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Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia
Brief Information on the Surfaces Not Included in the Basic Content of the Encyclopedia Brief information on some classes of the surfaces which cylinders, cones and ortoid ruled surfaces with a constant were not picked out into the special section in the encyclo- distribution parameter possess this property. Other properties pedia is presented at the part “Surfaces”, where rather known of these surfaces are considered as well. groups of the surfaces are given. It is known, that the Plücker conoid carries two-para- At this section, the less known surfaces are noted. For metrical family of ellipses. The straight lines, perpendicular some reason or other, the authors could not look through to the planes of these ellipses and passing through their some primary sources and that is why these surfaces were centers, form the right congruence which is an algebraic not included in the basic contents of the encyclopedia. In the congruence of the4th order of the 2nd class. This congru- basis contents of the book, the authors did not include the ence attracted attention of D. Palman [8] who studied its surfaces that are very interesting with mathematical point of properties. Taking into account, that on the Plücker conoid, view but having pure cognitive interest and imagined with ∞2 of conic cross-sections are disposed, O. Bottema [9] difficultly in real engineering and architectural structures. examined the congruence of the normals to the planes of Non-orientable surfaces may be represented as kinematics these conic cross-sections passed through their centers and surfaces with ruled or curvilinear generatrixes and may be prescribed a number of the properties of a congruence of given on a picture. -
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. -
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. -
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. -
An Approach on Simulation of Involute Tooth Profile Used on Cylindrical Gears
BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi Volumul 66 (70), Numărul 2, 2020 Secţia CONSTRUCŢII DE MAŞINI AN APPROACH ON SIMULATION OF INVOLUTE TOOTH PROFILE USED ON CYLINDRICAL GEARS BY MIHĂIȚĂ HORODINCĂ “Gheorghe Asachi” Technical University of Iaşi, Faculty of Machine Manufacturing and Industrial Management Received: April 6, 2020 Accepted for publication: June 10, 2020 Abstract. Some results on theoretical approach related by simulation of 2D involute tooth profiles on spur gears, based on rolling of a mobile generating rack around a fixed pitch circle are presented in this paper. A first main approach proves that the geometrical depiction of each generating rack position during rolling can be determined by means of a mathematical model which allows the calculus of Cartesian coordinates for some significant points placed on the rack. The 2D involute tooth profile appears to be the internal envelope bordered by plenty of different equidistant positions of the generating rack during a complete rolling. A second main approach allows the detection of Cartesian coordinates of this internal envelope as the best approximation of 2D involute tooth profile. This paper intends to provide a way for a better understanding of involute tooth profile generation procedure. Keywords: tooth profile; involute; generating rack; simulation. 1. Introduction Gear manufacturing is a major topic in industry due to a large scale utilization of gears for motion transmissions and speed reducers. Commonly the flank gear surface on cylindrical toothed wheels (gears) is described by two Corresponding author: e-mail: [email protected] 22 Mihăiţă Horodincă orthogonally curves: an involute as flank line (in a transverse plane) and a profile line. -
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. -
Plotting the Spirograph Equations with Gnuplot
Plotting the spirograph equations with gnuplot V´ıctor Lua˜na∗ Universidad de Oviedo, Departamento de Qu´ımica F´ısica y Anal´ıtica, E-33006-Oviedo, Spain. (Dated: October 15, 2006) gnuplot1 internal programming capabilities are used to plot the continuous and segmented ver- sions of the spirograph equations. The segmented version, in particular, stretches the program model and requires the emmulation of internal loops and conditional sentences. As a final exercise we develop an extensible minilanguage, mixing gawk and gnuplot programming, that lets the user combine any number of generalized spirographic patterns in a design. Article published on Linux Gazette, November 2006 (#132) I. INTRODUCTION S magine the movement of a small circle that rolls, with- ¢ O ϕ Iout slipping, on the inside of a rigid circle. Imagine now T ¢ 0 ¢ P ¢ that the small circle has an arm, rigidly atached, with a plotting pen fixed at some point. That is a recipe for β drawing the hypotrochoid,2 a member of a large family of curves including epitrochoids (the moving circle rolls on the outside of the fixed one), cycloids (the pen is on ϕ Q ¡ ¡ ¡ the edge of the rolling circle), and roulettes (several forms ¢ rolling on many different types of curves) in general. O The concept of wheels rolling on wheels can, in fact, be generalized to any number of embedded elements. Com- plex lathe engines, known as Guilloch´e machines, have R been used since the XVII or XVIII century for engrav- ing with beautiful designs watches, jewels, and other fine r p craftsmanships. Many sources attribute to Abraham- Louis Breguet the first use in 1786 of Gilloch´eengravings on a watch,3 but the technique was already at use on jew- elry. -
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. -
Deltoid* the Deltoid Curve Was Conceived by Euler in 1745 in Con
Deltoid* The Deltoid curve was conceived by Euler in 1745 in con- nection with his study of caustics. Formulas in 3D-XplorMath: x = 2 cos(t) + cos(2t); y = 2 sin(t) − sin(2t); 0 < t ≤ 2π; and its implicit equation is: (x2 + y2)2 − 8x(x2 − 3y2) + 18(x2 + y2) − 27 = 0: The Deltoid or Tricuspid The Deltoid is also known as the Tricuspid, and can be defined as the trace of a point on one circle that rolls inside * This file is from the 3D-XplorMath project. Please see: http://3D-XplorMath.org/ 1 2 another circle of 3 or 3=2 times as large a radius. The latter is called double generation. The figure below shows both of these methods. O is the center of the fixed circle of radius a, C the center of the rolling circle of radius a=3, and P the tracing point. OHCJ, JPT and TAOGE are colinear, where G and A are distant a=3 from O, and A is the center of the rolling circle with radius 2a=3. PHG is colinear and gives the tangent at P. Triangles TEJ, TGP, and JHP are all similar and T P=JP = 2 . Angle JCP = 3∗Angle BOJ. Let the point Q (not shown) be the intersection of JE and the circle centered on C. Points Q, P are symmetric with respect to point C. The intersection of OQ, PJ forms the center of osculating circle at P. 3 The Deltoid has numerous interesting properties. Properties Tangent Let A be the center of the curve, B be one of the cusp points,and P be any point on the curve. -
Unity Via Diversity 81
Unity via diversity 81 Unity via diversity Unity via diversity is a concept whose main idea is that every entity could be examined by different point of views. Many sources name this conception as interdisciplinarity . The mystery of interdisciplinarity is revealed when people realize that there is only one discipline! The division into multiple disciplines (or subjects) is just the human way to divide-and-understand Nature. Demonstrations of how informatics links real life and mathematics can be found everywhere. Let’s start with the bicycle – a well-know object liked by most students. Bicycles have light reflectors for safety reasons. Some of the reflectors are attached sideway on the wheels. Wheel reflector Reflectors notify approaching vehicles that there is a bicycle moving along or across the road. Even if the conditions prevent the driver from seeing the bicycle, the reflector is a sufficient indicator if the bicycle is moving or not, is close or far, is along the way or across it. Curve of the reflector of a rolling wheel When lit during the night, the wheel’s side reflectors create a beautiful luminous curve – a trochoid . 82 Appendix to Chapter 5 A trochoid curve is the locus of a fixed point as a circle rolls without slipping along a straight line. Depending on the position of the point a trochoid could be further classified as curtate cycloid (the point is internal to the circle), cycloid (the point is on the circle), and prolate cycloid (the point is outside the circle). Cycloid, curtate cycloid and prolate cycloid An interesting activity during the study of trochoids is to draw them using software tools. -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. -
The Nephroid
1/13 The Nephroid Breanne Perry and Brian Meyer Introduction:Definitions Cycloid:The locus of a point on the outside of a circle with a set radius (r) rolling along a straight line. 2/13 Epicycloid:An epicycloid is a cycloid rolling along the outside of a circle instead of along a straight line. The description of an epicycloid depends on the radius of the circle being traversed (b) and the radius of the rolling circle (a). Nephroid:A bicuspid epicycloid. Nephroid Literally means ”kidney shaped”. The nephroid arises when the radius of the center circle is twice that of the rolling circle (b=a/2). Catacaustic:The curve formed by reflecting light off of another curve. Catacaustic of the Cardiod 3/13 History of the Nephroid The Nephroid was first studied by Christiaan Huygens and Tshirn- hausen in 1678, they found it to be the catacaustic of a circle when the 4/13 light source is at infinity. George Airy proved this to be true through the wave theory of light in 1838. In 1878 Richard Proctor coined the term nephroid in his book The Geometry of cycloids. The term Nephroid replaced the existing two-cusped epicycloid. Some members of the epicycloid family 5/13 line 2 b=a Cardioid line 3 b=a/2 Nephroid line 4 b=a/3 three cusped epicycloid Deriving the formula for the Nephroid First to determine the relationship between the change in angle from the large circle to the small circle, we analyze the the relationship be- 6/13 tween point D and E as they traverse the outside of their respected circles.