Two Counterexamples of Global Differential Geometry for Polyhedra
|
|
- Frederick Anderson
- 5 years ago
- Views:
Transcription
1 Two Counterexamples of Global Differential Geometry for Polyhedra Abdênago Barros, Esdras Medeiros and Romildo Silva Departatamento de Matemática Universidade Federal do Ceará Fortaleza-Ceará, Brazil Abstract In this article we show, by giving counterexamples, that two classical results of global differential geometry are not valid for polyhedral surfaces. Considering the set of compact smooth surfaces we exhibit counterexamples concerning the Cohn-Vossen theorem and the existence of elliptical points. The key point on the construction of such polyhedral surfaces is the application of a nice discrete version of the Gaussian curvature. Keywords: Gauss-Bonnet, Cohn-Vossen, polyhedra, discrete curvature 2000 MSC: 52B70, 53C45 1. Introduction The study of polyhedral surfaces in the context of differential geometry finds its motivation in both pure and applied mathematics. Consider for example the pioneering work of Alexandrov [2] which investigates the geometry of surfaces by approximating by polyhedral metrics. In other fields such as computer graphics and numerics, we encounter a strong need for a discrete differential geometry of arbitrary meshes (see for instance Bobenko et al. [1] and Meyer et al. [7]). There are many classical results of differential geometry extended to other topological spaces. One fundamental theorem for a given smooth surface S is address: abbarros@mat.ufc.br, esdras@mat.ufc.br, rjs@mat.ufc.br (Abdênago Barros, Esdras Medeiros and Romildo Silva) Preprint submitted to Elsevier October 19, 2011
2 the celebrated Gauss-Bonnet theorem which links the topology of S and its total curvature. In the sixties, Banchoff [3] showed that this theorem is also valid for any compact polyhedral surface thanks to the nice discretization of the Gaussian curvature established by Alexandrov [2]. More recently, Forman [5] extended Morse Theory to cell complexes. Such examples of connections between smooth and discrete surfaces motivated us to investigate other similarities of global differential geometry and topology in polyhedral surfaces. Among then we are interested in studying two results intrinsically related to the Gaussian curvature. They are: A little variation of the Gauus-Bonnet theorem is the Cohn-Vossen theorem that relates the topology of S with its total absolute curvature. We build a simple polyhedral surface topologically equivalent to the torus for which the Cohn-Vossen theorem does not work. This is proved analytically after some elementary computations of the discrete Gaussian curvature. A fundamental theorem concerning the existence of elliptical points, more specifically, the existence one point with positive Gaussian curvature. By the Gauss-Bonnet theorem, this result is guaranteed for polyhedra with genus zero or one. However we can create a complicated geometric shape topologically equivalent to the 2-torus with no elliptical vertices, more than that, every vertex has strictly negative discrete Gaussian curvature. Our method follows a recent trend of computer assisted proofs [6] which means that the proof is established by numerical analysis tools. The article is organized as follows. The next section gives an explanation on smooth surfaces by giving some basic results. In the section 3 we explore the Gauss-Bonnet theorem for polyhedral surfaces and study the existence of elliptical points in surfaces with genus zero and one. In section 4 we show the Cohn-Vossen theorem counterexample. In section 5 we show the 2-torus counterexample for the non-existence of elliptical points the interval arithmetic method is applied to validate the model. We also give a simple generalization for genus greater than two. In section 6 we finish the article with a discussion. 2
3 2. Total curvature and Cohn-Vossen theorem The approach given originally by Gauss to define the total curvature of a neighborhood U on a smooth surface M 2 in the Euclidean space R 3 is the best way to deal with extrinsic curvature theory. Indeed, according to Gauss the total curvature of a small convex neighborhood U 1 on which the Gauss map g is injective and orientation-preserving is defined as the area of the spherical image g(u 1 ) on S 2, i.e., K(U1 ) = A(g(U 1 )), whereas, for non convex regions V on which g is injective, but orientation-reversing, its total curvature is minus the quoted area g(v), see e.g. Banchoff [3] and Do Carmo [4]. Withthisapproachwe mayhavethegaussiancurvatureatapointp U as the following limit: K(p) = lim di 0 K(U i ) A(U i ), where d i stands for the diameter of U i, while U i is a neighborhood of p. Then we also have K(U) = K(p)dA, U where da is the area measure of M 2. Finally the famous Gauss-Bonnet theorem states that the total curvature of M 2 is given by K(M 2 ) = K(p)dA = 2πχ(M 2 ), M 2 where χ(m 2 ) denotes the Euler characteristic of M 2. The Cohn-Vossen theorem for closed surfaces which deals with the total absolute curvature, i.e. K(p) da, is a beautiful remark of the Gauss- U Bonnet theory combined with the fact that the Gauss map is onto when restricted to the almost convex part of M 2. Indeed, if we let M + = {p M 2 : K(p) 0} and M = {p M 2 : K(p) < 0} we have that the Gauss map restrict to M + is onto S 2. As consequence we have K(p)dA 4π. (1) M + In order to deduce the above inequality we argue as follows: Given a direction ξ on R 3 we move a perpendicular plane to ξ so far from the surface until this touches the surface at the first time at a point p 0. At least at a point p 0 we have two nice consequences: The first one tell us that the 3
4 Gauss map is ±ξ, so we may choice our direction in such way that the Gauss map g(p 0 ) = ξ whereas the second one gives us that the Gaussian curvature K(p 0 ) > 0, since the surface at p 0 is more curved than the Euclidean sphere centered in the origin with radius p 0, i.e. we have an elliptic point in M 2. Next we write K(p)dA = K(p)dA+ K(p)dA (2) M 2 M + M and K(p) da = K(p)dA K(p)dA (3) M 2 M + M Adding equations (2) and (3) and using Gauss-Bonnet theorem we arrive at K(p) da = 2 K(p)dA 2πχ(M 2 ). (4) M 2 M + In order to deduce Cohn-Vossen theorem it is enough to apply inequality (1) to equation (4). Indeed, after that we obtain: M 2 K(p) da 2π(4 χ(m 2 )). (5) 3. Polyhedral Surfaces: Discrete curvatures A polyhedral surface M(V,E,F) is a set V of vertices, E edges and F faces such that each edge is the boundary of exactly two faces. The term polyhedron refers to a closed polyhedral surface. In this setting the polyhedron is closed, but might be not simple, i.e. non homeomorphic to the sphere. Without loss of generality we assume that the polyhedra are simplicial complexes which means that all the faces in F are triangular. The following definition will be useful to the next theorem: Definition 1. The star Str(v) of a vertex v is the union of all the faces and edges that contain this vertex. Apolyhedronembedded ineuclideanspace R 3 maybeseen asapiecewise linear surface. As we know, the non differentiability at the vertices restricts the computation of the curvatures by means of the tools from the differential geometry on smooth surfaces. 4
5 Despite the fact that the Gaussian curvature is not well defined for polyhedra, there is a definition which captures this concept in a natural way (see Banchoff [3]). Based on this work, we may establish a result which mimics the classical Gauss-Bonnet theorem for polyhedra: Theorem 1. Let P be a polyhedron. Then K(v) = 2πχ(P) v V where K(v) = 2π θ ( ) and θ = α i is the total angle around a vertex v, and α i are those angles of the faces in the Str(v) that are incident to v. The theorem above suggests that K(v) is simply the Gaussian curvature around the vertex v. Indeed, equation ( ) is the definition of discrete Gaussian curvature. It follows that the existence of elliptical points for polyhedra with genus zero holds since v V K(v) = 2πχ(P) = 4π. However, for toroidal polyhedra, we can only affirm that there no exists anyone with negative curvature at all vertices. The discrete Gaussian curvature gave us an important contribution in validating the Gauss-Bonnet theorem for polyhedra. Meanwhile, some results such as the Cohn-Vossen theorem and the existence of elliptical points for polyhedra with genus greater than one remain open. Next, we construct counterexamples for each one. 4. Tori of Small Total Curvature Let and be two concentrical equilateral triangles on the plane Γ = {(x,y,z) R 3 : z = 0 with vertices at z 01,z 02,z 03 and z 01,z 02,z 03, respectively, in such way that. Denote T t : R 3 R 3 the vertical translation given by T t (x,y,z) = (x,y,z + t). Let z λi = T λ (z 0i ) and z λi = T λ (z 0i), for i = 1,2,3. Notice that for each i, z λi and z λi lie in the plane z = λ. Now we consider the polyhedron T λ R 3 whose set of vertices is V λ = {z ±λi,z ±λi,i = 1,2,3}. We notice that the Gauß curvature at each vertex is given by while K(z ±λi ) = 2π ( π 2 + π 2 + π ) 2π = 3 3 5
6 Then we have while K(z ±λi) = 2π ( π 2 + π 2 +2π π ) 2π = 3 3. V K(v i ) = 6 2π 3 +6 ( 2π ) = 0 3 i=1 V K(v i ) = 6 2π 3 +6 ( 2π ) = 8π. 3 i=1 Now we consider the polyhedron T λ R3 (fig. 1) by degenerating the sides z λi z λi to the vertex z 0i, for i = 1,2,3. Hence the set of vertices of T λ is given by Vλ = {z 0i,z λi,z λi,i = 1,2,3} whose set of edges is given by E = {z 01 z 02,z 01 z λ1,z 01z λ1,z 02z 03,z 02 z λ2,z 02z λ2,z 03z 01,z 03 z λ3,z 03z λ3 }. Now let us denote by α the angle between z 01 z 02 and z 01 z λ1, as well as with respect to the other edges z 0i z 0i+1 and z 0i z ±λi. In order to compute the Gauß curvature at each vertex we notice that whereas K(v i ) = 2π 4α, for v i {z 0i } K(v i ) = 2π ( π 2 + π 2 +π α+π α) = 2α π, for v i {z ±λi}. Taking into account that α ( π 6, π 2) we arrive at while V K(v i ) = 3 ( 2π 4α ) +6 ( 2α π ) = 0 i=1 τ ( ) V T λ = K(v i ) = 3 ( 2π 4α ) +6 ( 2α π ) = 12 ( π 2α ). i=1 Fromwhere we deduce τ(α) = τ ( Tλ) = 12(π 2α). Therefore lim τ(α) = α π 6 8π while lim τ(t α π λ) =
7 Û Figure 1: T λ (left) and T λ (right) 5. 2-Torus with Strictly Negative Curvature Letusconsider asimplicial complex thatlookslikeablockwithtwo holes, i.e., the 2-torus B = (V,E,F) as showed on the left of figure 2 such that V = 24 and F = 52. The inner vertices, have negative curvature π 2 whereas the outer vertices have positive curvature π. In this basic model 2 the reader can easily verify the Gauss-Bonnet equation K(v) = 4π. Analogously to the previous section we transform the geometry of B by pushing up or down the inner vertices in the vertical direction. Let us again call such transformation as T a (x,y,z) = (x,y,z + a), a R. During this process, the curvatures of the outer vertices decrease whereas the inner vertices increase. Indeed, by the Gauss-Bonnet theorem the total curvature is invariant. A computer graphics system was implemented to solve a task too arduous ortootiringforanyhuman. Itallowedustoobserveinrealtimethecurvature changes of the vertices as they are moving through the transformation T a. After some attempts, we were able to find out the polyhedron with strictly negative curvatures at all vertices, here denoted as B. In figure 2 we illustrate this transformation process and the resulting polyhedron B. The figure 3 represents the wired and the unfolded versions of B. In the appendix at the end of the article we have table 1 and table 2 that present the geometry and combinatorial topology of B respectively. Surprisingly B is also another counterexample of the Cohn-Vossen theorem. Since, using the computer, we obtained τ ( B ) = K(v i ) = 2. Numerical Validation We computed numerically the curvature by using the dot product between two triangle edges and the arc cosine function (acos). All variables have 7
8 Û Figure 2: B (left) and B (right) (a) (b) Figure 3: wired (a) and unfolded (b) B double precision and, consequently, the results are accurate. However, the main concern for the 2-torus B relies on the difference of the true value x of the corresponding quantity and the computed value ˆx. To clarify it, in table 1, let us look at the smallest value (in modulo) of the curvatures ˆK(p) and notice that it is reasonably near to zero. Now, consider the following question: howfarfrom ˆK(p)isitstruevalueK(p)? Ifsuchdistance isgreater than ˆk(p) then the true value can be positive, invalidating our example. One of the simplest and most efficient models to validate numerical computations is interval arithmetic [8, 9]. In such model an interval is a pair of numbers which represents all the numbers between these two. The fundamental property of interval arithmetic is: if f is a function on a set of numbers, f can be extended to a new function defined on intervals. This new function f takes one interval argument and returns an interval such 8
9 Figure 4: Building a new handle. that: x [a,b],f(x) f([a,b]). Based on this simple numerical model, we evaluated the curvature intervals at all vertices. In our experimental results, the magnitude of the larger interval is which is highly accurate considering that the magnitude of the smallest curvature is only Such analysis proves, in fact, that all the vertices of B have negative curvature. Generalization for Genus 3 Now we construct more examples of polyhedra without elliptical points by adding handles as much as desired. This can be achieved by perforating B assuming that it is not hollow. For instance, we can perform a CSG operation by subtracting from B a rectangular parallelepiped between the two genus and, consequently, generating a new one (see figure 4). Such operation is enough because we can take advantage of two properties. First, the perforation does not affect the transformation T a, i. e., there is no selfintersections during the translation process of the vertices. Second, we may fine tune the size of the parallelepipeds to fit them appropriately between the two original genus. 6. Discussion In this paper, we showed two counterexamples of classical theorems in differential geometry for polyhedra. The first one was constructed analytically and contradicts the Cohn-Vossen theorem. On the other hand, the construction of a polyhedra without elliptical points (as conjectured by us initially) was a great challenge since all attempts to generate it analytically have failed. Indeed, the calculation of the curvature at each vertex demands much time. We even suspect that this conjecture was false, and we almost moved on to the proof of the existence of an elliptic point. Fortunately the 9
10 computer sped up the calculations of the discrete curvatures, which assisted us to find successfully the counterexample. In table 1 we observe that the ratio ρ between the maximal and minimal values of the curvatures at the vertices is of the order Hence, to advance in this work, we study the existence of a polyhedra with constant negative curvature or, more generally, look for the best ρ such that it is possible to generate a polyhedron with strictly negative curvature. Acknowledgments We would like to thank L. Jorge, J. Lira and C. Mota for discussions and suggestions to this paper. We also thank FUNCAP and CNPq for sponsoring our research. [1] John M. Sullivan Gnter M. Ziegler Alexander I. Bobenko, Peter Schrder. Discrete differential geometry. Birkhauser Verlag AG, [2] A. D. Alexandrov and V. A. Zalgaller. Intrinsic geometry of surfaces. Translation of Mathematical Monographs (AMS), 15, [3] T. F. Banchoff. Critical points and curvature for embedded polyhedral surfaces. The American Mathematical Monthly, 77(5): , [4] M. P. Do Carmo. Differential Geometry of curves and surfaces. Prentice- Hall, [5] R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90 145, [6] Michael Joswig and Konrad Polthier. Digital models and computer assisted proofs. Newsletter of the European Mathematical Society (EMS), [7] Mark Meyer, Mathieu Desbrun, Peter Schröder, and Alan H. Barr. Discrete differential-geometry operators for triangulated 2-manifolds. In Hans-Christian Hege and Konrad Polthier, editors, Visualization and Mathematics III, pages 35 57, Heidelberg, Springer-Verlag. [8] R. E. Moore. Interval Analysis. Prentice-Hall, [9] Ramon E. Moore and Fritz Bierbaum. Methods and applications of interval analysis (siam studies in applied and numerical mathematics)
11 Appendix The following tables describes the geometry and combinatorial topology of the 2-torus counterexample polyhedron B. index x y z curv index x y z curv Table 1: Vertices and curvatures Table 2: Faces. 11
Euler s Theorem. Brett Chenoweth. February 26, 2013
Euler s Theorem Brett Chenoweth February 26, 2013 1 Introduction This summer I have spent six weeks of my holidays working on a research project funded by the AMSI. The title of my project was Euler s
More informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
More informationShape Modeling and Geometry Processing
252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry
More informationGAUSS-BONNET FOR DISCRETE SURFACES
GAUSS-BONNET FOR DISCRETE SURFACES SOHINI UPADHYAY Abstract. Gauss-Bonnet is a deep result in differential geometry that illustrates a fundamental relationship between the curvature of a surface and its
More informationTHE UNIFORMIZATION THEOREM AND UNIVERSAL COVERS
THE UNIFORMIZATION THEOREM AND UNIVERSAL COVERS PETAR YANAKIEV Abstract. This paper will deal with the consequences of the Uniformization Theorem, which is a major result in complex analysis and differential
More informationHyperbolic structures and triangulations
CHAPTER Hyperbolic structures and triangulations In chapter 3, we learned that hyperbolic structures lead to developing maps and holonomy, and that the developing map is a covering map if and only if the
More information05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo
05 - Surfaces Acknowledgements: Olga Sorkine-Hornung Reminder Curves Turning Number Theorem Continuous world Discrete world k: Curvature is scale dependent is scale-independent Discrete Curvature Integrated
More informationGeodesic Paths on Triangular Meshes
Geodesic Paths on Triangular Meshes Dimas Martínez Luiz Velho Paulo Cezar Carvalho IMPA Instituto Nacional de Matemática Pura e Aplicada Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, RJ, Brasil
More informationLectures in Discrete Differential Geometry 3 Discrete Surfaces
Lectures in Discrete Differential Geometry 3 Discrete Surfaces Etienne Vouga March 19, 2014 1 Triangle Meshes We will now study discrete surfaces and build up a parallel theory of curvature that mimics
More informationIntroduction to geometry
1 2 Manifolds A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold Introduction to geometry The German way 2-manifold
More informationSimplicial Hyperbolic Surfaces
Simplicial Hyperbolic Surfaces Talk by Ken Bromberg August 21, 2007 1-Lipschitz Surfaces- In this lecture we will discuss geometrically meaningful ways of mapping a surface S into a hyperbolic manifold
More informationTwo Connections between Combinatorial and Differential Geometry
Two Connections between Combinatorial and Differential Geometry John M. Sullivan Institut für Mathematik, Technische Universität Berlin Berlin Mathematical School DFG Research Group Polyhedral Surfaces
More informationCS 177 Homework 1. Julian Panetta. October 22, We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F:
CS 177 Homework 1 Julian Panetta October, 009 1 Euler Characteristic 1.1 Polyhedral Formula We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F: V E + F = 1 First,
More information(Discrete) Differential Geometry
(Discrete) Differential Geometry Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties
More informationCurvature Berkeley Math Circle January 08, 2013
Curvature Berkeley Math Circle January 08, 2013 Linda Green linda@marinmathcircle.org Parts of this handout are taken from Geometry and the Imagination by John Conway, Peter Doyle, Jane Gilman, and Bill
More information7. The Gauss-Bonnet theorem
7. The Gauss-Bonnet theorem 7.1 Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationSPERNER S LEMMA, BROUWER S FIXED-POINT THEOREM, AND THE SUBDIVISION OF SQUARES INTO TRIANGLES
SPERNER S LEMMA, BROUWER S FIXED-POINT THEOREM, AND THE SUBDIVISION OF SQUARES INTO TRIANGLES AKHIL MATHEW Abstract These are notes from a talk I gave for high-schoolers at the Harvard- MIT Mathematics
More informationOn the Number of Tilings of a Square by Rectangles
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange University of Tennessee Honors Thesis Projects University of Tennessee Honors Program 5-2012 On the Number of Tilings
More informationEstimating normal vectors and curvatures by centroid weights
Computer Aided Geometric Design 21 (2004) 447 458 www.elsevier.com/locate/cagd Estimating normal vectors and curvatures by centroid weights Sheng-Gwo Chen, Jyh-Yang Wu Department of Mathematics, National
More informationGeodesic Paths on Triangular Meshes
Geodesic Paths on Triangular Meshes DIMAS MARTÍNEZ, LUIZ VELHO, PAULO CEZAR CARVALHO IMPA Instituto Nacional de Matemática Pura e Aplicada - Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, RJ, Brasil
More informationTutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass
Tutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass University of California, Davis, USA http://www.cs.ucdavis.edu/~koehl/ims2017/ What is a shape? A shape is a 2-manifold with a Riemannian
More informationA GENTLE INTRODUCTION TO THE BASIC CONCEPTS OF SHAPE SPACE AND SHAPE STATISTICS
A GENTLE INTRODUCTION TO THE BASIC CONCEPTS OF SHAPE SPACE AND SHAPE STATISTICS HEMANT D. TAGARE. Introduction. Shape is a prominent visual feature in many images. Unfortunately, the mathematical theory
More informationLecture 5: Simplicial Complex
Lecture 5: Simplicial Complex 2-Manifolds, Simplex and Simplicial Complex Scribed by: Lei Wang First part of this lecture finishes 2-Manifolds. Rest part of this lecture talks about simplicial complex.
More informationDon t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?
Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case?
More informationCLASSIFICATION OF SURFACES
CLASSIFICATION OF SURFACES JUSTIN HUANG Abstract. We will classify compact, connected surfaces into three classes: the sphere, the connected sum of tori, and the connected sum of projective planes. Contents
More informationResearch in Computational Differential Geomet
Research in Computational Differential Geometry November 5, 2014 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. Approximations Often
More informationHyplane Polyhedral Models of Hyperbolic Plane
Original Paper Forma, 21, 5 18, 2006 Hyplane Polyhedral Models of Hyperbolic Plane Kazushi AHARA Department of Mathematics School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku,
More informationTripod Configurations
Tripod Configurations Eric Chen, Nick Lourie, Nakul Luthra Summer@ICERM 2013 August 8, 2013 Eric Chen, Nick Lourie, Nakul Luthra (S@I) Tripod Configurations August 8, 2013 1 / 33 Overview 1 Introduction
More informationDISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU /858B Fall 2017
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B Fall 2017 LECTURE 10: DISCRETE CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B
More informationPRESCRIBING CURVATURE FOR PIECEWISE FLAT TRIANGULATED 3-MANIFOLDS
PRESCRIBING CURVATURE FOR PIECEWISE FLAT TRIANGULATED 3-MANIFOLDS HOWARD CHENG AND PETER MCGRATH Abstract. Here we study discrete notions of curvature for piecewise flat triangulated 3-manifolds. Of particular
More informationGeometric Modeling Mortenson Chapter 11. Complex Model Construction
Geometric Modeling 91.580.201 Mortenson Chapter 11 Complex Model Construction Topics Topology of Models Connectivity and other intrinsic properties Graph-Based Models Emphasize topological structure Boolean
More informationExact discrete Morse functions on surfaces. To the memory of Professor Mircea-Eugen Craioveanu ( )
Stud. Univ. Babeş-Bolyai Math. 58(2013), No. 4, 469 476 Exact discrete Morse functions on surfaces Vasile Revnic To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. In this paper,
More informationDISCRETE DIFFERENTIAL GEOMETRY
AMS SHORT COURSE DISCRETE DIFFERENTIAL GEOMETRY Joint Mathematics Meeting San Diego, CA January 2018 DISCRETE CONFORMAL GEOMETRY AMS SHORT COURSE DISCRETE DIFFERENTIAL GEOMETRY Joint Mathematics Meeting
More informationHeegaard splittings and virtual fibers
Heegaard splittings and virtual fibers Joseph Maher maher@math.okstate.edu Oklahoma State University May 2008 Theorem: Let M be a closed hyperbolic 3-manifold, with a sequence of finite covers of bounded
More informationGeometry and Gravitation
Chapter 15 Geometry and Gravitation 15.1 Introduction to Geometry Geometry is one of the oldest branches of mathematics, competing with number theory for historical primacy. Like all good science, its
More informationTHE DNA INEQUALITY POWER ROUND
THE DNA INEQUALITY POWER ROUND Instructions Write/draw all solutions neatly, with at most one question per page, clearly numbered. Turn in the solutions in numerical order, with your team name at the upper
More informationarxiv: v1 [math.co] 15 Apr 2018
REGULAR POLYGON SURFACES IAN M. ALEVY arxiv:1804.05452v1 [math.co] 15 Apr 2018 Abstract. A regular polygon surface M is a surface graph (Σ, Γ) together with a continuous map ψ from Σ into Euclidean 3-space
More informationNotes on Spherical Geometry
Notes on Spherical Geometry Abhijit Champanerkar College of Staten Island & The Graduate Center, CUNY Spring 2018 1. Vectors and planes in R 3 To review vector, dot and cross products, lines and planes
More informationThe Construction of a Hyperbolic 4-Manifold with a Single Cusp, Following Kolpakov and Martelli. Christopher Abram
The Construction of a Hyperbolic 4-Manifold with a Single Cusp, Following Kolpakov and Martelli by Christopher Abram A Thesis Presented in Partial Fulfillment of the Requirement for the Degree Master of
More informationWe have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance.
Solid geometry We have set up our axioms to deal with the geometry of space but have not yet developed these ideas much. Let s redress that imbalance. First, note that everything we have proven for the
More informationMA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces
MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces David L. Finn Today, we continue our discussion of subdivision surfaces, by first looking in more detail at the midpoint method and
More informationMonotone Paths in Geometric Triangulations
Monotone Paths in Geometric Triangulations Adrian Dumitrescu Ritankar Mandal Csaba D. Tóth November 19, 2017 Abstract (I) We prove that the (maximum) number of monotone paths in a geometric triangulation
More informationAn Introduction to Belyi Surfaces
An Introduction to Belyi Surfaces Matthew Stevenson December 16, 2013 We outline the basic theory of Belyi surfaces, up to Belyi s theorem (1979, [1]), which characterizes these spaces as precisely those
More informationAn Investigation of Closed Geodesics on Regular Polyhedra
An Investigation of Closed Geodesics on Regular Polyhedra Tony Scoles Southern Illinois University Edwardsville May 13, 2008 1 Introduction This paper was undertaken to examine, in detail, results from
More information274 Curves on Surfaces, Lecture 5
274 Curves on Surfaces, Lecture 5 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 5 Ideal polygons Previously we discussed three models of the hyperbolic plane: the Poincaré disk, the upper half-plane,
More informationGeometric structures on manifolds
CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory
More informationCAT(0)-spaces. Münster, June 22, 2004
CAT(0)-spaces Münster, June 22, 2004 CAT(0)-space is a term invented by Gromov. Also, called Hadamard space. Roughly, a space which is nonpositively curved and simply connected. C = Comparison or Cartan
More informationBands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes
Bands: A Physical Data Structure to Represent Both Orientable and Non-Orientable 2-Manifold Meshes Abstract This paper presents a physical data structure to represent both orientable and non-orientable
More informationSurfaces Beyond Classification
Chapter XII Surfaces Beyond Classification In most of the textbooks which present topological classification of compact surfaces the classification is the top result. However the topology of 2- manifolds
More informationEdge-disjoint Spanning Trees in Triangulated Graphs on Surfaces and application to node labeling 1
Edge-disjoint Spanning Trees in Triangulated Graphs on Surfaces and application to node labeling 1 Arnaud Labourel a a LaBRI - Universite Bordeaux 1, France Abstract In 1974, Kundu [4] has shown that triangulated
More informationSIMPLICIAL ENERGY AND SIMPLICIAL HARMONIC MAPS
SIMPLICIAL ENERGY AND SIMPLICIAL HARMONIC MAPS JOEL HASS AND PETER SCOTT Abstract. We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence
More informationMeasuring Lengths The First Fundamental Form
Differential Geometry Lia Vas Measuring Lengths The First Fundamental Form Patching up the Coordinate Patches. Recall that a proper coordinate patch of a surface is given by parametric equations x = (x(u,
More informationCLASSIFICATION OF SURFACES
CLASSIFICATION OF SURFACES YUJIE ZHANG Abstract. The sphere, Möbius strip, torus, real projective plane and Klein bottle are all important examples of surfaces (topological 2-manifolds). In fact, via the
More informationLecture 11 COVERING SPACES
Lecture 11 COVERING SPACES A covering space (or covering) is not a space, but a mapping of spaces (usually manifolds) which, locally, is a homeomorphism, but globally may be quite complicated. The simplest
More informationarxiv: v1 [math.co] 17 Jan 2014
Regular matchstick graphs Sascha Kurz Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, Germany Rom Pinchasi Mathematics Dept., Technion Israel Institute of Technology, Haifa 2000,
More informationCHAPTER 8. Essential surfaces
CHAPTER 8 Essential surfaces We have already encountered hyperbolic surfaces embedded in hyperbolic 3-manifolds, for example the 3-punctured spheres that bound shaded surfaces in fully augmented links.
More informationImpulse Gauss Curvatures 2002 SSHE-MA Conference. Howard Iseri Mansfield University
Impulse Gauss Curvatures 2002 SSHE-MA Conference Howard Iseri Mansfield University Abstract: In Riemannian (differential) geometry, the differences between Euclidean geometry, elliptic geometry, and hyperbolic
More informationLecture 5 CLASSIFICATION OF SURFACES
Lecture 5 CLASSIFICATION OF SURFACES In this lecture, we present the topological classification of surfaces. This will be done by a combinatorial argument imitating Morse theory and will make use of the
More informationpα i + q, where (n, m, p and q depend on i). 6. GROMOV S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD
6. GROMOV S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD of π 1 (M 2 )onπ 1 (M 4 ) by conjugation. π 1 (M 4 ) has a trivial center, so in other words the action of π 1 (M 4 ) on itself is effective.
More informationCombinatorial constructions of hyperbolic and Einstein four-manifolds
Combinatorial constructions of hyperbolic and Einstein four-manifolds Bruno Martelli (joint with Alexander Kolpakov) February 28, 2014 Bruno Martelli Constructions of hyperbolic four-manifolds February
More informationINTRODUCTION TO 3-MANIFOLDS
INTRODUCTION TO 3-MANIFOLDS NIK AKSAMIT As we know, a topological n-manifold X is a Hausdorff space such that every point contained in it has a neighborhood (is contained in an open set) homeomorphic to
More informationEULER S FORMULA AND THE FIVE COLOR THEOREM
EULER S FORMULA AND THE FIVE COLOR THEOREM MIN JAE SONG Abstract. In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler s formula and apply
More informationEmbeddability of Arrangements of Pseudocircles into the Sphere
Embeddability of Arrangements of Pseudocircles into the Sphere Ronald Ortner Department Mathematik und Informationstechnologie, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700-Leoben, Austria Abstract
More informationConvex Hulls (3D) O Rourke, Chapter 4
Convex Hulls (3D) O Rourke, Chapter 4 Outline Polyhedra Polytopes Euler Characteristic (Oriented) Mesh Representation Polyhedra Definition: A polyhedron is a solid region in 3D space whose boundary is
More informationA GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES
A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES Mohammed Mostefa Mesmoudi, Leila De Floriani, Paola Magillo Department of Computer Science and Information Science (DISI), University
More informationMathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.
Critical Points and Curvature for Embedded Polyhedral Surfaces Author(s): T. F. Banchoff Source: The American Mathematical Monthly, Vol. 77, No. 5 (May, 1970), pp. 475-485 Published by: Mathematical Association
More informationINTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES
INTRODUCTION TO THE HOMOLOGY GROUPS OF COMPLEXES RACHEL CARANDANG Abstract. This paper provides an overview of the homology groups of a 2- dimensional complex. It then demonstrates a proof of the Invariance
More informationA Flavor of Topology. Shireen Elhabian and Aly A. Farag University of Louisville January 2010
A Flavor of Topology Shireen Elhabian and Aly A. Farag University of Louisville January 2010 In 1670 s I believe that we need another analysis properly geometric or linear, which treats place directly
More informationBranched coverings and three manifolds Third lecture
J.M.Montesinos (Institute) Branched coverings Hiroshima, March 2009 1 / 97 Branched coverings and three manifolds Third lecture José María Montesinos-Amilibia Universidad Complutense Hiroshima, March 2009
More informationDiscrete Differential Geometry: An Applied Introduction
Discrete Differential Geometry: An Applied Introduction Eitan Grinspun with Mathieu Desbrun, Konrad Polthier, Peter Schröder, & Ari Stern 1 Differential Geometry Why do we care? geometry of surfaces Springborn
More informationDiscrete Surfaces. David Gu. Tsinghua University. Tsinghua University. 1 Mathematics Science Center
Discrete Surfaces 1 1 Mathematics Science Center Tsinghua University Tsinghua University Discrete Surface Discrete Surfaces Acquired using 3D scanner. Discrete Surfaces Our group has developed high speed
More informationTHE HALF-EDGE DATA STRUCTURE MODELING AND ANIMATION
THE HALF-EDGE DATA STRUCTURE MODELING AND ANIMATION Dan Englesson danen344@student.liu.se Sunday 12th April, 2011 Abstract In this lab assignment which was done in the course TNM079, Modeling and animation,
More informationPolyhedra inscribed in a hyperboloid & AdS geometry. anti-de Sitter geometry.
Polyhedra inscribed in a hyperboloid and anti-de Sitter geometry. Jeffrey Danciger 1 Sara Maloni 2 Jean-Marc Schlenker 3 1 University of Texas at Austin 2 Brown University 3 University of Luxembourg AMS
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics SIMPLIFYING TRIANGULATIONS OF S 3 Aleksandar Mijatović Volume 208 No. 2 February 2003 PACIFIC JOURNAL OF MATHEMATICS Vol. 208, No. 2, 2003 SIMPLIFYING TRIANGULATIONS OF S
More informationRigidity of ball-polyhedra via truncated Voronoi and Delaunay complexes
!000111! NNNiiinnnttthhh IIInnnttteeerrrnnnaaatttiiiooonnnaaalll SSSyyymmmpppooosssiiiuuummm ooonnn VVVooorrrooonnnoooiii DDDiiiaaagggrrraaammmsss iiinnn SSSccciiieeennnccceee aaannnddd EEEnnngggiiinnneeeeeerrriiinnnggg
More informationTopic: Orientation, Surfaces, and Euler characteristic
Topic: Orientation, Surfaces, and Euler characteristic The material in these notes is motivated by Chapter 2 of Cromwell. A source I used for smooth manifolds is do Carmo s Riemannian Geometry. Ideas of
More information751 Problem Set I JWR. Due Sep 28, 2004
751 Problem Set I JWR Due Sep 28, 2004 Exercise 1. For any space X define an equivalence relation by x y iff here is a path γ : I X with γ(0) = x and γ(1) = y. The equivalence classes are called the path
More informationGeodesic and curvature of piecewise flat Finsler surfaces
Geodesic and curvature of piecewise flat Finsler surfaces Ming Xu Capital Normal University (based on a joint work with S. Deng) in Southwest Jiaotong University, Emei, July 2018 Outline 1 Background Definition
More informationarxiv: v1 [math.ho] 7 Nov 2017
An Introduction to the Discharging Method HAOZE WU Davidson College 1 Introduction arxiv:1711.03004v1 [math.ho] 7 Nov 017 The discharging method is an important proof technique in structural graph theory.
More informationCurvatures of Smooth and Discrete Surfaces
Curvatures of Smooth and iscrete Surfaces John M. Sullivan The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous quantities for discrete curves and surfaces,
More informationGreedy Routing with Guaranteed Delivery Using Ricci Flow
Greedy Routing with Guaranteed Delivery Using Ricci Flow Jie Gao Stony Brook University Joint work with Rik Sarkar, Xiaotian Yin, Wei Zeng, Feng Luo, Xianfeng David Gu Greedy Routing Assign coordinatesto
More informationMatching and Planarity
Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of
More informationPartial Results on Convex Polyhedron Unfoldings
Partial Results on Convex Polyhedron Unfoldings Brendan Lucier 004/1/11 Abstract This paper is submitted for credit in the Algorithms for Polyhedra course at the University of Waterloo. We discuss a long-standing
More informationThree applications of Euler s formula. Chapter 10
Three applications of Euler s formula Chapter 10 A graph is planar if it can be drawn in the plane R without crossing edges (or, equivalently, on the -dimensional sphere S ). We talk of a plane graph if
More informationCHAPTER 8. Essential surfaces
CHAPTER 8 Essential surfaces We have already encountered hyperbolic surfaces embedded in hyperbolic 3-manifolds, for example the 3-punctured spheres that bound shaded surfaces in fully augmented links.
More informationInvariant Measures. The Smooth Approach
Invariant Measures Mathieu Desbrun & Peter Schröder 1 The Smooth Approach On this show lots of derivatives tedious expressions in coordinates For what? only to discover that there are invariant measures
More informationPlanar Graphs. 1 Graphs and maps. 1.1 Planarity and duality
Planar Graphs In the first half of this book, we consider mostly planar graphs and their geometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter
More informationIn this lecture we introduce the Gauss-Bonnet theorem. The required section is The optional sections are
Math 348 Fall 2017 Lectures 20: The Gauss-Bonnet Theorem II Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams.
More informationCYCLE DECOMPOSITIONS AND TRAIN TRACKS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume, Number, Pages S 000-999(0)07- Article electronically published on June, 00 CYCLE DECOMPOSITIONS AND TRAIN TRACKS CHARLES A. MATTHEWS AND DAVID J.
More informationGeometry of Flat Surfaces
Geometry of Flat Surfaces Marcelo iana IMPA - Rio de Janeiro Xi an Jiaotong University 2005 Geometry of Flat Surfaces p.1/43 Some (non-flat) surfaces Sphere (g = 0) Torus (g = 1) Bitorus (g = 2) Geometry
More informationAssignment 8; Due Friday, March 10
Assignment 8; Due Friday, March 10 The previous two exercise sets covered lots of material. We ll end the course with two short assignments. This one asks you to visualize an important family of three
More informationGeometric structures on manifolds
CHAPTER 3 Geometric structures on manifolds In this chapter, we give our first examples of hyperbolic manifolds, combining ideas from the previous two chapters. 3.1. Geometric structures 3.1.1. Introductory
More informationREINER HORST. License or copyright restrictions may apply to redistribution; see
MATHEMATICS OF COMPUTATION Volume 66, Number 218, April 1997, Pages 691 698 S 0025-5718(97)00809-0 ON GENERALIZED BISECTION OF n SIMPLICES REINER HORST Abstract. A generalized procedure of bisection of
More informationLecture notes for Topology MMA100
Lecture notes for Topology MMA100 J A S, S-11 1 Simplicial Complexes 1.1 Affine independence A collection of points v 0, v 1,..., v n in some Euclidean space R N are affinely independent if the (affine
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More information1 Introduction and Review
Figure 1: The torus. 1 Introduction and Review 1.1 Group Actions, Orbit Spaces and What Lies in Between Our story begins with the torus, which we will think of initially as the identification space pictured
More informationLecture 0: Reivew of some basic material
Lecture 0: Reivew of some basic material September 12, 2018 1 Background material on the homotopy category We begin with the topological category TOP, whose objects are topological spaces and whose morphisms
More informationSimplicial Complexes: Second Lecture
Simplicial Complexes: Second Lecture 4 Nov, 2010 1 Overview Today we have two main goals: Prove that every continuous map between triangulable spaces can be approximated by a simplicial map. To do this,
More informationBasics of Combinatorial Topology
Chapter 7 Basics of Combinatorial Topology 7.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union
More informationPortraits of Groups on Bordered Surfaces
Bridges Finland Conference Proceedings Portraits of Groups on Bordered Surfaces Jay Zimmerman Mathematics Department Towson University 8000 York Road Towson, MD 21252, USA E-mail: jzimmerman@towson.edu
More information