DOCSLIB.ORG

A Treatise on the Geometry of Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

BOSIDNCOUEGESCiENCEUBR/iri? £rf f2. Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/treatiseongeometOObass A TREATISE ON THE GEOMETRY OF SURFACES i,^.^ ^^U^i^^ ON THE GEOMETRY OF SUEFACES BY A. B. BASSET MA. F.R.S. TRINITY COLLEGE CAMBRIDGE MATH. DEPT. BOSTON COLLEGE LIBRARY CHESTNUT HILL, MASS, CAMBRIDGE DEIGHTON BELL AND CO. LONDON GEORGE BELL AND SONS I910 [AH Rights reserved'] : (Cambritige PRINTED BY JOHN CLAY, M.A. AT THE UNIVEKSITY PRESS. 1937DE PEEFACE THE last edition of Salmon's Analytic Geometry of Three Dimensions, which was published in 1884, has been out of print for some years ; and although there are several excellent works on Quadric Surfaces and other special branches of the subject, such as those of Mr Blythe on Cubic Surfaces and of the late Mr Hudson on Kummers Quartic Surface, yet there is no British treatise exclusively devoted to the theory of surfaces of higher degree than the second. I have therefore endeavoured to supply this want in the present work. The Theory of Surfaces is an extensive one, and a thoroughly comprehensive treatise would necessarily be voluminous. I have therefore decided to limit this work to the more elementary />^ a i^^ portions of the subject, and have abstained fronTrTntroducing ^>^^/ inves¥igations^"wliich require a knowledge of the Theory of Cm^^ Functions and of the higher branches of Modern Algebra. The|^^j^»</ ordinary methods of Analytical Geometry are quite sufficient to <S**'*^' enable the properties of cubic and quartic surfaces and twisted /Y^JC^^/ curves, and also the point and plane singularities of surfaces, to Mfm^i be discussed with tolerable completeness, and to demonstrate a ^^^vXi number of interesting and important theorems connected with ^rJiWW/i them ; but for the purpose of confining this treatise within a ^S^^*^ moderate compass, I have abstained from any general discussion of ^<«-^**' surfaces of higher degree than the fourth. The properties of a point-singularity may usually be examined I by means of a surface ofHow degree just as well as by one of the 1 nth degree ; but if the degree is less than a certain limit, which ) depends on the character of the singularity, the latter appears in an incomplete form on the surface. Thus the properties of a triple line cannot be fully investigated without employing a surface of the seventh degree, and this fact has rendered it necessary to partially discuss surfaces of higher degree than a quartic. VI PREFACE The resolution of a multiple point into its constituents has been discussed by Professor Segre of Turin, and other Italian mathematicians, in various papers jpublished in the Annali di important Matematica ; and these researches have shown that an analogy exists between the theories of plane curves and of surfaces. The class of an anautotomic plane curve of degree n, and also the reduction of class produced by a multiple point of order n, the — tangents at which are distinct, are both equal to n {n 1) ; whilst the constituents of the multiple point are \n{n — \) nodes. The class of an anautotomic surface of degree n, and also the reduction of class produced by a multiple point of order n, the tangent cone at which is anautotomic, are both equal to n{n — Vf; and from analogy I concluded that the constituents of the multiple point were ^n{n — lf conic nodes. In 1908 I succeeded in obtaining a formal proof of the last theorem, which enables a large number of singular points to be resolved into their constituent conic nodes and binodes. In the present treatise I have incorporated a variety of results, originally due to Italian and German mathematicians, many of which have been published since the last edition of Salmon's work ; and I have endeavoured to modernize the analysis and the terminology by discarding antiquated methods and inappropriate symbols and phrases. I have also to express my obligations to the late Professor Cayley's papers, references to which are denoted by the letters C. M. P. ; as well as to the Repertorio di Matematiche Superiori by Professor E. Pascal, which contains a valuable epitome of the subject, together with an exhaustive collection of references to the original papers of British and foreign mathematicians, who have studied this subject. Fledborough Hall, HoLYPORT, Berks. March, 1910. CONTENTS CHAPTER I. THEORY OF SURFACES. AKT. PAGE 1. Equation of a surface .... 1 2. Four distinct species of surfaces 2 3-4. Quadriplanar coordinates 2 5. Section by a tangent plane 3 6. Conic nodes 5 7. Binodes .... 5 8. Unodes .... 6 9. Tangential equation of a surface 7 10. The spinodal, flecnodal and bitangential curves 7 11. The six singular tangent planes 8 12. Boothian coordinates . .9 13. Polar surfaces 10 14-5. Class of a surface . 13 16. Tangent cones . 14 17-8. Nodal and cuspidal generators of tangent cone 15 19. Class of a surface which possesses C conic nodes and B binodes 16 20. Number of tangent planes which can be drawn through certain singular points . 17 21. Double and stationary tangent planes to tangent cone 17 22. Points of contact of stationary planes to tangent cone lie on the spinodal curve 18 23. Maximum number of double points 18 24. Segre's theorem. Multiple points 20 25-7. Different kinds of binodes 20 28. Three primary species of unodes 21 29. Tropes 23 30. Bitropes and unitropes 24 31. Singular lines, the tangent planes along which are fixed 25 32-3. A line lying in a surface is torsal or singular 25 34. Surfaces of a higher degree than the third cannot in general have lines lying in them 26 Vlll CONTENTS ABT. 35. Every tangent plane at a point on a torsal line is a multiple tangent plane .... 36. Three distinct species of nodal lines 37-9. Pinch points, and their properties 40. Tangent plane along a nodal line 41. Nodal lines of the third kind . 42. Number of points of intersection of three surfaces which possess a common multiple line .... 43. Reduction of class produced by a nodal line 44-5. Other properties of lines lying in a surface 46-8. On the intersections of surfaces 49. The Hessian 50. The Hessian intersects the surface in the spinodal curve 51. A conic node on a surface produces a conic node on the Hessian having the same nodal cone 52. The spinodal ciu-ve has a sextuple point at a conic node 53. A binode produces a cubic node on the Hessian and an octuple point on the spinodal curve .... 54. A unode produces a quartic node on the Hessian 55. Lines lying in a surface touch the spinodal curve 56. The curve of contact of a trope forms part of the spinodal curve ........... 57. AVhen a fixed plane touches a surface along a straight line, the latter twice repeated forms part of the spinodal curve 58-9. Cayley's theorems with respect to the degrees of the spinodal and flecnodal curves 60. The Hessian possesses conic nodes 61. The Steinerian .... CHAPTER II. CUBIC SURFACES. 62. Twenty-three different species of cubic surfaces 63. Equation of a cubic surface 64. Intersection of a quadric and a cubic .... 65. An anautotomic cubic surface possesses 27 lines lying in it 66. Such a surface possesses 45 triple tangent planes . 67. It has 54 singular planes, whose point of contact is a tacnode on the section 68. Double-sixes. ......... 69. The Hessian of a cubic surface. Conjugate poles . 70-3. Properties of conjugate poles 74-5. The polar quadric of the cubic with respect to a conic node on the Hessian, consists of two planes. 76. Sylvester's canonical form of a quaternary cubic 77. Sylvester's pentahedron ....... 78. Singularities of cubic surfaces CONTENTS IX ART. CONTENTS AKT. 120. Its mode of generation . 121. Reciprocal relations . 122. Unicursal curves usually possess cusps . 123. Other singularities of such curves 124. The singularities a, or, t, i and S do not in general exist 125. Deficiency of a twisted curve 126. The Pliicker-Cayley equations for certain twisted curves 127. Ditto, when the deficiency is zero 128- 9. Twisted cubic curves . 130. Equations of ditto . 131. Developable of which a twisted cubic is the edge of regression 132. A property of a twisted cubic 133. Twisted cubics are anautotomic curves . 134. Characteristics of a twisted cubic .... 135. Connection between plane and twisted cubics 136. Twisted quartic curves 137. Table of their characteristics ..... 138- 9. Quartics of the first species ..... 140. Connection between twisted quartics and plane cubics 141- 2. Connection between twisted quartics of the first species and plane binodal quartics . 143. Quartics of the first species may be the partial intersection of a quadric and a cubic surface 144. Quartics of the second species .... 145. Such a curve cannot possess actual double points 146. All autotomic quartics and quartics of the second species are included amongst the edges of regression of a certain sextic developable ..... 147- 8. Properties of those developables . 149- 50. Nodal quartics and their reciprocals . 151- 4. Cuspidal quartics and their reciprocals 155. Anautotomic quartics of the first species constitute a class of curves sioi generis .... 156. Quintic curves. Four primary species 157- 8. Quintic curves of the first species 159. „ „ „ second „ 160. „ „ „ third „ 161. fourth „ CHAPTER IV. COMPOUND SINGULARITIES OF PLANE CURVES. 162. Object of investigation 115 163. Point, line and mixed singularities 115 164. Determination of the point and line constituents of a singu- larity 115 CONTENTS XI ART. PAGE 165. Constituents of a multiple point 116 166.
Recommended publications
  • Algebraic Geometry Has Several Aspects to It

    Algebraic Geometry Has Several Aspects to It

    Lectures on Geometry of Plane Curves An Introduction to Algegraic Geometry ANANT R. SHASTRI Department of Mathematics Indian Institute of Technology, Mumbai Spring 1999 Contents 1 Introductory Remarks 3 2 Affine Spaces and Projective Spaces 6 3 Homogenization and De-homogenization 8 4 Defining Equation of a Curve 10 5 Relation Between Affine and Projective Curves 14 6 Resultant 17 7 Linear Transformations 21 8 Simple and Singular Points 24 9 Bezout’s Theorem 28 10 Basic Inequalities 32 11 Rational Curve 35 12 Co-ordinate Ring and the Quotient Field 37 13 Zariski Topology 39 14 Regular and Rational Maps 43 15 Closed Subspaces of Projective Spaces 48 16 Quasi Projective Varieties 50 17 Regular Functions on Quasi Projective Varieties 51 1 18 Rational Functions 55 19 Product of Quasi Projective Varieties 58 20 A Reduction Process 62 21 Study of Cubics 65 22 Inflection Points 69 23 Linear Systems 75 24 The Dual Curve 80 25 Power Series 85 26 Analytic Branches 90 27 Quadratic Transforms: 92 28 Intersection Multiplicity 94 2 Chapter 1 Introductory Remarks Lecture No. 1 31st Dec. 98 The present day algebraic geometry has several aspects to it. Let us illustrate two of the main aspects by some examples: (i) Solve geometric problems using algebraic techniques: Here is an example. To find the possible number of points of intersection of a circle and a straight line, we take the general equation of a circle and substitute for X (or Y ) using the general equation of a straight line and get a quadratic equation in one variable.
  • Combination of Cubic and Quartic Plane Curve

    Combination of Cubic and Quartic Plane Curve

    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis.
  • Polynomial Curves and Surfaces

    Polynomial Curves and Surfaces

    Polynomial Curves and Surfaces Chandrajit Bajaj and Andrew Gillette September 8, 2010 Contents 1 What is an Algebraic Curve or Surface? 2 1.1 Algebraic Curves . .3 1.2 Algebraic Surfaces . .3 2 Singularities and Extreme Points 4 2.1 Singularities and Genus . .4 2.2 Parameterizing with a Pencil of Lines . .6 2.3 Parameterizing with a Pencil of Curves . .7 2.4 Algebraic Space Curves . .8 2.5 Faithful Parameterizations . .9 3 Triangulation and Display 10 4 Polynomial and Power Basis 10 5 Power Series and Puiseux Expansions 11 5.1 Weierstrass Factorization . 11 5.2 Hensel Lifting . 11 6 Derivatives, Tangents, Curvatures 12 6.1 Curvature Computations . 12 6.1.1 Curvature Formulas . 12 6.1.2 Derivation . 13 7 Converting Between Implicit and Parametric Forms 20 7.1 Parameterization of Curves . 21 7.1.1 Parameterizing with lines . 24 7.1.2 Parameterizing with Higher Degree Curves . 26 7.1.3 Parameterization of conic, cubic plane curves . 30 7.2 Parameterization of Algebraic Space Curves . 30 7.3 Automatic Parametrization of Degree 2 Curves and Surfaces . 33 7.3.1 Conics . 34 7.3.2 Rational Fields . 36 7.4 Automatic Parametrization of Degree 3 Curves and Surfaces . 37 7.4.1 Cubics . 38 7.4.2 Cubicoids . 40 7.5 Parameterizations of Real Cubic Surfaces . 42 7.5.1 Real and Rational Points on Cubic Surfaces . 44 7.5.2 Algebraic Reduction . 45 1 7.5.3 Parameterizations without Real Skew Lines . 49 7.5.4 Classification and Straight Lines from Parametric Equations . 52 7.5.5 Parameterization of general algebraic plane curves by A-splines .
  • Construction of Rational Surfaces of Degree 12 in Projective Fourspace

    Construction of Rational Surfaces of Degree 12 in Projective Fourspace

    Construction of Rational Surfaces of Degree 12 in Projective Fourspace Hirotachi Abo Abstract The aim of this paper is to present two different constructions of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants. 1 Introduction Hartshone conjectured that only finitely many components of the Hilbert scheme of surfaces in P4 contain smooth rational surfaces. In 1989, this conjecture was positively solved by Ellingsrud and Peskine [7]. The exact bound for the degree is, however, still open, and hence the question concerning the exact bound motivates us to find smooth rational surfaces in P4. The goal of this paper is to construct five different families of smooth rational surfaces in P4 with degree 12 and sectional genus 13. The rational surfaces in P4 were previously known up to degree 11. In this paper, we present two different constructions of smooth rational surfaces in P4 with the given invariants. Both the constructions are based on the technique of “Beilinson monad”. Let V be a finite-dimensional vector space over a field K and let W be its dual space. The basic idea behind a Beilinson monad is to represent a given coherent sheaf on P(W ) as a homology of a finite complex whose objects are direct sums of bundles of differentials. The differentials in the monad are given by homogeneous matrices over an exterior algebra E = V V . To construct a Beilinson monad for a given coherent sheaf, we typically take the following steps: Determine the type of the Beilinson monad, that is, determine each object, and then find differentials in the monad.
  • A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

    A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by EPrints Complutense A brief note on the approach to the conic sections of a right circular cone from dynamic geometry Eugenio Roanes–Lozano Abstract. Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri Geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: z = 0) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic. Mathematics Subject Classification (2010). Primary 68U99; Secondary 97G40; Tertiary 68U05. Keywords. Conics, Dynamic Geometry, 3D Geometry, Computer Algebra. 1. Introduction Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5 [2], Calques 3D [7, 10] and Cabri Geometry 3D [11]. Focusing on the most widespread one (GeoGebra 5) and conics, there is a “conics” icon in the ToolBar.
  • Chapter 9: Conics Section 9.1 Ellipses

    Chapter 9: Conics Section 9.1 Ellipses

    Chapter 9: Conics Section 9.1 Ellipses ..................................................................................................... 579 Section 9.2 Hyperbolas ............................................................................................... 597 Section 9.3 Parabolas and Non-Linear Systems ......................................................... 617 Section 9.4 Conics in Polar Coordinates..................................................................... 630 In this chapter, we will explore a set of shapes defined by a common characteristic: they can all be formed by slicing a cone with a plane. These families of curves have a broad range of applications in physics and astronomy, from describing the shape of your car headlight reflectors to describing the orbits of planets and comets. Section 9.1 Ellipses The National Statuary Hall1 in Washington, D.C. is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to reflect from the walls in a special way. Two people standing in specific places are able to hear each other whispering even though they are far apart. To determine where they should stand, we will need to better understand ellipses. An ellipse is a type of conic section, a shape resulting from intersecting a plane with a cone and looking at the curve where they intersect. They were discovered by the Greek mathematician Menaechmus over two millennia ago. The figure below2 shows two types of conic sections. When a plane is perpendicular to the axis of the cone, the shape of the intersection is a circle. A slightly titled plane creates an oval-shaped conic section called an ellipse. 1 Photo by Gary Palmer, Flickr, CC-BY, https://www.flickr.com/photos/gregpalmer/2157517950 2 Pbroks13 (https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg), “Conic sections with plane”, cropped to show only ellipse and circle by L Michaels, CC BY 3.0 This chapter is part of Precalculus: An Investigation of Functions © Lippman & Rasmussen 2020.
  • Fundamental Theorems in Mathematics

    Fundamental Theorems in Mathematics

    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
  • View This Volume's Front and Back Matter

    View This Volume's Front and Back Matter

    Functions of Several Complex Variables and Their Singularities Functions of Several Complex Variables and Their Singularities Wolfgang Ebeling Translated by Philip G. Spain Graduate Studies in Mathematics Volume 83 .•S%'3SL"?|| American Mathematical Society s^s^^v Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz Originally published in the German language by Friedr. Vieweg & Sohn Verlag, D-65189 Wiesbaden, Germany, as "Wolfgang Ebeling: Funktionentheorie, Differentialtopologie und Singularitaten. 1. Auflage (1st edition)". © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden, 2001 Translated by Philip G. Spain 2000 Mathematics Subject Classification. Primary 32-01; Secondary 32S10, 32S55, 58K40, 58K60. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-83 Library of Congress Cataloging-in-Publication Data Ebeling, Wolfgang. [Funktionentheorie, differentialtopologie und singularitaten. English] Functions of several complex variables and their singularities / Wolfgang Ebeling ; translated by Philip Spain. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 83) Includes bibliographical references and index. ISBN 0-8218-3319-7 (alk. paper) 1. Functions of several complex variables. 2. Singularities (Mathematics) I. Title. QA331.E27 2007 515/.94—dc22 2007060745 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
  • (2019) Elliptic Mirror of the Quantum Hall Effect

    (2019) Elliptic Mirror of the Quantum Hall Effect

    PHYSICAL REVIEW B 99, 195152 (2019) Elliptic mirror of the quantum Hall effect C. A. Lütken Physics Deparment, University of Oslo, NO-0316 Oslo, Norway (Received 20 September 2018; revised manuscript received 22 March 2019; published 28 May 2019) Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a holomorphic modular symmetry, whose group structure is determined by the spin structure of the toroidal target space (an elliptic curve). Hall quantization is protected by the V σ ⊕ = μ ∈ topology of stable holomorphic vector bundles on this space, and plateau values H Q of the Hall conductivity are rational because such bundles are classified by their slope μ(V) = deg(V)/rk(V), where deg(V) is the degree and rk(V) is the rank of V. By exploiting a quantum equivalence called mirror symmetry,these models are mapped to tractable mirror models (also elliptic), in which topological protection is provided by more familiar winding numbers. Phase diagrams and scaling properties of elliptic models are compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One conspicuous model has a critical delocalization 2 4 exponent νtor = 18 ln 2/(π G ) = 2.6051 ... (G is Gauss’ constant) that is in excellent agreement with the value νnum = 2.607 ± .004 calculated in the numerical Chalker-Coddington model, suggesting that these models are in the same universality class.
  • A Conic Sections

    A Conic Sections

    A Conic Sections The ellipse, hyperbola, and parabola (and also the circle, which is a special case of the ellipse) are called the conic section curves (or the conic sections or just conics), since they can be obtained by cutting a cone with a plane (i.e., they are the intersections of a cone and a plane). The conics are easy to calculate and to display, so they are commonly used in applications where they can approximate the shape of other, more complex, geometric figures. Many natural motions occur along an ellipse, parabola, or hyperbola, making these curves especially useful. Planets move in ellipses; many comets move along a hyperbola (as do many colliding charged particles); objects thrown in a gravitational field follow a parabolic path. There are several ways to define and represent these curves and this section uses asimplegeometric definition that leads naturally to the parametric and the implicit representations of the conics. y F P D P D F x (a) (b) Figure A.1: Definition of Conic Sections. Definition: A conic is the locus of all the points P that satisfy the following: The distance of P from a fixed point F (the focus of the conic, Figure A.1a) is proportional 364 A. Conic Sections to its distance from a fixed line D (the directrix). Using set notation, we can write Conic = {P|PF = ePD}, where e is the eccentricity of the conic. It is easy to classify conics by means of their eccentricity: =1, parabola, e = < 1, ellipse (the circle is the special case e = 0), > 1, hyperbola.
  • Section 9.2 Hyperbolas 597

    Section 9.2 Hyperbolas 597

    Section 9.2 Hyperbolas 597 Section 9.2 Hyperbolas In the last section, we learned that planets have approximately elliptical orbits around the sun. When an object like a comet is moving quickly, it is able to escape the gravitational pull of the sun and follows a path with the shape of a hyperbola. Hyperbolas are curves that can help us find the location of a ship, describe the shape of cooling towers, or calibrate seismological equipment. The hyperbola is another type of conic section created by intersecting a plane with a double cone, as shown below5. The word “hyperbola” derives from a Greek word meaning “excess.” The English word “hyperbole” means exaggeration. We can think of a hyperbola as an excessive or exaggerated ellipse, one turned inside out. We defined an ellipse as the set of all points where the sum of the distances from that point to two fixed points is a constant. A hyperbola is the set of all points where the absolute value of the difference of the distances from the point to two fixed points is a constant. 5 Pbroks13 (https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg), “Conic sections with plane”, cropped to show only a hyperbola by L Michaels, CC BY 3.0 598 Chapter 9 Hyperbola Definition A hyperbola is the set of all points Q (x, y) for which the absolute value of the difference of the distances to two fixed points F1(x1, y1) and F2 (x2, y2 ) called the foci (plural for focus) is a constant k: d(Q, F1 )− d(Q, F2 ) = k .
  • Algebraic Plane Curves Deposited by the Faculty of Graduate Studies and Research

    Algebraic Plane Curves Deposited by the Faculty of Graduate Studies and Research

    PLUCKER'3 : NUMBERS: ALGEBRAIC PLANE CURVES DEPOSITED BY THE FACULTY OF GRADUATE STUDIES AND RESEARCH Iw • \TS • \t£S ACC. NO UNACC.'»"1928 PLITCOR'S FOOTERS IE THE THEORY 0? ALGEBRAIC PLAITS CURVES A Thesis presented in partial fulfilment for the degree of -Master of Arts, Department of Mathematics. April 28, 1928. by ALICiD WI1LARD TURIIPP. Acknowledgment Any commendation which this thesis may merit, is in large measure due to Dr. C..T. Sullivan, Peter Re&path Professor of Pure Mathematics; and I take this opportunity to express my indebtedness to him. April 25, 1928. I IT D A :: IITTPODIICTIOI? SECTIOH I. • pp.l-S4 Algebraic Curve defined. Intersection of two curves. Singular points on curves. Conditions for multiple points. Conditions determining an n-ic. Maximum number of double points. Examples. SECTION II ...... pp.25-39 Class of curve. Tangerfcial equations. Polar reciprocation. Singularities on a curve and its reciprocal. Superlinear branches, with application to intersections of two curves at singular points. Examples. 3SCTI0IT III pp.40-52 Polar Curves. Intersections of Curve and Pirst Polar. Hessian.Intersections of Hessian and Curve. Examples. 3ECTI0TT IV . .................. pp.52-69 Pliicker's numbers and Equations for simple singularities. Deficiency. Unicursal Curves. Classification of cubies and quartics. Extension of Plucker's relations to k-ple points with distinct tangents, and to ordinary superlinear branch points. Short discussion of Higher Singularities. Examples. I PhTJCPEirS PTE.3EE3 IE PEE EHSOHY CF ALGEBRAIC Ph-TITE CUPYE3. IPPEODUCPIOH Pith the advent of the nineteenth century,a new era dawned in the progress of analytic geometry. The appearance of poiiceletTs, "Traite des proprietes project ives des figures", in 1822, really initiated modern geometry.