DOCSLIB.ORG

An Elementary Treatise on Cubic and Quartic Curves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarytreatiOObassuoft 75-^/ 10^ AN ELEMENTARY TREATISE ON CUBIC AND QUARTIC CURVES AN ELEMENTARY TREATISE ON CUBIC AND QUARTIC CURVES BY i. ..^ A\ B? BASSET, M.A., F.R.S. TRINITY COLLEGE CAMBRIDGE t'-i^l.s CAMBRIDGE DEIGHTON BELL AND CO. LONDON GEORGE BELL AND SONS 1901 [All Rights reserved] PRINTED BY J. AND C. F. CLAY AT THE UNIVERSITY PRESS m 56 b PREFACE nnHE present work originated in certain notes, made about twenty- five years ago, upon the properties of some of the best-known higher plane curves; but upon attempting to revise them for the press, it appeared to me impossible to discuss the subject adequately without investigating the theory of the singularities of algebraic curves. I have accordingly included Plucker's equations, which determine the number and the species of the simple singularities of any algebraic curve ; and have also considered all the compound singularities which a quartic curve can possess. This treatise is intended to be an elementary one on the subject. I have therefore avoided investigations which would require a knowledge of Modern Algebra, such as the theory of the invariants, covariants and other concomitants of a ternary quantic ; and have assumed scarcely any further knowledge of analysis on the part of the reader, than is to be found in most of the ordinary text-books on the Differential Calculus and on Analytical Geometry. I have also endeavoured to give special prominence to geometrical methods, since the experience of many years has convinced me that a judicious combination of geometry and analysis is frequently capable of shortening and simplifying, what would otherwise be a tedious and lengthy investigation. ; VI PREFACE. The introductory Chapter contains a few algebraic definitions and propositions which are required in subsequent portions of the work. The second one deals with the elementary theory of the singularities of algebraic curves and the theory of polar curves. The third Chapter commences with an explanation of tangential coordinates and their uses, and then proceeds to discuss a variety of miscellaneous propositions connected with reciprocal polars, the circular points at infinity and the foci of curves. Chapter IV is devoted to Plucker's equations ; whilst Chapter V contains an account of the general theory of cubic curves, including the formal proof of the principal properties which are common to all curves of this degree. In this Chapter I have almost exclusively employed trilinear coordinates, since the introduction of a tnangle of reference, whose elements can be chosen at pleasure, constitutes a vast improvement on the antiquated methods of homogeneous coordinates and abridged notation. Chapter VI is devoted to the consideration of a variety of special cubics, including the particular class of circular cubics which are the inverses of conies with respect to their vertices and in this Chapter the method of Cartesian coordinates is the most appropriate. A short Chapter then follows on curves of the third class, after which the discussion of quartic curves commences. To adequately consider such an extensive subject as quartic curves would require a separate treatise. I have therefore confined the discussion to the simple and compound singulari- ties of curves of this degree, together with a few miscellaneous propositions; and in Chapter IX, I proceed to investigate the theory of bicircular quartics and cartesians, concluding with the general theory of circular cubics, which is better treated as a particular case of bicircular quartics than as a special case of cubic curves. Chapter X is devoted to the consideration of various well known quartic curves, most of which are bicircular PREFACE. vii or are cartesians ; whilst Chapter XI deals with cycloidal curves, together with a few miscellaneous curves which fre- quently occur in mathematical investigations. The theory of projection, which forms the subject of the last Chapter, is explained in most treatises on Conies ; but except in the case of conies, due weight has not always been given to the important fact that the projective properties of any special class of curves can be deduced from those of the simplest curve of the species. Thus all the projective properties of tricuspidal quartics can be obtained from those of the three-cusped hypocycloid or the cardioid ; those of quartics with a node and a pair of cusps from the lima^on ; those of quartics with three biflecnodes from the lemniscate of Bernoulli or the reciprocal polar of the four- cusped hypocycloid ; whilst the properties of binodal and bi- cuspidal quartics can be obtained from those of bicircular quartics and cartesians. Whenever the medical profession require a new word they usually have recourse to the Greek language, and mathematicians would do well to follow their example; since the choice of a suitable Greek word supplies a concise and pointed mode of expression which saves a great deal of circumlocution and verbosity. The old-fashioned phrase "a non-singular cubic or quartic curve " involves a contradiction of terms, since Plucker has shown that all algebraic curves except conies possess sin- gularities; and I have therefore introduced the words autotomic and anautotomic to designate curves which respectively do and do not possess multiple points. The words perigraphic, endo- dromic and exodromic, which are defined on page 14, are also useful; in fact a word such as aperigraphic is indispensable in order to avoid the verbose phrase "a curve which has branches extending to infinity." At the present day the subject of Analytical Geometry covers so vast a field that it is by no means easy to decide VIU PREFACE. what to insert and what to leave out. I trust, however, that the present work will form a useful introduction to the higher branches of the subject; and will facilitate the study of a variety of curves whose properties, by reason of their beauty and elegance, deserve at least as much attention as the well-worn properties of conies. Fledborouqh Hall, HoLYPORT, Berks. August, 1901. CONTENTS. CHAPTER L INTRODUCTION. ART. PAGE 1. Definition of a quantic 1 2. Discriminants 2 3. Eliminants 2 4. Discriminants of a binary quadric, cubic and quartic . 3 5-6. Theorems relating to the roots of an equation ... 3 7. Conditions that various equalities should exist between the roots of cubic and quartic equations ..... 4 8. Condition that the product of two of the roots of an equation should be equal to —1 5 9. The Hessian of a quantic. Invariants and covariants . 6 CHAPTER II. THEORY OF CURVES. 10. General equation of a curve of the nth degree . 8 11. Two curves of the mth and nth. degrees intersect in mn points . 8 12-13. Multiple points 9 14-16. Contact of two curves 10 17. Definition of tangency 10 18-19. Double points. Nodes and cusps 11 20. Definitions of deficiency, the class of a curve, points of inflexion and undulation, and multiple tangents . 13 21. Definitions of autotomic, perigraphic, endodromic, exodromic and multipartite curves 14 22. Conditions for a double point 14 23-24. Polar curves 16 17 25-28. -Number of tangents that can be drawn to a curve . CONTENTS. i^-^Si Vuwm propertiee of }x>lar curves S8-J8L Tbe HaaUn of a curve and its properties .... 30. The SleiMrian of a curve 40l AqrnplotaB ...•••••••• 41. QemnU aquatioD of a curve in trilinear coordinates 23 48. TIm HeHian paiwfr through every point of inflexion 49-46. Purthar propo8ition« connected with multiple points on a 24 curve, ita Hessian and its polars 25 4i-4a SingularitieB at infinity 27 fla Condition that the line at infinity should be a tangent . 61. IroaginHT}' iiinguUrities 28 CHAPTER III. TANGENTIAL COORDINATES. ftS. The boothian system of tangential coordinates ... 30 69L Taofeotaal equation of a curve whose Cartesian equation is given 30 54. fNirtfiiian equation of a curve whose tangential equation is given 31 6ft. On the curve (r/a)»+(y/fc)»=l 32 66. Eiaminatioo of the different tenus of a tangential equation . 32 67-66. Reciprocal ix>lans inverse and })edal curves ... 33 66. A mde oorraqionda to a double tangent on the I'eciprocal polar . 34 60l a ct»p correeponds to a stationary tangent on the reciprocal polar 35 61. The line at infinity 35 66-63L IfeUKx! of finding the nmltiple tangents to a curve . 36 64-66L Pedal cun'en. The tangential polar equation of a curve 38 67. On the curves r"* a* cos n^ 40 66. OrthopCk and iaoptic loci 40 66>70l The circular |M>inta at infinity 42 71-71. The trilinear system of tangential coordinates ... 43 7X. Taoffsotial equatif>n of a circle 44 74 Tanfntkl equation of the circular points .... 45 76. Pod of ennree 46 7& VMkkm^9 <ie6nition of a focus 47 77. Nunihv of foci when the line at infinity is a multiple tangent . 47 76. da when a curve paesos through the circular points 48 7fc da when the circular {mints are nodes ... 49 •^ <^ <Io. cusps or points of ifi^nitm 49 61. The iateiw of a focus is a focue of the inverse curve 50 66. Bqtaetlon of tangenta drawn from a ix)int (A, ^) to a curve . 51 of the pointe of intersection of tangents at the extremities of a line drswn through a fixed point .... 51 CONTENTS. xi CHAPTER IV. plucker's equations. ART. PAGE 84. Maximum number of double points of a curve of the nth degree 52 85-86. Maximum number of points of inflexion .... 53 87-88. Maximum and minimum values of the class of a curve of the nth degree , 54 89.
Recommended publications
  • Chapter 1 Geometry of Plane Curves

    Chapter 1 Geometry of Plane Curves

    Chapter 1 Geometry of Plane Curves Definition 1 A (parametrized) curve in Rn is a piece-wise differentiable func- tion α :(a, b) Rn. If I is any other subset of R, α : I Rn is a curve provided that α→extends to a piece-wise differentiable function→ on some open interval (a, b) I. The trace of a curve α is the set of points α ((a, b)) Rn. ⊃ ⊂ AsubsetS Rn is parametrized by α if and only if there exists set I R ⊂ ⊆ such that α : I Rn is a curve for which α (I)=S. A one-to-one parame- trization α assigns→ and orientation toacurveC thought of as the direction of increasing parameter, i.e., if s<t,the direction of increasing parameter runs from α (s) to α (t) . AsubsetS Rn may be parametrized in many ways. ⊂ Example 2 Consider the circle C : x2 + y2 =1and let ω>0. The curve 2π α : 0, R2 given by ω → ¡ ¤ α(t)=(cosωt, sin ωt) . is a parametrization of C for each choice of ω R. Recall that the velocity and acceleration are respectively given by ∈ α0 (t)=( ω sin ωt, ω cos ωt) − and α00 (t)= ω2 cos 2πωt, ω2 sin 2πωt = ω2α(t). − − − The speed is given by ¡ ¢ 2 2 v(t)= α0(t) = ( ω sin ωt) +(ω cos ωt) = ω. (1.1) k k − q 1 2 CHAPTER 1. GEOMETRY OF PLANE CURVES The various parametrizations obtained by varying ω change the velocity, ac- 2π celeration and speed but have the same trace, i.e., α 0, ω = C for each ω.
  • Tangent Lines

    Tangent Lines

    The Kabod Volume 5 Issue 1 Fall 2018 Article 1 September 2018 Tangent Lines Samuel Estep Liberty University, [email protected] Follow this and additional works at: https://digitalcommons.liberty.edu/kabod Part of the Mathematics Commons Recommended Citations MLA: Estep, Samuel "Tangent Lines," The Kabod 5. 1 (2018) Article 1. Liberty University Digital Commons. Web. [xx Month xxxx]. APA: Estep, Samuel (2018) "Tangent Lines" The Kabod 5( 1 (2018)), Article 1. Retrieved from https://digitalcommons.liberty.edu/kabod/vol5/iss1/1 Turabian: Estep, Samuel "Tangent Lines" The Kabod 5 , no. 1 2018 (2018) Accessed [Month x, xxxx]. Liberty University Digital Commons. This Individual Article is brought to you for free and open access by Scholars Crossing. It has been accepted for inclusion in The Kabod by an authorized editor of Scholars Crossing. For more information, please contact [email protected]. Estep: Tangent Lines Tangent Lines Sam Estep 2018-05-06 In [1] Leibniz published the first treatment of the subject of calculus. An English translation can be found in [2]; he says that to find a tangent is to draw a right line, which joins two points of the curve having an infinitely small difference, or the side of an infinite angled polygon produced, which is equivalent to the curve for us. Today, according to [3], a straight line is said to be a tangent line of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c; f(c)) on the curve and has slope f 0(c) where f 0 is the derivative of f.
  • Geometry of Algebraic Curves

    Geometry of Algebraic Curves

    Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2.
  • Curves with Rational Chord-Length Parametrization

    Curves with Rational Chord-Length Parametrization

    Curves with rational chord-length parametrization J. Sanchez-Reyes , L. Fernandez-Jambrina Instituto de Matemdtica Aplicada a la Ciencia e Ingenieria, ETS Ingenieros Industrials, Universidad de Castilla-La Mancha, Campus Universitario, 13071-Ciudad Real, Spain ETSI Navales, Universidad Politecnica de Madrid, Arco de la Victoria s/n, 28040-Madrid, Spain Abstract It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled. Keywords: Bezier circle; Bipolar coordinates; Chord-length parametrization; Equilateral hyperbola; Lemniscate of Bernoulli 1. Chord-length parametrization Given a parametric curve p(t) over a certain domain t e[a,b], its chord-length at a given point p(7) is defined as (Farm, 2001, 2006): IF|p(0 — A| chord(0 := y A = p(fl), B = p(Z?) (1) |p(0-A| + |p(0-B| where bars denote the modulus of a vector. The curve is said to be chord-length parameterized if chord(f) = t. Chord- length parametrization is clearly invariant with respect to linear transformations of the domain.
  • Generating Negative Pedal Curve Through Inverse Function – an Overview *Ramesha

    Generating Negative Pedal Curve Through Inverse Function – an Overview *Ramesha

    © 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) Generating Negative pedal curve through Inverse function – An Overview *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur. Abstract This paper attempts to study the negative pedal of a curve with fixed point O is therefore the envelope of the lines perpendicular at the point M to the lines. In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k2. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself. is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. The inverse function of f is also denoted.
  • Strophoids, a Family of Cubic Curves with Remarkable Properties

    Strophoids, a Family of Cubic Curves with Remarkable Properties

    Strophoids, a family of cubic curves with remarkable properties Hellmuth Stachel [email protected] — http://www.geometrie.tuwien.ac.at/stachel 6th International Conference on Engineering Graphics and Design University Transilvania, June 11–13, Brasov/Romania Table of contents 1. Definition of Strophoids 2. Associated Points 3. Strophoids as a Geometric Locus June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 1/29 replacements 1. Definition of Strophoids ′ F Definition: An irreducible cubic is called circular if it passes through the absolute circle-points. asymptote A circular cubic is called strophoid ′ if it has a double point (= node) with G orthogonal tangents. F g A strophoid without an axis of y G symmetry is called oblique, other- wise right. S S : (x 2 + y 2)(ax + by) − xy =0 with a, b ∈ R, (a, b) =6 (0, 0). In x N fact, S intersects the line at infinity at (0 : 1 : ±i) and (0 : b : −a). June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 2/29 1. Definition of Strophoids F ′ The line through N with inclination angle ϕ intersects S in the point asymptote 2 2 ′ sϕ c ϕ s ϕ cϕ G X = , . a cϕ + b sϕ a cϕ + b sϕ F g y G X This yields a parametrization of S. S ϕ = ±45◦ gives the points G, G′. ϕ The tangents at the absolute circle- N x points intersect in the focus F . June 12, 2015: 6th Internat. Conference on Engineering Graphics and Design, Brasov/Romania 3/29 1.
  • Sources and Studies in the History of Mathematics and Physical Sciences

    Sources and Studies in the History of Mathematics and Physical Sciences

    Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board J.Z. Buchwald J. Lützen G.J. Toomer Advisory Board P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Springer Science+Business Media, LLC Sources and Studies in the History of Mathematics and Physical Seiences K. Andersen Brook Taylor's Work on Linear Perspective H.l.M. Bos Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction 1. Cannon/S. Dostrovsky The Evolution of Dynamics: Vibration Theory from 1687 to 1742 B. ChandIerlW. Magnus The History of Combinatorial Group Theory A.I. Dale AHistory of Inverse Probability: From Thomas Bayes to Karl Pearson, Second Edition A.I. Dale Most Honourable Remembrance: The Life and Work of Thomas Bayes A.I. Dale Pierre-Simon Laplace, Philosophical Essay on Probabilities, Translated from the fifth French edition of 1825, with Notes by the Translator P. Damerow/G. FreudenthallP. McLaugWin/l. Renn Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development in Early Modem Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman, Second Edition P.l. Federico Descartes on Polyhedra: A Study of the De Solworum Elementis B.R. Goldstein The Astronomy of Levi ben Gerson (1288-1344) H.H. Goldstine A History of Numerical Analysis from the 16th Through the 19th Century H.H. Goldstine A History of the Calculus of Variations from the 17th Through the 19th Century G. Graßhoff The History of Ptolemy's Star Catalogue A.W. Grootendorst Jan de Witt's Eiementa Curvarum Linearum, über Primus Continued after Index The Arithmetic of Infinitesimals John Wallis 1656 Translated from Latin to English with an Introduction by Jacqueline A.
  • 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.
  • And Innovative Research of Emerging Technologies

    And Innovative Research of Emerging Technologies

    WWW.JETIR.ORG [email protected] An International Open Access Journal UGC and ISSN Approved | ISSN: 2349-5162 INTERNATIONAL JOURNAL OF EMERGING TECHNOLOGIES AND INNOVATIVE RESEARCH JETIR.ORG INTERNATIONAL JOURNAL OF EMERGING TECHNOLOGIES AND INNOVATIVE RESEARCH International Peer Reviewed, Open Access Journal ISSN: 2349-5162 | Impact Factor: 5.87 UGC and ISSN Approved Journals. Website: www. jetir.org Website: www.jetir.org JETIR INTERNATIONAL JOURNAL OF EMERGING TECHNOLOGIES AND INNOVATIVE RESEARCH (ISSN: 2349-5162) International Peer Reviewed, Open Access Journal ISSN: 2349-5162 | Impact Factor: 5.87 | UGC and ISSN Approved ISSN (Online): 2349-5162 This work is subjected to be copyright. All rights are reserved whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illusions, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication of parts thereof is permitted only under the provision of the copyright law, in its current version, and permission of use must always be obtained from JETIR www.jetir.org Publishers. International Journal of Emerging Technologies and Innovative Research is published under the name of JETIR publication and URL: www.jetir.org. © JETIR Journal Published in India Typesetting: Camera-ready by author, data conversation by JETIR Publishing Services – JETIR Journal. JETIR Journal, WWW. JETIR.ORG ISSN (Online): 2349-5162 International Journal of Emerging Technologies and Innovative Research (JETIR) is published in online form over Internet. This journal is published at the Website http://www.jetir.org maintained by JETIR Gujarat, India. © 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) Generating Negative pedal curve through Inverse function – An Overview *Ramesha.
  • Quadratic Transformations / Probability and Point Set Theory

    Quadratic Transformations / Probability and Point Set Theory

    QUADRATIC TRANSFORMATIONS QUADRATIC TRANSFORMATIONS I CREMONA TRANSFORMATIONS I II QUADRATIC TRANSFORMATIONS 6 III QUADRAIC INVERSION 7 IV DEGENERATE CASES 9 V SUCCESSIVE QUADRATIC TRANSFORMATIONS 12 VI GEOMETRIC CONSTRUCTIONS OF QUADRATIC TRANSFORMATIONS 15 VII APPLICATIONS 19 VIII SINGULARITIES OF A PLANE CURVE 20 IX HIGHER SINGULARITIES OF PLANE CURVES 24 « X ANALYSIS OF HIGHER SINGULARITIES BY QUADRATIC TRANSFORMATIONS 24 XI EXAMPLES 28 XII NOETHER'S THEOREM 31 BIBLIOGRAPHY QUADRATIC TRANSPORTATIONS I CR3MOHA TRAIISFBRMATIŒJS T3To consider the general problem of finding birational transformations between two spaces, with homogeneous coordinates (x,y,...) and (x',y* That is we want to find a transformation so that the relation 1) x':y*:... 3 X:Y*... where X,Y,... are rational,integral,h<$geneous functions, of the degree n, in (x,y,.,,)f implies the relation 2) x:y*... - X'sY':... where X’,Y*,... are rational, integral, homogeneous functions , of the degree n*, in (x*,y* That is,to each point (x,y,..«) there is to correspond one and only one point (x*,y*,...) and conversely. As an introductory example, we consider the transformation of the coordinates on a line. We are to have 3j xsy s X* sY* and 4) x*iy8 - X:Y with X,Y,X*,Y* all rational. The equation x'*y* s XsY must give a single variable value for the ratio x/y. But if X and Y are of the order n, there are n values of x/y for each value of xî/y*. Since but one of these is to be variable, we must have n - 1 values independent of x* and y*, this can only be the case if X and Y have a common factor of order n-1 .
  • Newton's Parabola Observed from Pappus

    Newton's Parabola Observed from Pappus

    Applied Physics Research; Vol. 11, No. 2; 2019 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education Newton’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Castillon’s Cardioid, and Ptolemy’s Circle (Hodograph) (09.02.2019) Jiří Stávek1 1 Bazovského, Prague, Czech Republic Correspondence: Jiří Stávek, Bazovského 1228, 163 00 Prague, Czech Republic. E-mail: [email protected] Received: February 2, 2019 Accepted: February 20, 2019 Online Published: February 25, 2019 doi:10.5539/apr.v11n2p30 URL: http://dx.doi.org/10.5539/apr.v11n2p30 Abstract Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia…” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician).
  • ED164303.Pdf

    ED164303.Pdf

    DOCUMENT RESUME -.. a ED 164.303 E 025. 458 AUTHOR Beck, A.; And Others ,TITLE Calculus, Part 3,'Studentl'Text, Unit No. 70. Revised Edition. , INSTITUTION Stanford Univ.,'CaliY. School Mathematics Study 't.-Group. SPONS AGENCY -National Science Foundation, Washington, D.C. PUB DATE ,65: NOTE 36bp.; For related documents, see SE 025 456-4:59; Contains light and braken'type EDRS PRICE MF-$0.83 HC-$19.41 Plus Postage. DESCRIPTORS *Calculus; *Cuiriculum; *Instructional Materials; *Mathematical Applications; Mathematics .Educatioi; S4Condary Education; *Secondary School Mathematics; *Textbooks IDENTIFIERS *School Mathematics Study Group ABSTRACT. This is part three of a'three-part SMSG calculuS text for high sahool students. One of the goals of the text is to present -calculus as a mathematical discipline -as yell, as presenting its practical uses. The authbrs emphasize the importance of being able to ipterpret the conceptS and theory interms of models to which they .apply. The text demonstrates the origins of the ideas of the calculus in pradtical problems; attempts to express these ideas precisely and_ ` develop them logically; and finally, returns to the problems and applies the theorems resulting from that development. Chapter topics include: (1)vectors and curves; (2) mechanics; (3) numerical analysis; (4) sequences and series; and (5) geometricl optics and waves. (MP) 1 - ********************************45*******************************/ * Reproductions supplied by EDRS are the best that can' be made ,* + ) - * from the original doCumente ../ * *****************************************************#*****,J U S DEPARTMENT lEDUCATIOPtilmi AI id NATIONALINS1 i I EDUCAT . THiSDOCUMENTHA! DUCEDEXACTLYAS TAE PERSONOR ORGAP ATIkr;T POINTSOF V STATEDDO NOT NECE SENTiOFFifiALNATION EDUCATIONPOSITION Calculus Part 3 Student's Text REVISED EDITION ' The following is a list of all thosewho.participiated in the preparation of this volume: A.