my sister and to Neha. Acknowledgements: A large number of people have had an influence on the writing and completion of this thesis. -
LOCAL RIGIDITY of INVERSIVE DISTANCE CIRCLE PACKING 3 Is Given by the Formula 2 2 2 L − R1 − R2 I(C1,C2)=
LOCAL RIGIDITY OF INVERSIVE DISTANCE CIRCLE PACKING REN GUO Abstract. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle. 1. Introduction 1.1. Andreev-Thurston Theorem. In his work on constructing hyperbolic met- rics on 3-manifolds, W. Thurston ([21], Chapter 13) studied a Euclidean (or hyper- bolic) circle packing on a closed triangulated surface with prescribed intersection angles. Thurston’s work generalizes Andreev’s result of circle packing on a sphere [1, 2]. The special case of tangent circle packing (every intersection angle is zero) on sphere was obtained by Koebe [14]. Suppose (Σ,T ) is a closed triangulated surface so that V,E,F are sets of all vertices, edges and triangles in T. We identify vertexes of T with indexes, edges of T with pairs of indexes and triangles of T with triples of indexes. This means V = {1, 2, ...|V |}, E = {ij | i, j ∈ V } and F = {△ijk | i, j, k ∈ V }. Fix a vector R|E| π Θ ∈ indexed by the set of edges E, such that Θij ∈ [0, 2 ] for the each ij ∈ E. This vector is call a weight on (Σ,T ). A circle packing on a closed triangulated weighted surface (Σ,T, Θ) is a con- figuration {ci,i ∈ V } of circles such that the intersection angle of ci and cj is Θij which is the prescribed weight on the edge ij. To obtain a Euclidean (or hyperbolic) R|V | circle packing for (Σ,T, Θ), we start with a radius vector r = (r1, r2, ..., r|V |) ∈ >0 which assigns each vertex i ∈ V a positive number ri. -
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DEMONSTRATIO MATHEMATICA Vol. XLII No 2 2009 Nilgun Sônmez SOME FEATURES IN CONNECTION WITH TRIANGLES IN THE POINCARÉ UPPER HALF PLANE Abstract. In this work, it is shown that the product of lengths that are segments reserved to altitudes of orthocenter is constant in the Poincaré triangle. I defined to Carnot theorem Poincaré upper half plane. 1. Introduction The Poincaré upper half plane geometry has been introduced by Henri Poincaré. Poincaré upper half plane is the upper half plane of the Euclidean analytical plane M2. Although the points in the Poincaré upper half plane are the same as the points in the upper half plane of the Euclidean analytical plane R2, the lines and the distance function between any two points are different. The lines in the Poincaré upper half plane are defined by 2 aL = {(x, y) € M | x = a, y > 0, a € M, a constant} half lines and 2 2 2 2 c:L r = {(x, y) G M I (x - c) + y = r , y > 0, c, r G R, c, r constant, r > 0} half circles. If A={x\, y\) and B—(x2, 2/2 ) are any two points in H then the Poincaré distance between these points is given by |ln(y2/yi)|, if X\ = X2 dH(A,B) = i lnfy2{xi-c + r)\ In if Xl ± X2 \yi(x2 - c + r)J ' 1991 Mathematics Subject Classification: 51K05, 51K99. Key words and phrases: Poincaré upper half plane, Poincaré circle, Poincaré triangle. 404 N. Sònmez where c={yl~y\ + xl- X\)/2{X2 - xi), 2 2 r = ^(xi - c) + y\ = yj(x2 ~ c) + y\. -
A ROUNDABOUT INTRODUCTION to HYPERBOLIC AREA Contents 1
A ROUNDABOUT INTRODUCTION TO HYPERBOLIC AREA SARAH ZHANG Abstract. The Gauss-Bonnet theorem gives a simple formula for the area of any \reasonable" hyperbolic polygon based on its internal angle measures. To approach this result, we give an abbreviated overview of M¨obiustransforma- tions, two models of hyperbolic space, convexity in the hyperbolic plane, and related formulas for hyperbolic area. Contents 1. Introduction 1 2. M¨obiustransformations 1 3. The upper half plane H and the Poincar´edisc D 6 4. Convexity 9 5. Hyperbolic polygons and the Gauss-Bonnet formula 10 Acknowledgments 15 References 16 1. Introduction When it comes to two dimensional spaces, the usual go-to is the Euclidean plane, where lines are straight, shapes are flat, and the curvature of the space is uniformly zero. Another is the surface of a sphere, where the shortest paths between points are portions of great circles (potentially nonunique if the points are at polar opposite locations) and the curvature of the space is uniformly positive. Another still is the hyperbolic plane, a space with constant negative curvature often called \wrinkly" for (among other crimes) \having too much area everywhere." We contend with this one. (For a hands-on visualization, see [2] or look up some of the online tutorials for crocheting/tiling/etc. the hyperbolic plane.) The report is organized as follows: Section 2 explores the properties of M¨obius transformations on the Riemann sphere. Section 3 defines the hyperbolic metric and introduces two models of the hyperbolic plane. Section 4 explores convexity in the hyperbolic plane. Section 5 defines hyperbolic area and works towards a proof of the generalized Gauss-Bonnet theorem, which states that a reasonable hyperbolic n-gon has area (n − 2)π minus the sum of its internal angles.