The Geometry of Map Equations for Trochoids
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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. -
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]. -
The Math in “Laser Light Math”
The Math in “Laser Light Math” When graphed, many mathematical curves are beautiful to view. These curves are usually brought into graphic form by incorporating such devices as a plotter, printer, video screen, or mechanical spirograph tool. While these techniques work, and can produce interesting images, the images are normally small and not animated. To create large-scale animated images, such as those encountered in the entertainment industry, light shows, or art installations, one must call upon some unusual graphing strategies. It was this desire to create large and animated images of certain mathematical curves that led to our design and implementation of the Laser Light Math projection system. In our interdisciplinary efforts to create a new way to graph certain mathematical curves with laser light, Professor Lessley designed the hardware and software and Professor Beale constructed a simplified set of harmonic equations. Of special interest was the issue of graphing a family of mathematical curves in the roulette or spirograph domain with laser light. Consistent with the techniques of making roulette patterns, images created by the Laser Light Math system are constructed by mixing sine and cosine functions together at various frequencies, shapes, and amplitudes. Images created in this fashion find birth in the mathematical process of making “roulette” or “spirograph” curves. From your childhood, you might recall working with a spirograph toy to which you placed one geared wheel within the circumference of another larger geared wheel. After inserting your pen, then rotating the smaller gear around or within the circumference of the larger gear, a graphed representation emerged of a certain mathematical curve in the roulette family (such as the epitrochoid, hypotrochoid, epicycloid, hypocycloid, or perhaps the beautiful rose family). -
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’. -
The Project Gutenberg Ebook #31061: a History of Mathematics
The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith. -
On the Theory of Rolling Curves. by Mr JAMES CLERK MAXWELL. Communicated by the Rev. Professor KELLAND. There Is
( 519 ) XXXV.—On the Theory of Rolling Curves. By Mr JAMES CLERK MAXWELL. Communicated by the Rev. Professor KELLAND. (Read, 19th February 1849.) There is an important geometrical problem which proposes to find a curve having a given relation to a series of curves described according to a given law. This is the problem of Trajectories in its general form. The series of curves is obtained from the general equation to a curve by the variation of its parameters. In the general case, this variation may change the form of the curve, but, in the case which we are about to consider, the curve is changed only in position. This change of position takes place partly by rotation, and partly by trans- ference through space. The rolling of one curve on another is an example of this compound motion. As examples of the way in which the new curve may be related to the series of curves, we may take the following:— 1. The new curve may cut the series of curves at a given angle. When this angle becomes zero, the curve is the envelope of the series of curves. 2. It may pass through corresponding points in the series of curves. There are many other relations which may be imagined, but we shall confine our atten- tion to this, partly because it affords the means of tracing various curves, and partly on account of the connection which it has with many geometrical problems. Therefore the subject of this paper will be the consideration of the relations of three curves, one of which is fixed, while the second rolls upon it and traces the third. -
Representations of Epitrochoids and Hypotrochoids
REPRESENTATIONS OF EPITROCHOIDS AND HYPOTROCHOIDS by Michelle Natalie Rene Marie Bouthillier Submitted in partial fulfillment of the requirements for the degree of Master of Science at Dalhousie University Halifax, Nova Scotia March 2018 © Copyright by Michelle Natalie Rene Marie Bouthillier, 2018 Table of Contents List of Figures ..................................... iii Abstract ......................................... iv Acknowledgements .................................. v Chapter 1 Introduction ............................. 1 Chapter 2 The Implicitization Problem .................. 5 2.1 Introduction . 5 2.2 Resultants . 9 2.3 Gr¨obnerBases . 18 2.4 Resultants versus Gr¨obnerBasis . 47 Chapter 3 Implicitization of Hypotrochoids and Epitrochoids ... 51 3.1 Implicitization Method . 51 3.2 Results for Small Values of m and n . 56 3.3 Examples . 64 Chapter 4 Envelopes ............................... 79 4.1 Introduction . 79 4.2 Epicycloids . 80 4.3 Hypocycloids . 87 Chapter 5 Conclusion .............................. 94 Bibliography ....................................... 95 Appendix ......................................... 97 ii List of Figures 1.1 Epitrochoid with R = 1, r = 3, d = 4. 1 1.2 Hypotrochoid with R = 4, r = 1, d = 3~2. 2 1.3 A trochoid with a = 2r ........................ 3 2.1 Example of a curve that we may wish to implicitize . 8 3.1 Epicycloids with implicit forms described by Conjecture 2 . 66 3.2 Epicycloids with implicit forms described by Conjecture 3 . 67 3.3 Hypocycloids with implicit forms described by Conjecture 4 . 69 3.4 Hypocycloids with implicit forms described by Conjecture 5 . 71 3.5 Hypocycloids with implicit forms described by Conjecture 6 . 73 3.6 Hypocycloids with implicit forms described by Conjecture 7 . 74 3.7 Hypocycloids with implicit forms described by Conjecture 8 . 76 3.8 Hypocycloids with implicit forms described by Conjecture 9 . -
Encyclopaedia Britannica, 11Th Edition, by Various 1
Encyclopaedia Britannica, 11th Edition, by Various 1 Encyclopaedia Britannica, 11th Edition, by Various The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 6, by Various This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 6 "English Language" to "Epsom Salts" Author: Various Release Date: February 17, 2011 [EBook #35306] Language: English Character set encoding: ASCII *** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA *** Produced by Marius Masi, Don Kretz and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber's notes: (1) Numbers following letters (without space) like C2 were originally printed in subscript. Letter subscripts are preceded by an underscore, like Cn. Encyclopaedia Britannica, 11th Edition, by Various 2 (2) Characters following a carat (^) were printed in superscript. (3) Side-notes were relocated to function as titles of their respective paragraphs. (4) Macrons and breves above letters and dots below letters were not inserted. (5) Small and capital EZH letters are subtituted with [gh] and [Gh] respectively. Thorn is subtituted with th or Th, and eth is subtituted with dh. (6) [root] stands for the root symbol; [alpha], [beta], etc. for greek letters. (7) The following typographical errors have been corrected: ARTICLE ENGLISH LANGUAGE: "The writers of each district wrote in the dialect familiar to them; and between extreme forms the difference was so great as to amount to unintelligibility ..." 'familiar' amended from 'familar'. -
NEW HORIZONS in GEOMETRY 2010 Mathematics Subject Classification
DOLCIANI MATHEMATICAL EXPOSITIONS 47 10.1090/dol/047 NEW HORIZONS IN GEOMETRY 2010 Mathematics Subject Classification. Primary 51-01. Originally published by The Mathematical Association of America, 2012. ISBN: 978-1-4704-4335-1 LCCN: 2012949754 Copyright c 2012, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2018 The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 23 22 21 20 19 18 The Dolciani Mathematical Expositions NUMBER FORTY-SEVEN NEW HORIZONS IN GEOMETRY Tom M. Ap ostol California Institute of Technology and Mamikon A. Mnatsakanian California Institute of Technology Providence, Rhode Island The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initi- ating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. -
ICTCM.COM ICTCM 28Th International Conference on Technology in Collegiate Mathematics
ICTCM 28th International Conference on Technology in Collegiate Mathematics THE BRACHISTOCHRONE ON A ROTATING EARTH J Villanueva Florida Memorial University 15800 NW 42nd Ave Miami, FL 33054 [email protected] 1. Introduction 1.1 The brachistochrone problem 1.2 The cycloid 2. Free fall through the Earth 2.1 Homogeneous stationary Earth: simple harmonic motion 2.2 Linear gravity rotating Earth: trochoid 3. Fall trajectory 3.1 Trochoid: epicycles 3.2 Trochoid: hypocycles 4. Examples 4.1 Free fall in the equatorial plane 4.2 Free fall in the northern hemisphere References: 1. R Calinger, R, 1995. Classics of Mathematics. Englewood Cliffs, NJ: Prentice-Hall 2. Hawking, S, 1988. A Brief History of Time. NY: Bantam. 3. Maor, E, 1998. Trigonometric Delights. Princeton, NJ: Princeton University Press. 4. Simoson, A, 2010. Voltaire’s Riddle. Washington, DC: Mathematical Association of America. 5. Simoson, A, 2007. Hesiod’s Anvil. Washington, DC: Mathematical Association of America. 6. Stewart, J, 2012. Single Variable Calculus. Belmont, CA: Brooks/Cole. 7. Thomas, G, 1972. Calculus. Reading, MA: Addison-Wesley. In 1696 Johann Bernoulli challenged the mathematicians of his day to find the curve of quickest descent for a body to fall through a frictionless homogeneous Earth between two given points in a vertical plane [Calinger, 1995]. The problem languished for months until Newton chanced 408 ICTCM.COM ICTCM 28th International Conference on Technology in Collegiate Mathematics upon the problem, quickly solved it, and published the solution anonymously. But Bernoulli recognized the “lion by its claw” – the path is a cycloid. We have shown the solution of this problem by various authors in previous communcications.