The Math in “Laser Light Math”
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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. -
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. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
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. -
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. -
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. -
Hypocycloid Motion in the Melvin Magnetic Universe
Hypocycloid motion in the Melvin magnetic universe Yen-Kheng Lim∗ Department of Mathematics, Xiamen University Malaysia, 43900 Sepang, Malaysia May 19, 2020 Abstract The trajectory of a charged test particle in the Melvin magnetic universe is shown to take the form of hypocycloids in two different regimes, the first of which is the class of perturbed circular orbits, and the second of which is in the weak-field approximation. In the latter case we find a simple relation between the charge of the particle and the number of cusps. These two regimes are within a continuously connected family of deformed hypocycloid-like orbits parametrised by the magnetic flux strength of the Melvin spacetime. 1 Introduction The Melvin universe describes a bundle of parallel magnetic field lines held together under its own gravity in equilibrium [1, 2]. The possibility of such a configuration was initially considered by Wheeler [3], and a related solution was obtained by Bonnor [4], though in today’s parlance it is typically referred to as the Melvin spacetime [5]. By the duality of electromagnetic fields, a similar solution consisting of parallel electric fields can be arXiv:2004.08027v2 [gr-qc] 18 May 2020 obtained. In this paper, we shall mainly be interested in the magnetic version of this solution. The Melvin spacetime has been a solution of interest in various contexts of theoretical high-energy physics. For instance, the Melvin spacetime provides a background of a strong magnetic field to induce the quantum pair creation of black holes [6, 7]. Havrdov´aand Krtouˇsshowed that the Melvin universe can be constructed by taking the two charged, accelerating black holes and pushing them infinitely far apart [8]. -
Engineering Curves and Theory of Projections
ME 111: Engineering Drawing Lecture 4 08-08-2011 Engineering Curves and Theory of Projection Indian Institute of Technology Guwahati Guwahati – 781039 Distance of the point from the focus Eccentrici ty = Distance of the point from the directric When eccentricity < 1 Ellipse =1 Parabola > 1 Hyperbola eg. when e=1/2, the curve is an Ellipse, when e=1, it is a parabola and when e=2, it is a hyperbola. 2 Focus-Directrix or Eccentricity Method Given : the distance of focus from the directrix and eccentricity Example : Draw an ellipse if the distance of focus from the directrix is 70 mm and the eccentricity is 3/4. 1. Draw the directrix AB and axis CC’ 2. Mark F on CC’ such that CF = 70 mm. 3. Divide CF into 7 equal parts and mark V at the fourth division from C. Now, e = FV/ CV = 3/4. 4. At V, erect a perpendicular VB = VF. Join CB. Through F, draw a line at 45° to meet CB produced at D. Through D, drop a perpendicular DV’ on CC’. Mark O at the midpoint of V– V’. 3 Focus-Directrix or Eccentricity Method ( Continued) 5. With F as a centre and radius = 1–1’, cut two arcs on the perpendicular through 1 to locate P1 and P1’. Similarly, with F as centre and radii = 2– 2’, 3–3’, etc., cut arcs on the corresponding perpendiculars to locate P2 and P2’, P3 and P3’, etc. Also, cut similar arcs on the perpendicular through O to locate V1 and V1’. 6. Draw a smooth closed curve passing through V, P1, P/2, P/3, …, V1, …, V’, …, V1’, … P/3’, P/2’, P1’. -
Polar Graphs
POLAR EQUATIONS AND THEIR GRAPHS It is often necessary to transform from rectangular to polar form or vice versa. The following polar-rectangular relationships are useful in this regard. Strategy for Changing Equations in Rectangular Form to Polar Form 2 2 2 • Use the conversions r= x + y , x= r cos θ , and x= r sin θ to find a polar equation. • Write the polar equation as r in terms of θ . If the equation contains an r 2 you MUST show factoring in the solution path! DO NOT EVER divide by r to arrive at the final answer! Strategy for Changing Equations in Polar Form to Rectangular Form • If the equation contains r and sines or cosines, multiply both sides of the equation by r unless the sine or cosine is in the denominator. If the equation contains a trigonometric ratio other than sine or cosine, use the Reciprocal or Quotient Identities to change to sines and cosines first. • If the equation contains r and a constant, square both sides of the equation. • If the equation contains an angle θ and a constant, turn both sides into a tangent ratio remembering that y tan θ = . x Polar equations can also be graphed by hand in the Polar Coordinate System , however, this task often becomes extremely difficult. That is why we need to know the shape of some of the graphs in polar coordinates. 1 For the following "special" polar equations you need to be able to associate their name with their characteristics and graphs! Equations of Circles Circles with center along one of the coordinate axes and radius a Circle with center at the origin and radius a.