Cycloids and Paths
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Engineering Curves – I
Engineering Curves – I 1. Classification 2. Conic sections - explanation 3. Common Definition 4. Ellipse – ( six methods of construction) 5. Parabola – ( Three methods of construction) 6. Hyperbola – ( Three methods of construction ) 7. Methods of drawing Tangents & Normals ( four cases) Engineering Curves – II 1. Classification 2. Definitions 3. Involutes - (five cases) 4. Cycloid 5. Trochoids – (Superior and Inferior) 6. Epic cycloid and Hypo - cycloid 7. Spiral (Two cases) 8. Helix – on cylinder & on cone 9. Methods of drawing Tangents and Normals (Three cases) ENGINEERING CURVES Part- I {Conic Sections} ELLIPSE PARABOLA HYPERBOLA 1.Concentric Circle Method 1.Rectangle Method 1.Rectangular Hyperbola (coordinates given) 2.Rectangle Method 2 Method of Tangents ( Triangle Method) 2 Rectangular Hyperbola 3.Oblong Method (P-V diagram - Equation given) 3.Basic Locus Method 4.Arcs of Circle Method (Directrix – focus) 3.Basic Locus Method (Directrix – focus) 5.Rhombus Metho 6.Basic Locus Method Methods of Drawing (Directrix – focus) Tangents & Normals To These Curves. CONIC SECTIONS ELLIPSE, PARABOLA AND HYPERBOLA ARE CALLED CONIC SECTIONS BECAUSE THESE CURVES APPEAR ON THE SURFACE OF A CONE WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES. OBSERVE ILLUSTRATIONS GIVEN BELOW.. Ellipse Section Plane Section Plane Hyperbola Through Generators Parallel to Axis. Section Plane Parallel to end generator. COMMON DEFINATION OF ELLIPSE, PARABOLA & HYPERBOLA: These are the loci of points moving in a plane such that the ratio of it’s distances from a fixed point And a fixed line always remains constant. The Ratio is called ECCENTRICITY. (E) A) For Ellipse E<1 B) For Parabola E=1 C) For Hyperbola E>1 Refer Problem nos. -
Polar Coordinates and Complex Numbers Infinite Series Vectors and Matrices Motion Along a Curve Partial Derivatives
Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER 12 Motion Along a Curve I [ 12.1 The Position Vector I-, This chapter is about "vector functions." The vector 2i +4j + 8k is constant. The vector R(t) = ti + t2j+ t3k is moving. It is a function of the parameter t, which often represents time. At each time t, the position vector R(t) locates the moving body: position vector =R(t) =x(t)i + y(t)j + z(t)k. -
Leibniz's Differential Calculus Applied to the Catenary
Leibniz’s differential calculus applied to the catenary Olivier Keller, agrégé in mathematics, PhD (EHESS) Study of the catenary was a response to a challenge laid down by Jacques Bernoulli, and which was successfully met by Leibniz as well as by Jean Bernoulli and Huygens: to find the curve described by a piece of strung suspended from its two ends. Stimulated by the success of this initial research, Jean Bernoulli put forward and solved similar problems: the form taken by a horizontal blade immobilised on one side, with a weight attached to the other; the form taken by a piece of linen filled with liqueur; the curve of a sail. Challenges among scholars In the 17th century, it was customary for scholars to set each other challenges in alternate issues of journals. Leibniz, for example, challenged the Cartesians – as part of the controversy surrounding the laws of collision, known as the “querelle des forces vives” (vis viva controversy) – to find the curve down which a body falls at constant vertical velocity (isochrone curve). Put forward in the Nouvelle République des Lettres of September 1687, this problem received Huygens’ solution in October of the same year, while Leibniz’s appeared in 1689 in the journal he had created, the Acta Eruditorum. Another famous example is that of the brachistochrone, or the “curve of the fastest descent”, which, thanks to the new differential calculus, proved to be not a circle, as Galileo had believed, but a cycloid: the challenge had been set by Jean Bernoulli in the Acta Eruditorum in June 1696, and was resolved by Leibniz in May 1697. -
Arts Revealed in Calculus and Its Extension
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/265557431 Arts revealed in calculus and its extension Article · August 2014 CITATIONS READS 9 455 1 author: Hanna Arini Parhusip Universitas Kristen Satya Wacana 95 PUBLICATIONS 89 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Pusnas 2015/2016 View project All content following this page was uploaded by Hanna Arini Parhusip on 16 September 2014. The user has requested enhancement of the downloaded file. [Type[Type aa quotequote fromfrom thethe documentdocument oror thethe International Journal of Statistics and Mathematics summarysummary ofof anan interestinginteresting point.point. YouYou cancan Vol. 1(3), pp. 016-023, August, 2014. © www.premierpublishers.org, ISSN: xxxx-xxxx x positionposition thethe texttext boxbox anywhereanywhere inin thethe IJSM document. Use the Text Box Tools tab to document. Use the Text Box Tools tab to changechange thethe formattingformatting ofof thethe pullpull quotequote texttext box.]box.] Review Arts revealed in calculus and its extension Hanna Arini Parhusip Mathematics Department, Science and Mathematics Faculty, Satya Wacana Christian University (SWCU)-Jl. Diponegoro 52-60 Salatiga, Indonesia Email: [email protected], Tel.:0062-298-321212, Fax: 0062-298-321433 Motivated by presenting mathematics visually and interestingly to common people based on calculus and its extension, parametric curves are explored here to have two and three dimensional objects such that these objects can be used for demonstrating mathematics. Epicycloid, hypocycloid are particular curves that are implemented in MATLAB programs and the motifs are presented here. The obtained curves are considered to be domains for complex mappings to have new variation of Figures and objects. -
Introduction to the Modern Calculus of Variations
MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result. -
ON the BRACHISTOCHRONE PROBLEM 1. Introduction. The
ON THE BRACHISTOCHRONE PROBLEM OLEG ZUBELEVICH STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA, 119899, MOSCOW, MGU [email protected] Abstract. In this article we consider different generalizations of the Brachistochrone Problem in the context of fundamental con- cepts of classical mechanics. The correct statement for the Brachis- tochrone problem for nonholonomic systems is proposed. It is shown that the Brachistochrone problem is closely related to vako- nomic mechanics. 1. Introduction. The Statement of the Problem The article is organized as follows. Section 3 is independent on other text and contains an auxiliary ma- terial with precise definitions and proofs. This section can be dropped by a reader versed in the Calculus of Variations. Other part of the text is less formal and based on Section 3. The Brachistochrone Problem is one of the classical variational prob- lems that we inherited form the past centuries. This problem was stated by Johann Bernoulli in 1696 and solved almost simultaneously by him and by Christiaan Huygens and Gottfried Wilhelm Leibniz. Since that time the problem was discussed in different aspects nu- merous times. We do not even try to concern this long and celebrated history. 2000 Mathematics Subject Classification. 70G75, 70F25,70F20 , 70H30 ,70H03. Key words and phrases. Brachistochrone, vakonomic mechanics, holonomic sys- tems, nonholonomic systems, Hamilton principle. The research was funded by a grant from the Russian Science Foundation (Project No. 19-71-30012). 1 2 OLEG ZUBELEVICH This article is devoted to comprehension of the Brachistochrone Problem in terms of the modern Lagrangian formalism and to the gen- eralizations which such a comprehension involves. -
On the Topology of Hypocycloid Curves
Física Teórica, Julio Abad, 1–16 (2008) ON THE TOPOLOGY OF HYPOCYCLOIDS Enrique Artal Bartolo∗ and José Ignacio Cogolludo Agustíny Departamento de Matemáticas, Facultad de Ciencias, IUMA Universidad de Zaragoza, 50009 Zaragoza, Spain Abstract. Algebraic geometry has many connections with physics: string theory, enu- merative geometry, and mirror symmetry, among others. In particular, within the topo- logical study of algebraic varieties physicists focus on aspects involving symmetry and non-commutativity. In this paper, we study a family of classical algebraic curves, the hypocycloids, which have links to physics via the bifurcation theory. The topology of some of these curves plays an important role in string theory [3] and also appears in Zariski’s foundational work [9]. We compute the fundamental groups of some of these curves and show that they are in fact Artin groups. Keywords: hypocycloid curve, cuspidal points, fundamental group. PACS classification: 02.40.-k; 02.40.Xx; 02.40.Re . 1. Introduction Hypocycloid curves have been studied since the Renaissance (apparently Dürer in 1525 de- scribed epitrochoids in general and then Roemer in 1674 and Bernoulli in 1691 focused on some particular hypocycloids, like the astroid, see [5]). Hypocycloids are described as the roulette traced by a point P attached to a circumference S of radius r rolling about the inside r 1 of a fixed circle C of radius R, such that 0 < ρ = R < 2 (see Figure 1). If the ratio ρ is rational, an algebraic curve is obtained. The simplest (non-trivial) hypocycloid is called the deltoid or the Steiner curve and has a history of its own both as a real and complex curve. -
Around and Around ______
Andrew Glassner’s Notebook http://www.glassner.com Around and around ________________________________ Andrew verybody loves making pictures with a Spirograph. The result is a pretty, swirly design, like the pictures Glassner EThis wonderful toy was introduced in 1966 by Kenner in Figure 1. Products and is now manufactured and sold by Hasbro. I got to thinking about this toy recently, and wondered The basic idea is simplicity itself. The box contains what might happen if we used other shapes for the a collection of plastic gears of different sizes. Every pieces, rather than circles. I wrote a program that pro- gear has several holes drilled into it, each big enough duces Spirograph-like patterns using shapes built out of to accommodate a pen tip. The box also contains some Bezier curves. I’ll describe that later on, but let’s start by rings that have gear teeth on both their inner and looking at traditional Spirograph patterns. outer edges. To make a picture, you select a gear and set it snugly against one of the rings (either inside or Roulettes outside) so that the teeth are engaged. Put a pen into Spirograph produces planar curves that are known as one of the holes, and start going around and around. roulettes. A roulette is defined by Lawrence this way: “If a curve C1 rolls, without slipping, along another fixed curve C2, any fixed point P attached to C1 describes a roulette” (see the “Further Reading” sidebar for this and other references). The word trochoid is a synonym for roulette. From here on, I’ll refer to C1 as the wheel and C2 as 1 Several the frame, even when the shapes Spirograph- aren’t circular. -
Additive Manufacturing Application for a Turbopump Rotor
EDITORIAL Identificarea nevoii societale (1) Fie că este vorba de ştiinţă sau de politică, Max Weber viza acelaşi scop: să extragă etica specifică unei activităţi pe care o dorea conformă cu finalitatea sa Raymond ARON Viața oricărei persoane moderne este asociată, mai mult sau mai puțin conștient, așteptării societale. Putem spune că suntem imersați în așteptare, că depindem de ea și o influențăm. Cumpărăm mărfuri din magazine mici ori hipermarketuri, ne instruim în școli și universități, folosim produse realizate în fabrici, avem relații cu băncile. Ca o concluzie, suntem angajați ai unor instituții/organizații și clienți ai altora. Toate aceste lucruri determină o creștere mai mare ca niciodată a valorii organizațiilor și a activităților organizaționale. Teoria organizației este o știință bine formată, fondator al acestei discipline fiind considerat sociologul, avocatul, economistul și istoricului german Max Weber (1864-1920) cel care a trasat așa-numita direcție birocratică în dezvoltarea teoriei managementului și organizării. În scrierile sale despre raționalizarea societății, căutând un răspuns la întrebarea ce trebuie făcut pentru ca întreaga organizație să funcționeze ca o mașină, acesta a subliniat că ordinea, susținută de reguli relevante, este cea mai eficientă metodă de lucru pentru orice grup organizat de oameni. El a considerat că organizația poate fi descompusă în părțile componente și se poate normaliza activitatea fiecăreia dintre aceste părți, a propus astfel să se reglementeze cu precizie numărul și funcțiile angajaților și ale organizațiilor, a subliniat că organizația trebuie administrată pe o bază rațională/impersonală. În acest punct, aspectul social al organizației este foarte important, oamenii fiind recunoscuți pentru ideile, caracterul, relațiile, cultura și deloc de neglijat, așteptările lor. -
On the Topology of Hypocycloids
ON THE TOPOLOGY OF HYPOCYCLOIDS ENRIQUE ARTAL BARTOLO AND JOSE´ IGNACIO COGOLLUDO-AGUST´IN Abstract. Algebraic geometry has many connections with physics: string theory, enumerative geometry, and mirror symmetry, among others. In par- ticular, within the topological study of algebraic varieties physicists focus on aspects involving symmetry and non-commutativity. In this paper, we study a family of classical algebraic curves, the hypocycloids, which have links to physics via the bifurcation theory. The topology of some of these curves plays an important role in string theory [3] and also appears in Zariski’s foundational work [9]. We compute the fundamental groups of some of these curves and show that they are in fact Artin groups. 1. Introduction Hypocycloid curves have been studied since the Renaissance (apparently D¨urer in 1525 described epitrochoids in general and then Roemer in 1674 and Bernoulli in 1691 focused on some particular hypocycloids, like the astroid, see [5]). Hypocy- cloids are described as the roulette traced by a point P attached to a circumference S of radius r rolling about the inside of a fixed circle C of radius R, such that r 1 0 < ρ = R < 2 (see Figure 1). If the ratio ρ is rational, an algebraic curve is ob- tained. The simplest (non-trivial) hypocycloid is called the deltoid or the Steiner curve and has a history of its own both as a real and complex curve. S r C P arXiv:1703.08308v1 [math.AG] 24 Mar 2017 R Figure 1. Hypocycloid Key words and phrases. hypocycloid curve, cuspidal points, fundamental group. -
The Cycloid: Tangents, Velocity Vector, Area, and Arc Length
The Cycloid: Tangents, Velocity Vector, Area, and Arc Length [This is Chapter 2, section 13 of Historical Perspectives for the Reform of Mathematics Curriculum: Geometric Curve Drawing Devices and their Role in the Transition to an Algebraic Description of Functions; http://www.quadrivium.info/mathhistory/CurveDrawingDevices.pdf Interactive applets for the figures can also be found at Mathematical Intentions.] The circle is the curve with which we all have the most experience. It is an ancient symbol and a cultural icon in most human societies. It is also the one curve whose area, tangents, and arclengths are discussed in our mathematics curriculum without the use of calculus, and indeed long before students approach calculus. This discussion can take place, because most people have a lot of experience with circles, and know several ways to generate them. Pascal thought that, second only to the circle, the curve that he saw most in daily life was the cycloid (Bishop, 1936). Perhaps the large and slowly moving carriage wheels of the seventeenth century were more easily observed than those of our modern automobile, but the cycloid is still a curve that is readily generated and one in which many students of all ages easily take an interest. In a variety of settings, when I have mentioned, for example, the path of an ant riding on the side of a bicycle tire, some immediate interest has been sparked (see Figure 2.13a). Figure 2.13a The cycloid played an important role in the thinking of the seventeenth century. It was used in architecture and engineering (e.g. -
UTOPIAE Optimisation and Uncertainty Quantification CPD for Teachers of Advanced Higher Physics and Mathematics
UTOPIAE Optimisation and Uncertainty Quantification CPD for Teachers of Advanced Higher Physics and Mathematics Outreach Material Peter McGinty Annalisa Riccardi UTOPIAE, UNIVERSITY OF STRATHCLYDE GLASGOW - UK This research was funded by the European Commission’s Horizon 2020 programme under grant number 722734 First release, October 2018 Contents 1 Introduction ....................................................5 1.1 Who is this for?5 1.2 What are Optimisation and Uncertainty Quantification and why are they im- portant?5 2 Optimisation ...................................................7 2.1 Brachistochrone problem definition7 2.2 How to solve the problem mathematically8 2.3 How to solve the problem experimentally 10 2.4 Examples of Brachistochrone in real life 11 3 Uncertainty Quantification ..................................... 13 3.1 Probability and Statistics 13 3.2 Experiment 14 3.3 Introduction of Uncertainty 16 1. Introduction 1.1 Who is this for? The UTOPIAE Network is committed to creating high quality engagement opportunities for the Early Stage Researchers working within the network and also outreach materials for the wider academic community to benefit from. Optimisation and Uncertainty Quantification, whilst representing the future for a large number of research fields, are relatively unknown disciplines and yet the benefit and impact they can provide for researchers is vast. In order to try to raise awareness, UTOPIAE has created a Continuous Professional Development online resource and training sessions aimed at teachers of Advanced Higher Physics and Mathematics, which is in line with Scotland’s Curriculum for Excellence. Sup- ported by the Glasgow City Council, the STEM Network, and Scottish Schools Education Resource Centre these materials have been published online for teachers to use within a classroom context.