DOCSLIB.ORG

Differential Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Differential Topology Bjørn Ian Dundas January 10, 2013 2 Contents 1 Preface 7 2 Introduction 9 2.1 Arobot’sarm: ................................. 9 2.2 The configuration space of two electrons . ..... 14 2.3 Statespacesandfiberbundles . .. 15 2.4 Furtherexamples ................................ 17 3 Smooth manifolds 25 3.1 Topologicalmanifolds. .. 25 3.2 Smoothstructures ............................... 28 3.3 Maximalatlases................................. 33 3.4 Smoothmaps .................................. 37 3.5 Submanifolds .................................. 41 3.6 Productsandsums ............................... 45 4 The tangent space 49 4.1 Germs ...................................... 51 4.2 Thetangentspace ............................... 57 4.3 Thecotangentspace .............................. 62 4.4 Derivations ................................... 67 5 Regular values 73 5.1 Therank..................................... 73 5.2 Theinversefunctiontheorem . .. 76 5.3 Theranktheorem................................ 77 5.4 Regularvalues.................................. 80 5.5 Transversality .................................. 87 5.6 Sard’stheorem ................................. 88 5.7 Immersionsandimbeddings . 90 6 Vector bundles 95 6.1 Topologicalvectorbundles . ... 96 3 4 CONTENTS 6.2 Transitionfunctions. 100 6.3 Smoothvectorbundles ............................. 102 6.4 Pre-vectorbundles ............................... 105 6.5 Thetangentbundle............................... 107 6.6 Thecotangentbundle.............................. 113 7 Constructions on vector bundles 117 7.1 Subbundles and restrictions . 117 7.2 Theinducedbundle............................... 122 7.3 Whitneysumofbundles ............................ 124 7.4 More general linear algebra on bundles . ..... 126 7.5 Normalbundles ................................. 131 7.6 Orientations................................... 133 7.7 ThegeneralizedGaussmap . 134 8 Integrability 137 8.1 Flowsandvelocityfields . 138 8.2 Integrability: compactcase. .... 143 8.3 Localflows.................................... 147 8.4 Integrability................................... 149 8.5 Second order differential equations . ..... 150 9 Local phenomena that go global 153 9.1 Refinementsofcovers.............................. 153 9.2 Partitionofunity ................................ 155 9.3 Riemannianstructures . 159 9.4 Normalbundles ................................. 161 9.5 Ehresmann’sfibrationtheorem. 164 10 Appendix: Point set topology 171 10.1 Topologies: openandclosedsets. ..... 172 10.2Continuousmaps ................................ 173 10.3 Basesfortopologies. 174 10.4Separation ................................... 174 10.5Subspaces .................................... 175 10.6Quotientspaces ................................ 176 10.7Compactspaces ................................ 177 10.8Productspaces ................................. 178 10.9Connectedspaces ................................ 178 10.10Settheoreticalstuff . 179 11 Hints or solutions to the exercises 181 References 204 CONTENTS 5 Index 205 6 CONTENTS Chapter 1 Preface 1.0.1 What version is this, and how stable is it? The version you are looking at right now is a β-release resulting from the major revision on Kistrand, Northern Norway in June 2012. The last stable manuscript: August 2007. If you have any comments or suggestion, I will be more than happy to hear from you so that the next stable release of these notes will be maximally helpful. The plan is to keep the text available on the net, also in the future, and I have occa- sionally allowed myself to provide links to interesting sites. If any of these links are dead, please inform me so that I can change them in the next edition. 1.0.2 Acknowledgments First and foremost, I am indebted to the students who have used these notes and given me invaluable feedback. Special thanks go to Håvard Berland, Elise Klaveness and Karen Sofie Ronæss. I owe a couple of anonymous referees much for their diligent reading and many helpful comments. I am also grateful to the Department of Mathematics for allowing me to do the 2012 revision in an inspiring environment. 1.0.3 The history of manifolds Although a fairly young branch of mathematics, the history behind the theory of manifolds is rich and fascinating. The reader should take the opportunity to check up some of the biographies at The MacTutor History of Mathematics archive or at the Wikipedia of the mathematicians that actually are mentioned by name in the text (I have occasionally pro- vided direct links). There is also a page called History Topics: Geometry and Topology Index which is worthwhile spending some time with. Of printed books, I have found Jean Dieudonné’s book [4] especially helpful (although it is mainly concerned with topics beyond the scope of this book). 7 8 CHAPTER 1. PREFACE 1.0.4 Notation We let N = 0, 1, 2,... , Z = ..., 1, 0, 1,... , Q, R and C be the sets of natural numbers, integers,{ rational} numbers,{ real− numbers} and complex numbers. If X and Y are two sets, X Y is the set of ordered pairs (x, y) with x an element in X and y an element in Y . If n is a natural× number, we let Rn and Cn be the vector space of ordered n-tuples of real n 2n n or complex numbers. Occasionally we may identify C with R . If x =(x1,...,xn) R , 2 2 n ∈ n+1 we let x be the norm x1 + + xn. The sphere of dimension n is the subset S R | | n+1··· 0 1⊆ of all x = (x0,...,xn)q R with x = 1 (so that S = 1, 1 R, and S can be viewed as all the complex∈ numbers e|iθ |of unit length). Given{− functions} ⊆ f : X Y and g : Y Z, we write gf for the composite, and g f only if the notation is cluttered→ and → ◦ the improves readability. The constellation g f will occur in the situation where f and g are◦ functions with the same source and target,· and where multiplication makes sense in the target. 1.0.5 How to start reading The text proper starts with chapter 3 on smooth manifolds. If you are weak on point set topology, you will probably want to read the appendix 10 in parallel with chapter 3. The introduction 2 is not strictly necessary for highly motivated readers who can not wait to get to the theory, but provides some informal examples and discussions that may put the theme of these notes in some perspective. You should also be aware of the fact that chapter 6 and 5 are largely independent, and apart from a few exercises can be read in any order. Also, at the cost of removing some exercises and examples, the section on derivations 4.4, the section on the cotangent space/bundle 4.3/6.6 can be removed from the curriculum without disrupting the logical development of ideas. Do the exercises, and only peek at the hints if you really need to. Kistranda January 10, 2013 Chapter 2 Introduction The earth is round. This may at one point have been hard to believe, but we have grown accustomed to it even though our everyday experience is that the earth is (fairly) flat. Still, the most effective way to illustrate it is by means of maps: a globe is a very neat device, but its global(!) A globe character makes it less than practical if you want to represent fine details. This phenomenon is quite common: locally you can represent things by means of “charts”, but the global character can’t be represented by one single chart. You need an entire atlas, and you need to know how the charts are to be assembled, or even better: the charts overlap so that we know how they all fit together. The mathematical framework for working with such situations is manifold theory. These notes are about manifold the- ory, but before we start off with the details, let us take an informal look at some examples illustrating the basic structure. 2.1 A robot’s arm: To illustrate a few points which will be important later on, we discuss a concrete situation in some detail. The features that appear are special cases of general phenomena, and hopefully the example will provide the reader with some deja vue experiences later on, when things are somewhat more obscure. Consider a robot’s arm. For simplicity, assume that it moves in the plane, has three joints, with a telescopic middle arm (see figure). 9 10 CHAPTER 2. INTRODUCTION y x z Call the vector defining the inner arm x, the second arm y and the third arm z. Assume x = z = 1 and y [1, 5]. Then the robot can reach anywhere inside a circle of radius 7.| | But| most| of these| | ∈ positions can be reached in several different ways. In order to control the robot optimally, we need to understand the various configura- tions, and how they relate to each other. As an example, place the robot in the origin and consider all the possible positions of the arm that reach the point P = (3, 0) R2, i.e., look at the set T of all triples (x, y, z) R2 R2 R2 such that ∈ ∈ × × x + y + z = (3, 0), x = z =1, and y [1, 5]. | | | | | | ∈ We see that, under the restriction x = z = 1, x and z can be chosen arbitrarily, and | | | | determine y uniquely. So T is “the same as” the set (x, z) R2 R2 x = z =1 . { ∈ × || | | | } Seemingly, our space T of configurations resides in four-dimensional space R2 R2 = R4, × ∼ but that is an illusion – the space is two-dimensional and turns out to be a familiar shape. We can parametrize x and z by angles if we remember to identify the angles 0 and 2π. So T is what you get if you consider the square [0, 2π] [0, 2π] and identify the edges as in × the picture below. A B B A 2.1. A ROBOT’S ARM: 11 See http://www.it.brighton.ac.uk/staff/jt40/MapleAnimations/Torus.html for a nice animation of how the plane model gets glued. In other words: The set T of all positions such that the robot reaches P = (3, 0) may be identified with the torus. This is also true topologically in the sense that “close configurations” of the robot’s arm correspond to points close to each other on the torus. 2.1.1 Question What would the space S of positions look like if the telescope got stuck at y = 2? | | Partial answer to the question: since y = (3, 0) x z we could try to get an idea of what points of T satisfy y = 2 by means of inspection− − of the graph of y .
Recommended publications
  • Understanding Euler Angles 1

    Understanding Euler Angles 1

    Welcome Guest View Cart (0 items) Login Home Products Support News About Us Understanding Euler Angles 1. Introduction Attitude and Heading Sensors from CH Robotics can provide orientation information using both Euler Angles and Quaternions. Compared to quaternions, Euler Angles are simple and intuitive and they lend themselves well to simple analysis and control. On the other hand, Euler Angles are limited by a phenomenon called "Gimbal Lock," which we will investigate in more detail later. In applications where the sensor will never operate near pitch angles of +/‐ 90 degrees, Euler Angles are a good choice. Sensors from CH Robotics that can provide Euler Angle outputs include the GP9 GPS‐Aided AHRS, and the UM7 Orientation Sensor. Figure 1 ‐ The Inertial Frame Euler angles provide a way to represent the 3D orientation of an object using a combination of three rotations about different axes. For convenience, we use multiple coordinate frames to describe the orientation of the sensor, including the "inertial frame," the "vehicle‐1 frame," the "vehicle‐2 frame," and the "body frame." The inertial frame axes are Earth‐fixed, and the body frame axes are aligned with the sensor. The vehicle‐1 and vehicle‐2 are intermediary frames used for convenience when illustrating the sequence of operations that take us from the inertial frame to the body frame of the sensor. It may seem unnecessarily complicated to use four different coordinate frames to describe the orientation of the sensor, but the motivation for doing so will become clear as we proceed. For clarity, this application note assumes that the sensor is mounted to an aircraft.
  • Hirschdx.Pdf

    Hirschdx.Pdf

    130 5. Degrees, Intersection Numbers, and the Euler Characteristic. 2. Intersection Numbers and the Euler Characteristic M c: Jr+1 Exercises 8. Let be a compad lHlimensionaJ submanifold without boundary. Two points x, y E R"+' - M are separated by M if and only if lkl(x,Y}.MI * O. lSee 1. A complex polynomial of degree n defines a map of the Riemann sphere to itself Exercise 7.) of degree n. What is the degree of the map defined by a rational function p(z)!q(z)? 9. The Hopfinvariant ofa map f:5' ~ 5' is defined to be the linking number Hlf) ~ 2. (a) Let M, N, P be compact connected oriented n·manifolds without boundaries Lk(g-l(a),g-l(b)) (see Exercise 7) where 9 is a c~ map homotopic 10 f and a, b are and M .!, N 1. P continuous maps. Then deg(fg) ~ (deg g)(deg f). The same holds distinct regular values of g. The linking number is computed in mod 2 if M, N, P are not oriented. (b) The degree of a homeomorphism or homotopy equivalence is ± 1. f(c) * a, b. *3. Let IDl. be the category whose objects are compact connected n-manifolds and whose (a) H(f) is a well-defined homotopy invariant off which vanishes iff is nuD homo- topic. morphisms are homotopy classes (f] of maps f:M ~ N. For an object M let 7t"(M) (b) If g:5' ~ 5' has degree p then H(fg) ~ pH(f). be the set of homotopy classes M ~ 5".
  • Euler Quaternions

    Euler Quaternions

    Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.
  • New Ideas in Algebraic Topology (K-Theory and Its Applications)

    New Ideas in Algebraic Topology (K-Theory and Its Applications)

    NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems.
  • On the Homology of Configuration Spaces

    On the Homology of Configuration Spaces

    TopologyVol. 28, No. I. pp. I II-123, 1989 Lmo-9383 89 53 al+ 00 Pm&d I” Great Bntam fy 1989 Pcrgamon Press plc ON THE HOMOLOGY OF CONFIGURATION SPACES C.-F. B~DIGHEIMER,~ F. COHEN$ and L. TAYLOR: (Receiued 27 July 1987) 1. INTRODUCTION 1.1 BY THE k-th configuration space of a manifold M we understand the space C”(M) of subsets of M with catdinality k. If c’(M) denotes the space of k-tuples of distinct points in M, i.e. Ck(M) = {(z,, . , zk)eMklzi # zj for i #j>, then C’(M) is the orbit space of C’(M) under the permutation action of the symmetric group Ik, c’(M) -+ c’(M)/E, = Ck(M). Configuration spaces appear in various contexts such as algebraic geometry, knot theory, differential topology or homotopy theory. Although intensively studied their homology is unknown except for special cases, see for example [ 1, 2, 7, 8, 9, 12, 13, 14, 18, 261 where different terminology and notation is used. In this article we study the Betti numbers of Ck(M) for homology with coefficients in a field IF. For IF = IF, the rank of H,(C’(M); IF,) is determined by the IF,-Betti numbers of M. the dimension of M, and k. Similar results were obtained by Ldffler-Milgram [ 171 for closed manifolds. For [F = ff, or a field of characteristic zero the corresponding result holds in the case of odd-dimensional manifolds; it is no longer true for even-dimensional manifolds, not even for surfaces, see [S], [6], or 5.5 here.
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology

    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
  • Evaluation of MEMS Accelerometer and Gyroscope for Orientation Tracking Nutrunner Functionality

    Evaluation of MEMS Accelerometer and Gyroscope for Orientation Tracking Nutrunner Functionality

    EXAMENSARBETE INOM ELEKTROTEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2017 Evaluation of MEMS accelerometer and gyroscope for orientation tracking nutrunner functionality Utvärdering av MEMS accelerometer och gyroskop för rörelseavläsning av skruvdragare ERIK GRAHN KTH SKOLAN FÖR TEKNIK OCH HÄLSA Evaluation of MEMS accelerometer and gyroscope for orientation tracking nutrunner functionality Utvärdering av MEMS accelerometer och gyroskop för rörelseavläsning av skruvdragare Erik Grahn Examensarbete inom Elektroteknik, Grundnivå, 15 hp Handledare på KTH: Torgny Forsberg Examinator: Thomas Lind TRITA-STH 2017:115 KTH Skolan för Teknik och Hälsa 141 57 Huddinge, Sverige Abstract In the production industry, quality control is of importance. Even though today's tools provide a lot of functionality and safety to help the operators in their job, the operators still is responsible for the final quality of the parts. Today the nutrunners manufactured by Atlas Copco use their driver to detect the tightening angle. There- fore the operator can influence the tightening by turning the tool clockwise or counterclockwise during a tightening and quality cannot be assured that the bolt is tightened with a certain torque angle. The function of orientation tracking was de- sired to be evaluated for the Tensor STB angle and STB pistol tools manufactured by Atlas Copco. To be able to study the orientation of a nutrunner, practical exper- iments were introduced where an IMU sensor was fixed on a battery powered nutrunner. Sensor fusion in the form of a complementary filter was evaluated. The result states that the accelerometer could not be used to estimate the angular dis- placement of tightening due to vibration and gimbal lock and therefore a sensor fusion is not possible.
  • Differential Topology from the Point of View of Simple Homotopy Theory

    Differential Topology from the Point of View of Simple Homotopy Theory

    PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. BARRY MAZUR Differential topology from the point of view of simple homotopy theory Publications mathématiques de l’I.H.É.S., tome 15 (1963), p. 5-93 <http://www.numdam.org/item?id=PMIHES_1963__15__5_0> © Publications mathématiques de l’I.H.É.S., 1963, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CHAPTER 1 INTRODUCTION It is striking (but not uncharacteristic) that the « first » question asked about higher dimensional geometry is yet unsolved: Is every simply connected 3-manifold homeomorphic with S3 ? (Its original wording is slightly more general than this, and is false: H. Poincare, Analysis Situs (1895).) The difficulty of this problem (in fact of most three-dimensional problems) led mathematicians to veer away from higher dimensional geometric homeomorphism-classificational questions. Except for Whitney's foundational theory of differentiable manifolds and imbed- dings (1936) and Morse's theory of Calculus of Variations in the Large (1934) and, in particular, his analysis of the homology structure of a differentiable manifold by studying critical points of 0°° functions defined on the manifold, there were no classificational results about high dimensional manifolds until the era of Thorn's Cobordisme Theory (1954), the beginning of" modern 9? differential topology.
  • Topics in Low Dimensional Computational Topology

    Topics in Low Dimensional Computational Topology

    THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés.
  • Dirac's Belt Trick, Gyroscopes, and the Ipad

    Dirac's Belt Trick, Gyroscopes, and the Ipad

    Dirac's Belt Trick, Gyroscopes, and the iPad Richard Koch May 19, 2012 Dirac's Belt Trick P. A. M. Dirac, 1902 - 1984 Nobel Prize (with Erwin Schrodinger) in 1933 Formulated Dirac equation, a relativistically correct quantum mechanical description of the electron, which predicted the existence of antiparticles. My Home Page: http://pages.uoregon.edu/koch TeXShop for Macintosh; TeX by Donald Knuth for Everything 2 \documentclass[11pt]{amsart} 2 25 Using T X, we can typeset 1+x+x and the matrix . \usepackage[paper width = 6in, paperheight = 7in]{geometry} E e2x+p5 p10 7 \usepackage[parfill]{parskip} According to calculus q ✓ − ◆ \usepackage{graphicx} 1 2 1 x2 p⇡ 2x +3x dx = 2 and e− dx = \begin{document} 2 Z0 Z0 Using \TeX, we can typeset $\sqrt{ {{1 + x + x^2} The path of a particle in a gravitational field is given by γi(t) \over {e^{2x + \sqrt{5}}}}}$ where and the matrix $\left( \begin{array}{cc} 2 & 5 \\ d2γ dγ dγ i + Γi i j =0 \sqrt{10} & -7 \end{array} \right)$. dt2 jk dt dt According to calculus Xjk $$\int_0^1 {2x + 3x^2}\ dx = 2 \hspace{.2in} \mbox{and} \hspace{.2in} \int_0^\infty e^{- x^2} \ dx = {{\sqrt{\pi}} \over 2}$$ The path of a particle in a gravitational field is given by $\gamma_i(t)$ where $${{d^2 \gamma_i} \over {d t^2}} + \sum_{jk} \Gamma^i_{jk} {{d \gamma_i} \over {dt}} {{d \gamma_j} \over {dt}} = 0$$ \begin{figure}[htbp] \centering \includegraphics[width=2in]{MoveTest-math.jpg} \end{figure} \end{document} 1 WWDC, Apple's Worldwide Developer's Conference Gyroscopes in the iPhone and iPad 1.
  • Baumgarte Stabilisation Over the SO(3) Rotation Group for Control

    Baumgarte Stabilisation Over the SO(3) Rotation Group for Control

    2015 IEEE 54th Annual Conference on Decision and Control (CDC) December 15-18, 2015. Osaka, Japan Baumgarte Stabilisation over the SO(3) Rotation Group for Control Sebastien Gros, Marion Zanon, Moritz Diehl Abstract— Representations of the SO(3) rotation group are quaternion and the Direction Cosine Matrix (DCM). For a crucial for airborne and aerospace applications. Euler angles complete survey, we refer to [2]. is a popular representation in many applications, but yield Quaternions are an extension of complex numbers to models having singular dynamics. This issue is addressed via non-singular representations, operating in dimensions higher a four-dimensional space, which allows for describing the than 3. Unit quaternions and the Direction Cosine Matrix are SO(3) rotation group. They can be construed as four- the best known non-singular representations, and favoured in dimensional vectors q R4. The SO(3) rotation group is challenging aeronautic and aerospace applications. All non- 2 4 then covered by q Q := q R q>q = 1 with: singular representations yield invariants in the model dynamics, 2 2 j i.e. a set of nonlinear algebraic conditions that must be fulfilled R(q) = E(q)G(q)> SO(3); by the model initial conditions, and that remain fulfilled over 2 2 3 q1 q0 q3 q2 time. However, due to numerical integration errors, these condi- − − tions tend to become violated when using standard integrators, G(q) = 4 q2 q3 q0 q1 5; − − making the model inconsistent with the physical reality. This q3 q2 q1 q0 issue poses some challenges when non-singular representations 2 − − 3 q1 q0 q3 q2 are deployed in optimal control.
  • PDF Download

    PDF Download

    International Journal of Latest Research in Engineering and Technology (IJLRET) ISSN: 2454-5031 www.ijlret.com || Volume 04 - Issue 09 || September 2018 || PP. 25-32 The Controlled Human Gyroscope – Virtual Reality Motion Base Simulator Randy C. Arjunsingh1, Matthew J. Jensen1, Razvan Rusovici2, Ondrej Doule3 1(Mechanical and Civil Engineering Department, Florida Institute of Technology, United States) 2(Aerospace, Physics and Space Sciences Department, Florida Institute of Technology, United States) 3(Computer Engineering and Sciences Department, Florida Institute of Technology, United States) Abstract: Flight motion simulators are currently used for flight training and research, but there are many limitations to these existing systems. This paper presents a low-cost design for a rotational motion platform titled, ‘The Controlled Human Gyroscope’. It uses a 4-axis system instead of the conventional 3-axis system to avoid gimbal lock and prevent the unnecessary motion of the user. The Human Gyroscope features unlimited rotation about the roll, pitch and yaw axes regardless of the occupant’s orientation. It will therefore provide high fidelity motion simulation and if it is paired with a translational motion platform, it can provide up to 6 degrees of freedom. Equations of motion for this specific system are presented in this paper and can be used to develop a control algorithm. Keywords: Flight motion simulator, Gimbal lock, Human gyroscope, Full rotational control, Rotational Motion I. INTRODUCTION Motion simulation provides a cheap and safe alternative to field training. It submerges the operator into a virtual environment and synchronizes this virtual motion with actual movement of the operator. Mistakes in the field can therefore be avoided, situations can be more accurately replicated than in static simulators and dangerous training situations in real-world no longer have consequences, as they are not in an actual aircraft.