DOCSLIB.ORG

A New Approach to Find a Generalized Evolute and Involute Curve in 4

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A New Approach to Find a Generalized Evolute and Involute Curve in 4 Palestine Journal of Mathematics Vol. 8(1)(2019) , 397–411 © Palestine Polytechnic University-PPU 2019 A new approach to find a generalized evolute and involute curve in 4-dimensional Minkowski space-time Muhammad Hanif and Zhong Hua Hou Communicated by Zafar Ahsan MSC 2010 Classifications: : Primary 53-XX, 53Axx; Secondary 53A35. Keywords and phrases: Evolute, Involute, Null Cartan curve, Minkowski space. This work is supported by the National Natural Science Foundation of China (No.61473059). Abstract. In this study, we introduce a new approach to find a special kind of generalized 4 evolute and involute curve in Minkowski space-time E1 . Necessary and sufficient condition for the curve possessing generalized evolute as well as involute curve were obtained. Furthermore, Cartan null curve is also discussed in detail. 1 Introduction Many mathematicians did work about the general theory of the curves in Euclidean space. Now, we have much understanding on their local geometry as well as their global geometry. Identi- fication of a regular curve is one of the important and interesting complication in the theory of curves in Euclidean space. Using two different approaches this complication can be solved, the relation between the frenet vectors of the curve [1], and determining the shape and size of the curve by curvatures κ1 and κ2. In differential geometry an evolute is the envelope of the normals of the specific curve. An evolute and its involute are defined in mutual pairs. The evolute of any curve is defined as the locus of the centers of curvatures of the curve. The original curve is then described as the involute of the evolute. Evolutes and involutes (also known as evolvents) were studied by C. Huygens [2]. Later, in [3] the author explore that if evolute occur then the evolute of parallel arc also occur. In [4] the author fixed that evolute Frenet apparatus can be establish by involute apparatus in four dimensional Euclidean space so by this approach another orthonormal of the same space is acquired. In [5] author resolved that the iteration of involutes create a pair of sequences of curves with respect to Minkowski metric and its dual. In (1845), Saint Venant [6] suggested a question either the principle normal of a curve is the principle normal of another on the surface produced by the principle normal of the specific curve. Bertrand answered this question in [7]. He proved that a necessary and sufficient condition for the existence of such a second curve is required. Using this method we define a kind of generalized evolute-involute curve in Minkowski space time. We acquire the necessary and sufficient conditions for the curves with spacelike (1,3)- normal plane to be (1,3)-Evolute curves and we also prove its converse by using the condition of Evolute curve that is spanned by principle normal and the second binormal. In the end we give some examples for these curves. Evolute curves and their identification were studied by some researchers in Minkowski space [8], [9], [10], [11], [12], [13], [14], [15], [16] as well as in Euclidean space. We see that mainly evolute-involute curves have been studied but not so much research has been carried out to find the mate curves of Cartan null curves. In this paper, a kind of generalized evolute and involute curve is considered for Cartan null curve in Minkowski space-time. The necessary and sufficient conditions for a curve possessing generalized evolute as well as involute mate curves is obtained. 2 Preliminaries 4 4 Consider the Minkowski space-time (E1 ;H) where E1 = fz = (z1; z2; z3; z4)jzi 2 Rg and 2 2 2 2 H = −dz1 + dz2 + dz3 + dz4. For any U = (x1; x2; x3; x4) and V = (y1; y2; y3; y4) 2 TzE, we 398 Muhammad Hanif and Zhong Hua Hou denote U · V = H(U; V ) = −x1y1 + x2y2 + x3y3 + x4y4. 4 4 Let I be an open interval in R and G : I ! E1 be a regular curve in E1 parameterized by arc length parameter s and fT;N1;N2;N3g be a moving Frenet frame along G, consisting of tangent vector T , principal normal vector N1, the first binormal vector N2 and the second binormal vector 4 N3 respectively, so that T ^ N1 ^ N2 ^ N3 coincides with the standard orientation of E1 . From [17] Frenet seret formula. 2 0 3 2 3 2 3 T 0 2κ1 0 0 T 6N 07 6− κ 0 κ 0 7 6N 7 6 17 = 6 1 1 3 2 7 6 17 ; (2.1) 6 07 6 7 6 7 4N25 4 0 −2κ2 0 −123κ35 4N25 0 N3 0 0 −3κ3 0 N3 where H(T;T ) = 1;H(N1;N1) = 2;H(N2;N2) = 3; H(N3;N3) = 4; 1234 = −1; i 2 f1; −1g; i 2 f1; 2; 3; 4g: In specific, the succeeding conditions exist: H(T;N1) = H(T;N2) = H(T;N3) = H(N1;N2) = H(N1;N3) = H(N2;N3) = 0: 4 0 A curve G(s) in E1 can be spacelike, timelike, or null if its velocity vectors G (s) are com- monly spacelike, timelike, or null [18]. A null curve G is parametrized by pseudo-arc s if H(G00(s); G00(s)) = 1 [19]. Further more nonnull curve G, we have this condition H(G0(s); G0(s)) = ±1. From [19, 20] if G is null Cartan curve, the Cartan Frenet frame is given by 2 0 3 2 3 2 3 T 0 κ1 0 0 T 6N 07 6 κ 0 −κ 0 7 6N17 6 17 = 6 2 1 7 6 7 ; (2.2) 6 07 6 7 6 7 4N25 4 0 −κ2 0 κ35 4N2 5 0 N3 −κ3 0 0 0 N3 where κ1(s) = 0 if G(s) is a null straight line or κ1(s) = 1 in all other cases. In this case T · T = N2 · N2 = 0;N1 · N1 = N3 · N3 = 1, T · N1 = T · N3 = N1 · N2 = N1 · N3 = N2 · N3 = 0, T · N2 = 1. We established some terminologies in this study. At any point of G, the plane spanned by fT;N2g is called the (0; 2)-tangent plane of G. The plane spanned by fN1;N3g is called the (1; 3)-normal plane of G. 4 ∗ 4 4 Let G : I ! E and G : I ! E1 be two regular curves in E1 where s is the arc-length parameter of G. Denote s∗ = f(s) to be the arc-length parameters of G∗. For any s 2 I, if the (0; 2)-tangent plane of G at G(s) of coincides with the (1; 3)-normal plane at G∗(s) of G∗, then ∗ 4 ∗ G is called the (0; 2)-involute curve of G in E1 and G is called the (1; 3)-evolute curve of G in 4 E1 . 4 2.1 (1,3)-involute curve of a given curve in E1 In this section, we proceed to study the existence and expression of the (1; 3)-evolute curve of a 4 given curve in E1 . 4 Let G : I ! E1 be a regular curve with arc-length parameter s so thatκ1, κ2 and κ3 are not ∗ 4 ∗ ∗ ∗ ∗ zero. Let G : I ! E1 be the (1; 3)-evolute curve of G. Denote fT ;N1 ;N2 ;N3 g to be the ∗ ∗ ∗ ∗ ∗ Frenet frame along G and κ1 , κ2 and κ3 to be the curvatures of G . Then ∗ ∗ ∗ ∗ spanfT;N2g = spanfN1 ;N3 g; spanfN1;N3g = spanfT ;N2 g: 4 Generalized involute-evolute curves in E1 399 3 Theorem 4 Let G : I ! E1 be a regular curve with arc-length parameter s so that κ1, κ2 and κ3 are not zero. ∗ 4 ∗ ∗ ∗ ∗ Let G : I ! E1 be the (1; 3)-evolute curve of G. Denote fT ;N1 ;N2 ;N3 g to be the Frenet ∗ ∗ ∗ ∗ ∗ frame along G and κ1 , κ2 and κ3 to be the curvatures of G if and only if there there exists scalar functions F, Y of arc-length parameter s and real constant numbers α =6 ±1, β satisfying 0 F0(s) =6 0; Y (s) =6 0; (3.1) 0 0 F (s) = αY (s); (3.2) βαk1(s) = αk2(s) − k3(s); (3.3) 2 2 2 2 [−κ2κ3(α − 1) + α(κ3 − κ1 − κ2)] =6 0 (3.4) for all s 2 I. 4 Proof. Let G : I ! E1 be a regular curve with arc-length parameter s so that κ1, κ2 and κ3 are ∗ 4 ∗ ∗ ∗ ∗ not zero. Let G : I ! E1 be the (1; 3)-evolute curve of G. Denote fT ;N1 ;N2 ;N3 g to be the ∗ ∗ ∗ ∗ ∗ Frenet frame along G and κ1 , κ2 and κ3 to be the curvatures of G . Then ∗ ∗ ∗ ∗ spanfT;N2g = spanfN1 ;N3 g; spanfN1;N3g = spanfT ;N2 g: Moreover, we can write the curve G∗ as follows ∗ ∗ G (s ) = G(s) + F(s)N1(s) + Y(s)N3; (3.5) for all s∗ 2 I∗, s 2 I where F(s) and Y(s) are C1 functions on I. Taking derivative of (3:5), using the equation (2:1), we get ∗ 0 0 0 T f = (1 − F1κ1)T (s) + F (s)N1(s) + Y (s)N3 + 3(F(s)κ2 − Y(s)κ3)N2: (3.6) ∗ ∗ 1 fT ;N g?fT;N2g so 1 − a1κ1 = 0 and 3(aκ2 − bκ3) = 0 from these we have F = ,Y = 2 1κ1 1κ2 . κ1κ3 Equation(3.6) gets the form 0 ∗ 0 0 f T = F (s)N1 + Y (s)N3: (3.7) So (3:6) gets the form 0 0 F Y T ∗ = N + N : (3.8) f 0 1 f 0 3 Multiplying (3:7) by itself, we get ∗ 0 2 0 2 0 2 1 (f ) = 2(F ) + 4(Y ) : (3.9) If we denote 0 0 F Y η = ; ζ = : (3.10) f 0 f 0 Using equation (3.10) in (3.7), we get ∗ T = ηN1 + ζN3: (3.11) Taking derivative of equation (3.11) using (2.1), we get 0 ∗ ∗ 0 0 f κ1 N1 = η N1 − 1ηκ1T + ζ N3 + 3(ηκ2 − ζκ3)N2: (3.12) Taking inner product on both side of (3.12) by N1 and N3 respectively, we get η0 = 0; ζ0 = 0: (3.13) 400 Muhammad Hanif and Zhong Hua Hou 0 ∗ ∗ f κ1 N1 = −1ηκ1T + 3(ηκ2 − ζκ3)N2: (3.14) Multiplying (3.14) by itself, we get 0 2 ∗ 2 2 2 2 (f ) (κ1 ) = η κ1 + (ηκ2 − ζκ3) : (3.15) ∗ ∗ ∗ We know that curvatures κ1 , κ2 , and κ3 =6 0 , so we obtained result (3.1) 0 F0 =6 0; Y =6 0: From (3:10), we get the result (3.2) 0 0 F = αY : (3.16) F−η Integrating (3.16), we get F = αY + η and Y = α .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-Space E3
    Fundamentals of Contemporary Mathematical Sciences (2020) 1(2) 63 { 70 Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-space E3 Abdullah Yıldırım 1,∗ Feryat Kaya 2 1 Harran University, Faculty of Arts and Sciences, Department of Mathematics S¸anlıurfa, T¨urkiye 2 S¸ehit Abdulkadir O˘guzAnatolian Imam Hatip High School S¸anlıurfa, T¨urkiye, [email protected] Received: 29 February 2020 Accepted: 29 June 2020 Abstract: In this study, evolute-involute curves are researched. Characterization of evolute-involute curves lying on the surface are examined according to Darboux frame and some curves are obtained. Keywords: Curve, surface, geodesic, curvature, frame. 1. Introduction The interest of special curves has increased recently. Some of these are associated curves. They are curves where one of the Frenet vectors at opposite points is linearly dependent to the other curve. One of the best examples of these curves is the evolute-involute partner curves. An involute thought known to have been used in his optical work came up in 1658 by C. Huygens. C. Huygens discovered involute curves while trying to make more accurate measurement studies [5]. Many researches have been conducted about evolute-involute partner curves. Some of them conducted recently are Bilici and C¸alı¸skan [4], Ozyılmaz¨ and Yılmaz [9], As and Sarıo˘glugil[2]. Bekta¸sand Y¨uceconsider the notion of the involute-evolute curves lying on the surfaces for a special situation. They determine the special involute-evolute partner D−curves in E3: By using the Darboux frame of the curves they obtain the necessary and sufficient conditions between κg , ∗ − ∗ ∗ τg; κn and κn for a curve to be the special involute partner D curve.
    [Show full text]
  • Computer-Aided Design and Kinematic Simulation of Huygens's
    applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock.
    [Show full text]
  • The Pope's Rhinoceros and Quantum Mechanics
    Bowling Green State University ScholarWorks@BGSU Honors Projects Honors College Spring 4-30-2018 The Pope's Rhinoceros and Quantum Mechanics Michael Gulas [email protected] Follow this and additional works at: https://scholarworks.bgsu.edu/honorsprojects Part of the Mathematics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons, and the Physics Commons Repository Citation Gulas, Michael, "The Pope's Rhinoceros and Quantum Mechanics" (2018). Honors Projects. 343. https://scholarworks.bgsu.edu/honorsprojects/343 This work is brought to you for free and open access by the Honors College at ScholarWorks@BGSU. It has been accepted for inclusion in Honors Projects by an authorized administrator of ScholarWorks@BGSU. The Pope’s Rhinoceros and Quantum Mechanics Michael Gulas Honors Project Submitted to the Honors College at Bowling Green State University in partial fulfillment of the requirements for graduation with University Honors May 2018 Dr. Steven Seubert Dept. of Mathematics and Statistics, Advisor Dr. Marco Nardone Dept. of Physics and Astronomy, Advisor Contents 1 Abstract 5 2 Classical Mechanics7 2.1 Brachistochrone....................................... 7 2.2 Laplace Transforms..................................... 10 2.2.1 Convolution..................................... 14 2.2.2 Laplace Transform Table.............................. 15 2.2.3 Inverse Laplace Transforms ............................ 15 2.3 Solving Differential Equations Using Laplace
    [Show full text]
  • The Fractal Theory of the Saturn Ring
    523.46/.481 THE FRACTAL THEORY OF THE SATURN RING M.I.Zelikin Abstract The true reason for partition of the Saturn ring as well as rings of other plan- ets into great many of sub-rings is found. This reason is the theorem of Zelikin- Lokutsievskiy-Hildebrand about fractal structure of solutions to generic piece-wise smooth Hamiltonian systems. The instability of two-dimensional model of rings with continues surface density of particles distribution is proved both for Newtonian and for Boltzmann equations. We do not claim that we have solved the problem of stability of Saturn ring. We rather put questions and suggest some ideas and means for researches. 1 Introduction Up to the middle of XX century the astronomy went hand in hand with mathematics. So, it is no wonder that such seemingly purely astronomical problems as figures of equilibrium for rotating, gravitating liquids or the stability of Saturn Ring came to the attention of the most part of outstanding mathematicians. It will be suffice to mention Galilei, Huygens, Newton, Laplace, Maclaurin, Maupertuis, Clairaut, d’Alembert, Legendre, Liouville, Ja- cobi, Riemann, Poincar´e, Chebyshev, Lyapunov, Cartan and numerous others (see, for example, [1]). It is pertinent to recall the out of sight discussion between Poincar´eand Lyapunov regarding figures of equilibrium for rotating, gravitating liquids. Poincar´e, who obtained his results using not so rigorous reasoning and often by simple analogy, wrote, “It is possible to make many objections, but the same rigor as in pure analysis is not de- manded in mechanics”. Lyapunov had entirely different position: “It is impermissible to use doubtful reasonings when solving a problem in mechanics or physics (it is the same) if it is formulated quite definitely as a mathematical one.
    [Show full text]
  • The Cycloid Scott Morrison
    The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on.
    [Show full text]
  • Mathematics, the Language of Watchmaking
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE 90LEARNINGprovidedL by Infoscience E- École polytechniqueA fédérale de LausanneRNINGLEARNINGLEARNING Mathematics, the language of watchmaking Morley’s theorem (1898) states that if you trisect the angles of any triangle and extend the trisecting lines until they meet, the small triangle formed in the centre will always be equilateral. Ilan Vardi 1 I am often asked to explain mathematics; is it just about numbers and equations? The best answer that I’ve found is that mathematics uses numbers and equations like a language. However what distinguishes it from other subjects of thought – philosophy, for example – is that in maths com- plete understanding is sought, mostly by discover- ing the order in things. That is why we cannot have real maths without formal proofs and why mathe- maticians study very simple forms to make pro- found discoveries. One good example is the triangle, the simplest geometric shape that has been studied since antiquity. Nevertheless the foliot world had to wait 2,000 years for Morley’s theorem, one of the few mathematical results that can be expressed in a diagram. Horology is of interest to a mathematician because pallet verge it enables a complete understanding of how a watch or clock works. His job is to impose a sequence, just as a conductor controls an orches- tra or a computer’s real-time clock controls data regulating processing. Comprehension of a watch can be weight compared to a violin where science can only con- firm the preferences of its maker.
    [Show full text]
  • Publication Named Lunar Formations by Blagg and Müller (Published in 1935)
    Lunar and Planetary Information Bulletin January 2018 • Issue 151 Contents Feature Story From the Desk of Jim Green News from Space Meeting Highlights Opportunities for Students Spotlight on Education In Memoriam Milestones New and Noteworthy Calendar Copyright 2018 The Lunar and Planetary Institute Issue 151 Page 2 of 76 January 2018 Feature Story Planetary Nomenclature: A Brief History and Overview Note from the Editors: This issue’s lead article is the ninth in a series of reports describing the history and current activities of the planetary research facilities partially funded by NASA and located nationwide. This issue provides a brief history and overview of planetary nomenclature, an important activity that provides structure and coordination to NASA’s planetary exploration program and the scientific analysis of planetary data. — Paul Schenk and Renée Dotson Humans have been naming the stars and planets for thousands of years, and many of these ancient names are still in use. For example, the Romans named the planet Mars for their god of war, and the satellites Phobos and Deimos, discovered in 1877, were named for the twin sons of Ares, the Greek god of war. In this age of orbiters, rovers, and high-resolution imagery, modern planetary nomenclature is used to uniquely identify a topographical, morphological, or albedo feature on the surface of a planet or satellite so that the feature can be easily located, described, and discussed by scientists and laypeople alike. History of Planetary Nomenclature With the invention of the telescope in 1608, astronomers from many countries began studying the Moon and other planetary bodies.
    [Show full text]
  • Some Characterizations of Involute-Evolute Curve Couple In
    Computa & tio d n ie a l l Abdel-Aziz et al., J Appl Computat Math 2019, 8:2 p M p a Journal of A t h f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access Some Characterizations of Involute-Evolute Curve Couple in the Hyperbolic Space Abdel-Aziz HS1*, Saad MK2 and Abdel-Salam AA1 1Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt 2Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Madinah, Saudi Arabia Abstract This paper aims to show that Frenet apparatus of an evolute curve can be formed according to apparatus of the involute curve in hyperbolic space. We establish relationships among Frenet frame on involute-evolute curve couple in hyperbolic 2-space, hyperbolic 3-space and de Sitter 3-space. Finally, we illustrate some numerical examples in support our main results. Keywords: Frenet frames; Involute-evolute curves; Hyperbolic equations and for more details see [1,6]. In this paper, we calculate space; De Sitter space. the Frenet apparatus of the evolute curve by apparatus of the involute curve in hyperbolic 2-space, hyperbolic 3-space, and de Sitter 3-space. Mathematics Subject Classification (2010). 53A25, 53C50. Our results can be seen as refinement and generalization of many Introduction corresponding results exist in the literature and useful in mathematical modeling and some other applications. The idea of a string involute is due to, who is also known for his work in optics [1]. He discovered involutes while trying to develop a Preliminaries more accurate clock [2].
    [Show full text]
  • 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).
    [Show full text]
  • Is the Tautochrone Curve Unique? Pedro Terra,A) Reinaldo De Melo E Souza,B) and C
    Is the tautochrone curve unique? Pedro Terra,a) Reinaldo de Melo e Souza,b) and C. Farinac) Instituto de Fısica, Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ 21945-970, Brazil (Received 22 February 2016; accepted 9 September 2016) We show that there are an infinite number of tautochrone curves in addition to the cycloid solution first obtained by Christiaan Huygens in 1658. We begin by reviewing the inverse problem of finding the possible potential energy functions that lead to periodic motions of a particle whose period is a given function of its mechanical energy. There are infinitely many such solutions, called “sheared” potentials. As an interesting example, we show that a Poschl-Teller€ potential and the one-dimensional Morse potentials are sheared relative to one another for negative energies, clarifying why they share the same oscillation periods for their bounded solutions. We then consider periodic motions of a particle sliding without friction over a track around its minimum under the influence of a constant gravitational field. After a brief historical survey of the tautochrone problem we show that, given the oscillation period, there is an infinity of tracks that lead to the same period. As a bonus, we show that there are infinitely many tautochrones. VC 2016 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4963770] 3 I. INTRODUCTION instance, Pippard presents a nice graphical demonstration of sheared potentials, whereas Osypowski and Olsson4 provide In classical mechanics, we find essentially two kinds of a different demonstration with the aid of Laplace transforms. problems: (i) fundamental (or direct) problems, in which the Recent developments on this topic, and analogous quantum forces on a system are given and we must obtain the possible cases, can be found in Asorey et al.5 motions, and (ii) inverse problems, in which we know the In this article, our main goal is to analyze inverse prob- possible motions and must determine the forces that caused lems for periodic motions of a particle sliding on a friction- them.
    [Show full text]
  • Prediction of Uncertainties in Atmospheric Properties Measured by Radio Occultation Experiments
    Available online at www.sciencedirect.com Advances in Space Research 46 (2010) 58–73 www.elsevier.com/locate/asr Prediction of uncertainties in atmospheric properties measured by radio occultation experiments Paul Withers Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA Received 3 September 2008; received in revised form 9 November 2009; accepted 4 March 2010 Abstract Refraction due to gradients in ionospheric electron density, N e, and neutral number density, nn, can shift the frequency of radio sig- nals propagating through a planetary atmosphere. Radio occultation experiments measure time series of these frequency shifts, from which N e and nn can be determined. Major contributors to uncertainties in frequency shift are phase noise, which is controlled by the Allan Deviation of the experiment, and thermal noise, which is controlled by the signal-to-noise ratio of the experiment. We derive expressions relatingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uncertainties in atmospheric properties to uncertainties in frequency shift. Uncertainty in N e is approximately 2 ð4prDf fcme0=Ve Þ 2pH p=R where rDf is uncertainty in frequency shift, f is the carrier frequency, c is the speed of light, me is the electron mass, 0 is the permittivity of free space, V isp speed,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie is the elementary charge, H p is a plasma scale height and R is planetary radius. Uncertainty in nn is approximately ðrDf c=Vf jÞ H n=2pR where j and H n are the refractive volume and scale height of the neutral atmo- sphere. Predictions from these expressions are consistent with the uncertainties of the radio occultation experiment on Mars Global Sur- veyor.
    [Show full text]