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Toponogov.V.A.Differential.Geometry
Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhauser¨ Boston • Basel • Berlin Victor A. Toponogov (deceased) With the editorial assistance of: Department of Analysis and Geometry Vladimir Y. Rovenski Sobolev Institute of Mathematics Department of Mathematics Siberian Branch of the Russian Academy University of Haifa of Sciences Haifa, Israel Novosibirsk-90, 630090 Russia Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: 2005048111 ISBN-10 0-8176-4384-2 eISBN 0-8176-4402-4 ISBN-13 978-0-8176-4384-3 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and re- trieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/EB) 987654321 www.birkhauser.com Contents Preface ....................................................... vii About the Author ............................................. -
Differential Geometry
Differential Geometry J.B. Cooper 1995 Inhaltsverzeichnis 1 CURVES AND SURFACES—INFORMAL DISCUSSION 2 1.1 Surfaces ................................ 13 2 CURVES IN THE PLANE 16 3 CURVES IN SPACE 29 4 CONSTRUCTION OF CURVES 35 5 SURFACES IN SPACE 41 6 DIFFERENTIABLEMANIFOLDS 59 6.1 Riemannmanifolds .......................... 69 1 1 CURVES AND SURFACES—INFORMAL DISCUSSION We begin with an informal discussion of curves and surfaces, concentrating on methods of describing them. We shall illustrate these with examples of classical curves and surfaces which, we hope, will give more content to the material of the following chapters. In these, we will bring a more rigorous approach. Curves in R2 are usually specified in one of two ways, the direct or parametric representation and the implicit representation. For example, straight lines have a direct representation as tx + (1 t)y : t R { − ∈ } i.e. as the range of the function φ : t tx + (1 t)y → − (here x and y are distinct points on the line) and an implicit representation: (ξ ,ξ ): aξ + bξ + c =0 { 1 2 1 2 } (where a2 + b2 = 0) as the zero set of the function f(ξ ,ξ )= aξ + bξ c. 1 2 1 2 − Similarly, the unit circle has a direct representation (cos t, sin t): t [0, 2π[ { ∈ } as the range of the function t (cos t, sin t) and an implicit representation x : 2 2 → 2 2 { ξ1 + ξ2 =1 as the set of zeros of the function f(x)= ξ1 + ξ2 1. We see from} these examples that the direct representation− displays the curve as the image of a suitable function from R (or a subset thereof, usually an in- terval) into two dimensional space, R2. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
The Math in “Laser Light Math”
The Math in “Laser Light Math” When graphed, many mathematical curves are beautiful to view. These curves are usually brought into graphic form by incorporating such devices as a plotter, printer, video screen, or mechanical spirograph tool. While these techniques work, and can produce interesting images, the images are normally small and not animated. To create large-scale animated images, such as those encountered in the entertainment industry, light shows, or art installations, one must call upon some unusual graphing strategies. It was this desire to create large and animated images of certain mathematical curves that led to our design and implementation of the Laser Light Math projection system. In our interdisciplinary efforts to create a new way to graph certain mathematical curves with laser light, Professor Lessley designed the hardware and software and Professor Beale constructed a simplified set of harmonic equations. Of special interest was the issue of graphing a family of mathematical curves in the roulette or spirograph domain with laser light. Consistent with the techniques of making roulette patterns, images created by the Laser Light Math system are constructed by mixing sine and cosine functions together at various frequencies, shapes, and amplitudes. Images created in this fashion find birth in the mathematical process of making “roulette” or “spirograph” curves. From your childhood, you might recall working with a spirograph toy to which you placed one geared wheel within the circumference of another larger geared wheel. After inserting your pen, then rotating the smaller gear around or within the circumference of the larger gear, a graphed representation emerged of a certain mathematical curve in the roulette family (such as the epitrochoid, hypotrochoid, epicycloid, hypocycloid, or perhaps the beautiful rose family). -
Parsing Images Into Regions, Curves, and Curve Groups
Submitted to Int’l J. of Computer Vision, October 2003. First Review March 2005. Revised June 2005. Parsing Images into Regions, Curves, and Curve Groups Zhuowen Tu1 and Song-Chun Zhu2 Lab of Neuro Imaging, Department of Neurology1, Departments of Statistics and Computer Science2, University of California, Los Angeles, Los Angeles, CA 90095. emails: {ztu,sczhu}@stat.ucla.edu Abstract In this paper, we present an algorithm for parsing natural images into middle level vision representations – regions, curves, and curve groups (parallel curves and trees). This al- gorithm is targeted for an integrated solution to image segmentation and curve grouping through Bayesian inference. The paper makes the following contributions. (1) It adopts a layered (or 2.1D-sketch) representation integrating both region and curve models which compete to explain an input image. The curve layer occludes the region layer and curves observe a partial order occlusion relation. (2) A Markov chain search scheme Metropolized Gibbs Samplers (MGS) is studied. It consists of several pairs of reversible jumps to tra- verse the complex solution space. An MGS proposes the next state within the jump scope of the current state according to a conditional probability like a Gibbs sampler and then accepts the proposal with a Metropolis-Hastings step. This paper discusses systematic de- sign strategies of devising reversible jumps for a complex inference task. (3) The proposal probability ratios in jumps are factorized into ratios of discriminative probabilities. The latter are computed in a bottom-up process, and they drive the Markov chain dynamics in a data-driven Markov chain Monte Carlo framework. -
Geometry of Curves and Surfaces Notes C J M Speight 2011
Geometry of Curves and Surfaces Notes c J M Speight 2011 Contact Details Lecturer: Dr. Lamia Alqahtani E-mail [email protected] This note was made by Prof. Martin Speight, School of Mathematics, The University of Leeds, E-mail: [email protected] 0 Why study geometry? Curves and surfaces are all around us in the natural world, and in the built environ- ment. The first step in understanding these structures is to find a mathematically natural way to describe them. That is the primary aim of this course, which will focus particularly on the concept of curvature. As a side effect, you will develop some very useful transferable skills. Key among these is the ability to translate a mathematical system (a set of equations, or formulae, or inequalities) into a visual picture. Your geometric intuition about the picture can then give you useful insight into the original mathematical system. This trick of visualizing mathematical systems can be very powerful and is, unfortunately, not strongly emphasized in the teaching of maths. Example 0 How does the number of solutions of the pair of simultaneous equations xy = 1 x2 + y2 = a2 depend on the constant a > 0? 0 x2 So for a < a0 (in fact, a0 = √2), the system has 0 solutions, for a = a0 it has 2 and for a > a0 it has 4. One could easily verify this by solving the equations explicitly, but the point is that visualizing x the system gave us a very quick (in fact, 1 almost instantaneous) short cut. The above example featured a pair of curves, each associated with an algebraic equation. -
On Geodesic Behavior of Some Special Curves
S S symmetry Article On Geodesic Behavior of Some Special Curves Savin Trean¸t˘a Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania; [email protected] Received: 2 March 2020; Accepted: 17 March 2020; Published: 1 April 2020 Abstract: In this paper, geometric structures on an open subset D ⊆ R2 are investigated such that the graphs associated with the solutions of some special functions to become geodesics. More precisely, we determine the Riemannian metric g such that Bessel (Hermite, harmonic oscillator, Legendre and Chebyshev) ordinary differential equation (ODE) is identified with the geodesic ODEs produced by the Riemannian metric g. The technique is based on the Lagrangian (the energy of the curve) 1 L = k x˙(t) k2, the associated Euler–Lagrange ODEs and their identification with the considered 2 special ODEs. Keywords: auto-parallel curve; geodesic; Euler–Lagrange equations; Lagrangian; special functions MSC: 34A26; 53B15; 53C22 1. Introduction and Preliminaries The concept of connection plays an important role in geometry and, depending on what sort of data one wants to transport along some trajectories, a variety of kinds of connections have been introduced in modern geometry. Crampin et al. [1], in a certain vector bundle, described the construction of a linear connection associated with a second-order differential equation field and, moreover, the corresponding curvature was computed. Ermakov [2] established that linear second-order equations with variable coefficients can be completely integrated only in very rare cases. Further, some aspects of time-dependent second-order differential equations and Berwald-type connections have been studied, with remarkable results, by Sarlet and Mestdag [3]. -
Polar Graphs
POLAR EQUATIONS AND THEIR GRAPHS It is often necessary to transform from rectangular to polar form or vice versa. The following polar-rectangular relationships are useful in this regard. Strategy for Changing Equations in Rectangular Form to Polar Form 2 2 2 • Use the conversions r= x + y , x= r cos θ , and x= r sin θ to find a polar equation. • Write the polar equation as r in terms of θ . If the equation contains an r 2 you MUST show factoring in the solution path! DO NOT EVER divide by r to arrive at the final answer! Strategy for Changing Equations in Polar Form to Rectangular Form • If the equation contains r and sines or cosines, multiply both sides of the equation by r unless the sine or cosine is in the denominator. If the equation contains a trigonometric ratio other than sine or cosine, use the Reciprocal or Quotient Identities to change to sines and cosines first. • If the equation contains r and a constant, square both sides of the equation. • If the equation contains an angle θ and a constant, turn both sides into a tangent ratio remembering that y tan θ = . x Polar equations can also be graphed by hand in the Polar Coordinate System , however, this task often becomes extremely difficult. That is why we need to know the shape of some of the graphs in polar coordinates. 1 For the following "special" polar equations you need to be able to associate their name with their characteristics and graphs! Equations of Circles Circles with center along one of the coordinate axes and radius a Circle with center at the origin and radius a. -
Unit 3. Plane Curves ______
UNIT 3. PLANE CURVES ____________ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Explicit formulas for plane curves, rotation number of a closed curve, osculating circle, evolute, involute, parallel curves, "Umlaufsatz". Convex curves and their characterization, the Four Vertex Theorem. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- This unit and the following one are devoted to the study of curves in low dimensional spaces. We start with plane curves. A plane curve g :[a,b]----------L R2 is given by two coordinate functions. g(t) = ( x(t) , y(t) ) t e [a,b]. The curve g is of general position if the vector g’ is a linearly independent "system of vectors". Since a single vector is linearly independent if and only if it is non-zero, the condition of being a curve of general type is equivalent to regularity for plane curves. From this point on we suppose that g is regular. -
Envelopes of One-Parameter Families of Framed Curves in the Euclidean Space
J. Geom. (2019) 110:48 c 2019 The Author(s) 0047-2468/19/030001-31 published online September 7, 2019 Journal of Geometry https://doi.org/10.1007/s00022-019-0503-1 Envelopes of one-parameter families of framed curves in the Euclidean space Donghe Pei, Masatomo Takahashi, and Haiou Yu Abstract. As a one-parameter family of singular space curves, we con- sider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on one- parameter families of framed curves in the Euclidean 3-space. As appli- cations, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and one- parameter families of spherical Legendre curves, respectively. Mathematics Subject Classification. 58K05, 53A40, 57R45. Keywords. Envelope, one-parameter families of framed curves, singularity. 1. Introduction For regular plane curves, envelopes are classical objects in differential geome- try [1,2,6,7,10,12]. An envelope of a family of curves is a new curve which is tangent to curves of the given family. An envelope of a family of submanifolds in the Euclidean space were studied in [3,15]. Using the notion of stability for germs of family, they investigate the singularities of the envelope of the sub- manifolds in the Euclidean space via Legendrian singularity theory. However, by using implicit functions, the definition and calculation of the envelope of space curves are complicated. On the other hand, if we look at the classical concept of envelope we see that the envelope is the set of characteristic points and avoiding singularities of the elements of the family. -
DIFFERENTIAL GEOMETRY: a First Course in Curves and Surfaces
DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Fall, 2015 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend c 2015 Theodore Shifrin No portion of this work may be reproduced in any form without written permission of the author, other than duplication at nominal cost for those readers or students desiring a hard copy. CONTENTS 1. CURVES.................... 1 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. SomeGlobalResults 23 2. SURFACES: LOCAL THEORY . 35 1. Parametrized Surfaces and the First Fundamental Form 35 2. The Gauss Map and the Second Fundamental Form 44 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57 4. Covariant Differentiation, Parallel Translation, and Geodesics 66 3. SURFACES: FURTHER TOPICS . 79 1. Holonomy and the Gauss-Bonnet Theorem 79 2. An Introduction to Hyperbolic Geometry 91 3. Surface Theory with Differential Forms 101 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107 Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118 SOLUTIONS TO SELECTED EXERCISES . 121 INDEX ................... 124 Problems to which answers or hints are given at the back of the book are marked with an asterisk (*). Fundamental exercises that are particularly important (and to which reference is made later) are marked with a sharp (]). August, 2015 CHAPTER 1 Curves 1. Examples, Arclength Parametrization We say a vector function f .a; b/ R3 is Ck (k 0; 1; 2; : : :) if f and its first k derivatives, f , f ,..., W ! D 0 00 f.k/, exist and are all continuous. -
On the Fibonacci and Lucas Curves
NTMSCI 3, No. 3, 1-10 (2015) 1 New Trends in Mathematical Sciences http://www.ntmsci.com On the Fibonacci and Lucas curves Mahmut Akyigit, Tulay Erisir and Murat Tosun Department of Mathematics, Faculty of Arts and Sciences Sakarya University, 54187 Sakarya-Turkey Received: 3 March 2015, Revised: 9 March 2015, Accepted: 19 March 2015 Published online: 16 June 2015 Abstract: In this study, we have constituted the Frenet vector fields of Fibonacci and Lucas curves which are created with the help of Fibonacci and Lucas sequences hold a very important place in nature defined by Horadam and Shannon, [1]. Based on these vector fields, we have investigated notions of evolute, pedal and parallel curves and obtained the graphics of these curves. Keywords: Fibonacci and Lucas curves, evolute curve, parallel curve, pedal curve. 1 Introduction One of the most fundamental structure of differential geometry is the theory of curves. An increasing interest of this theory makes a development of special curves to be investigated. A way for classification and characterization of curves is the relationship between the Frenet vectors of the curves. By means of this notions, characterizations of special curves have been revealed, ([3], [5], [8], [9]). Some of these are involute-evolute, parallel and pedal curves etc. C. Boyer discovered involutes while trying to build a more accurate clock, [6]. The involute of a plane curve is constructed from the unit tangent at each point of F(s), multiplied by the difference between the arc length to a constant and the arc length to F(s). The evolute and involute of a plane curve are inverse operations similar to Differentiation and Integration.