DOCSLIB.ORG

Circle Packings, Triangulations, and Rigidity for Tom Hales' Birthday

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Circle Packings, Triangulations, and Rigidity for Tom Hales' Birthday First Section Circle Packings, Triangulations, and Rigidity For Tom Hales' Birthday Bob Connelly With Steven Gortler, Evan Solomonides, and Maria Yampolskaya. Cornell University June 2018 1 / 25 First Section Part I - The Packing Problem A packing of circular disks is a collection of such disks with disjoint interiors. The density of such a packing is the ratio of total area of the disks divided by the area of the container that holds them. (For an infinite packing, you take a limit.) The Packing Problem: For a given container and collection of disks, find a packing of those disks with greatest density. Example: Forp the plane and equal radius disks the largest density is: δ = π= 12 = 0:9068 ::: . 2 / 25 First Section History \Axel Thue provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to L´aszl´oFejes T´othin 1940." (From Wikipedia). But L´aszloFejes T´othdid more: He found a whole series of packings with a range of radii that he proposed as candidates for the most dense packings given a particular range of ratios of the radii. For example, if there are just two radii, with a ratio of p 2 − 1 = 0:414 ::: Al´adar Heppes (2000) proved that the following configuration is p the most dense at δ = π(2 − 2)=2 = 0:92015 ::: . 3 / 25 r = 2-1 = .414.. δ = 0.92015.. First Section Compact Packings = Triangulated Packings The graph of a circle packing is obtained by connecting the centers of each pair of touching disks by a line segment. When that graph of a packing is a triangulation Fejes T´othcalled the packing a compact packing. In many cases such \compact packings" were candidates for the most dense packings with those radii sizes. For a range between two sizes of radii, the following are his candidates for the maximum density. Most packings that come up with respect to density questions are periodic, and so it is natural to simply assume that they are periodic, which is the same as assuming they live on a (flat) torus. 4 / 25 First Section Fejes Toth's packings From \Regular Figures" by L´aszl´oFejes T´oth 5 / 25 First Section Fejes Toth's density estimates The function s above is an upper bound for the density, by August Florian (1960), where q is the minimum ratio ri =rj of the circle radii. πq2 + 2(1 − ρ2) arcsin( q ) s = p 1+q : 2q 2q + 1 6 / 25 First Section Koebe, Andreev, Thurston Theory Meanwhile, Koebe, Andreev, and Thurston have the following theorem: Theorem (KAT) Given a triangulation T of a 2-manifold M with constant curvature, there is a circle packing of M whose graph is T , which is unique up to circle preserving maps (Moebius transformations). Keep in mind that circle packing defines the metric, that is the lattice that defines M, up to Moebius transformations. 7 / 25 First Section Examples of KAT theory for M = torus One way to create periodic triangulated packings is to start with a packing, and subdivide an edge as below. The yellow point is on the edge to be subdivided for the next configuration. KAT insures that there is a corresponding packing with that graph. Only a fundamental region is shown. radii 1 1: 1- 2 1: 2 :3 1 number of circles 1 2 3 4 density π/ 12 = 0.9068.. π(1- 2 /2 )= 0.92015 7π/24 = 0.91629.. π/ 12 = 0.9068.. 8 / 25 First Section Edge flipping Another method is to flip an edge of the triangulation, which is where an edge is removed and replaced as the other diagonal in the resulting quadrilateral. Any edge can be flipped, as long as you don't flip away a degree 3 vertex. The packing on the right is obtained by flipping the green edge on the 4 vertex lattice triangulation. The symmetry of the abstract triangulation and the uniqueness of the packing implies that the symmetry persists in the metric configuration of the packing, which implies that there are just two radii in this flipped packing. This is Laszlo Fejes T´oth'spacking #3. 9 / 25 First Section Triangulated Packings with Two Sizes of Radii Considering Fejes T´oth'sdensity estimates using triangulated packings, Tom Kennedy found all possible two sized radii packings, that were triangulated packings. He found one (or two) more that was not in Fejes T´oth'slist. These are the following: 10 / 25 First Section Triangulated Packings with Two Sizes of Radii Compact Packings of the Plane with Two Sizes of Discs 257 (a) r c1 0.637556 (b) r c2 0.545151 . (c) r c3 0.533296 = = ··· = = ··· = = ··· e b d f c b a c j f h g e d g a i j i h m k l (d) r c4 0.414214 (e) r c5 0.386106 (f) r c6 0.349198 = = ··· = = ··· = = ··· (g) r c7 0.280776 (h) r c8 0.154701 (i) r c9 0.101021 11 / 25 = = ··· = = ··· = = ··· Fig. 1. A compact packing is shown for each of the nine values of r which admit compact packings. and non-negative integers l, m, n such that lα′ mβ nπ/3 2π.(4) + + = For every value of r there is a trivial solution of both of these equations, namely i j l m 0, k n 6. In a compact packing that contains discs of both radii, there= must= be= at least= one= pair= of discs of different radii that are tangent, and so there must be at least one solution of (3) other than the trivial one and at least one solution of (4) other than the trivial one. We start by determining when (3) has solutions. First Section Problems for triangulated packings of a torus 1 What can you say about the change in density with an edge subdivision or edge flip? 2 Any triangulation can be obtained by edge flips from a triangular lattice with the same number of vertices. 3 What is the least dense triangulated packing other than those coming from the triangular lattice? (Anyp triangulated packing has a density strictly greater than π= 12 if the disks are different sizes.) 4 Florian showed that if a packing is such that the minimum radiusp ratios ri =rj > 0:73, then their maximum density is only π= 12, which is when the respective circles with a fixed radiusp are segregated into their own regions with density π= 12. Can the 0:73 bound be improved? 12 / 25 First Section Problem for Tom Is there a similar result in dimension 3? Namely is there a ratio 3 ρ < 1 such that if there is a packing of R by spherical balls with radii ri such the minimump ratio ri =rj ≥ ρ, then the packing has density at most π=3 2? 13 / 25 First Section Conjecture A packing is saturated if it is not possible to move a disk to another location while fixing the others. Conjecture: Fix the number n of disks in a packing. Fix the radii r1; r2;:::; rn of the disks. Suppose that there is a saturated triangulated packing of a torus with n circles with those radii having density δ. Then any other torus packing with n circles and those radii has density at most δ. In Kennedy's list, any cover of packings a, b, c, d, f, g, h, have been shown to satisfy the conjecture, all proved by Al´adar Heppes, except packing b was shown by Kennedy. 14 / 25 First Section Nazarov's example The condition that the packing be saturated is necessary. Fedja Nazarov pointed out the following example. This is a fundamental region of a triangulated packing of a torus, where the six colored packing disks can be removed and reinserted to places shown with dashed boundaries in the triangular regions to the right and left. The packing disks can be freed with small motions from contact with their neighbors one at a time. At the end of the motion, all the packing disks have room to grow and increase density. 15 / 25 First Section Part II - Jammed Packings Fix a lattice Λ that defines a torus. Place n circular disks in the torus with fixed radius ratios. Let them grow maintaining the radius ratios until they jam and can't grow anymore. When that happens, there will be at least a portion of the packing that is rigid and can only move under a global translation as in the figure below. This is the most dense packing of 8 equal disks in a square torus, proved by Musin and Nikitenko (2016) 16 / 25 First Section Infinitesimal Rigidity 0 0 0 0 An infinitesimal flex p = (p1; p2;:::; pn) of a tensegrity (G; p) is a 0 vector pi assigned to the i-th vertex such that 0 0 (pi − pj ) · (pi − pj ) = 0; ≤ 0; ≥ 0; for a bar, cable, strut. 0 0 0 0 An infinitesimal flex p = (p1; p2;:::; pn) is trivial if it is the time 0 derivative of a one parameter family of global isometries of the ambient space. If a tensegrity (G; p) has only trivial infinitesimal flexes, then it is called infinitesimally rigid. Theorem Infinitesimal rigidity implies local rigidity. 17 / 25 First Section Equilibrium stress Dual to the notion of an infinitesimal flex is the idea of an equilibrium stress, which is scalar !ij = !ji , all the same sign for packings, associated to each edge of the packing graph, such that for each disk i, X !ij (pi − pj ) = 0: j The distribution of the stresses in a packing are important for the study of forces in granular materials, for example.
Load more

Recommended publications
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Mobius-Invariant Lorentz Geometry of the Space of Circles on S2
    Mobius-invariant Lorentz geometry of the space of circles on S2 Ben Johnsrude Advisor: David Barrett Abstract The space of circles and lines on the Riemann sphere has a natural topological 3- manifold structure. Coxeter observed that a geometry on this collection exists such that Mobius transformations are isometries. Properly defined, this geometry is double covered by (1+2)-dimensional de Sitter space. We discuss some of the properties circle space inherits from de Sitter space, as well as features original to the circle space. Let C denote the orbit of the unit circle in the Riemann sphere Cb under action of the Mobius group M ; thus C is the \space of circles on Cb," equivalently defined as the space of planar circles together with lines union the point at infinity. We may parameterize C as pairs (ζ; ρ) of spherical center and spherical radius, respectively, with the caveat that, as C consists of unoriented circles, we identify (ζ; ρ) ∼ (−ζ; 1 − ρ) (1) Here we understand radius in the sense of the chordal metric jz − wj q(z; w) := p1 + jzj2p1 + jwj2 With this, C takes the topological structure of a 3-manifold ∼ 2 C = (S × R)= ∼ where ∼ follows (1) above. An equivalent topology may be found by considering triples 3 (z1; z2; z3) 2 Cb ; any three such distinct points determine a unique c 2 C , and for each triple of points determining a circle, each of the points may be varied along the circular arc to determine the same circle. Classical inversive geometry [1] considers this collection with a group of rigid motions known as circle inversions: for a given circle c 2 C , this is the complex-analytic reflection across c, given in the case jzj = 1 by w w 7! jwj2 Note that M~ := hM ; z 7! z¯i is exactly the group generated by circle inversions.
    [Show full text]
  • Quasihyperbolic Distance, Pointed Gromov-Hausdorff Distance, and Bounded Uniform Convergence
    Quasihyperbolic Distance, Pointed Gromov-Hausdorff Distance, and Bounded Uniform Convergence A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematical Sciences College of Arts and Sciences University of Cincinnati, May 2019 Author: Abigail Richard Chair: Dr. David Herron Degrees: B.A. Mathematics, 2011, Committee: Dr. Michael Goldberg University of Indianapolis Dr. Nageswari Shanmugalingam M.S. Mathematics, 2013, Dr. Marie Snipes Miami University Dr. Gareth Speight Abstract We present relationships between various types of convergence. In particular, we examine several types of convergence of sets including Hausdorff convergence, Gromov-Hausdorff convergence, etc. In studying these relationships, we also gain a better understanding of the necessary conditions for certain conformal metric distances to pointed Gromov- Hausdorff converge. We use pointed Gromov-Hausdorff convergence to develop approx- imations of the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances via spaces with only finitely many boundary points. ii c 2019 by Abigail Richard. All rights reserved. Acknowledgments I would like to thank God, my family, Dr. Sourav Saha, Dr. David Herron, Dr. Marie Snipes, Dr. David Minda, my committee, Dr. Lakshmi Dinesh, Dr. Fred Ahrens, and Dr. Crystal Clough for all of their help and support while I have been a graduate student at UC. I would also like to thank and recognize the Taft Center for their financial support. iv Contents Abstract ii Copyright iii Acknowledgments iv 1 Introduction 1 2 Background 5 2.1 Preliminary Notation . 5 2.2 Conformal Metrics . 7 2.2.1 Quasihyperbolic Metric . 8 2.2.2 Ferrand Metric .
    [Show full text]
  • On the Global Rigidity of Sphere Packings on 3-Dimensional Manifolds
    On the global rigidity of sphere packings on 3-dimensional manifolds Xu Xu Abstract In this paper, we prove the global rigidity of sphere packings on 3-dimensional man- ifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin in [5]. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author in [13]. MSC (2010): 52C25; 52C26 Keywords: Global rigidity; Sphere packing; Combinatorial scalar curvature 1 Introduction In his investigation of hyperbolic metrics on 3-manifolds, Thurston ([30], Chapter 13) introduced the circle packing with prescribed intersection angles and proved the Andreev- Thurston Theorem, which consists of two parts. The first part is on the existence of circle packing for a given triangulation. The second part is on the rigidity of circle packings, which states that a circle packing is uniquely determined by its discrete Gauss curvature (up to scaling for the Euclidean background geometry). For a proof of Andreev-Thurston Theorem, see [4, 6, 20, 26, 28, 30]. To study the 3-dimensional analogy of the circle packing on surfaces, Cooper and Rivin [5] introduced the sphere packing on 3-dimensional manifolds. Suppose M is a 3- dimensional closed manifold with a triangulation = V; E; F; T , where the symbols arXiv:1611.08835v4 [math.GT] 27 Dec 2018 T f g V; E; F; T represent the sets of vertices, edges, faces and tetrahedra, respectively. Definition 1.1 ([5]). A Euclidean (hyperbolic respectively) sphere packing metric on (M; ) is a map r : V (0; + ) such that (1) the length of an edge ij E with T ! 1 f g 2 vertices i; j is l = r + r and (2) for each tetrahedron i; j; k; l T , the lengths ij i j f g 2 lij; lik; lil; ljk; ljl; lkl form the edge lengths of a Euclidean (hyperbolic respectively) tetra- hedron.
    [Show full text]
  • Notes on Euclidean Geometry Kiran Kedlaya Based On
    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 Powerofapoint.................................. 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 .
    [Show full text]
  • Rigidity of Configurations of Balls and Points in the $ N $-Sphere
    RIGIDITY OF CONFIGURATIONS OF BALLS AND POINTS IN THE N-SPHERE EDWARD CRANE AND IAN SHORT Abstract. We answer two questions of Beardon and Minda that arose from their study of the conformal symmetries of circular regions in the complex plane. We show that a configuration of closed balls in the N-sphere is determined up to M¨obius trans- formations by the signed inversive distances between pairs of its elements, except when the boundaries of the balls have a point in common, and that a configuration of points in the N-sphere is determined by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic (N + 1)-space. 1. Introduction N N N Let R∞ denote the one point extension of R . A sphere in R∞ refers to either an (N 1)-dimensional Euclidean sphere in RN , or an (N 1)-dimensional Euclidean plane− in RN with the point attached. An inversion in− a sphere refers to either a Euclidean inversion, if the sphere∞ is a Euclidean sphere, or a Euclidean reflection, if N the sphere is a Euclidean plane. The group of M¨obius transformations on R∞ is the N group generated by inversions in spheres. By an open ball B in R∞ we mean one of the connected components of the complement of a sphere. We write B for the other N component. Given two distinct open balls B1 and B2 in R∞, we denote the signed inversive distance between them by [B1, B2]. For any four distinct pointsbp1, p2, p3, and N p4 in R∞ we let p1,p2,p3,p4 denote their absolute cross-ratio.
    [Show full text]
  • Packing Disks by Flipping and Flowing
    Packing Disks by Flipping and Flowing Robert Connelly∗ Steven J. Gortlery Abstract We provide a new type of proof for the Koebe-Andreev-Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge e− of this graph and then continuously flow the disks in the plane, such that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge e− in G. This flow is parameterized by a single inversive distance. 1 Introduction The well known Koebe{Andreev{Thurston (KAT) (planar) circle packing theorem states that for every planar graph G with n vertices, there is a corresponding packing of n disks (with mutually disjoint interiors) in the plane, whose contact graph is isomorphic to G. Moreover if G is maximal (ie. all faces triangular), then this packing is unique up to M¨obius transformations and reflections [2, 19, 33]. This theorem is important in conformal geometry [27] and has been generalized in numerous directions [26, 30, 33]. In [14], a packing whose graph is a triangulation of the plane is called a compact packing. There are a variety of techniques that have been used to establish the KAT theorem, These include methods based on conformal geometry [19], combinatorial analysis of hyperbolic polyhe- dra [2, 28] circle geometry, cone-singularities and topology of continuous maps [24, 33], variational methods [3, 13, 26], iterative “flattening" algorithms [9, 25, 32], and combinatorial Ricci Flow [8].
    [Show full text]
  • H.S.M. Coxeter, Angles and Arcs in the Hyperbolic Plane, P 17-34Mathschron009-004.Pdf
    ANGLES AND ARCS IN THE HYPERBOLIC PLANE H.S.M. Coxeter Dedicated to H.G. Forder on his 90th birthday (received 17 May, 1979) 1. Introduction This essay may be regarded as a sequel to Chapter V of H.G. Forder's charming little book, Geometry [Forder 1950, pp. 80-95], which is summarized in Sections 2 and 3. Section 4 gives a fuller account of Lobachevsky's approach to hyperbolic trigonometry. Section 5 contains a simple proof that the natural unit of measurement is the length of a horocyclic arc such that the tangent at one end is parallel to the diameter at the other end. Section 6 frees Poincare's half-plane model from its dependence on cross-ratio, by introducing a more natural concept: the inversive distance between two disjoint circles. Section 7 shows how an equidistant-curve of altitude a is represented by lines crossing the absolute line at angles ±n(a) . Finally, Section 8 provides an easy deduction of Poincare's formula \dz\/y for the hyperbolic line- element. B Math. Chronicle 9(1980) 17-33. 17 2. Asymptotic triangles Felix Klein once said that "non-Euclidean geometry ... forms one of the few parts of mathematics which ... is talked about in wide circles, so that any teacher may be asked about it at any moment" [Klein 1939, p. 135]. The richest kind of non-Euclidean geometry is the one discovered independently about 1826 by J. Bolyai (1802-1860) and N.I. Lobachevsky (1793-1856). After 1858, when B. Riemann and L. Schlafli had discovered another kind, Klein named the old kind hyperbolic.
    [Show full text]