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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. -
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. -
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. -
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 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. -
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’. -
Distance Minimizing Paths on Certain Simple Flat Surfaces with Singularities
Millersville University Distance Minimizing Paths on Certain Simple Flat Surfaces with Singularities A Senior Thesis Submitted to the Department of Mathematics & The University Honors Program In Partial Fulfillment of the Requirements for the University & Departmental Honors Baccalaureate By Joel B. Mohler Millersville, Pennsylvania December 2002 1 This Senior Thesis was completed in the Department of Mathematics, defended before and approved by the following members of the Thesis committee Ronald Umble, Ph.D (Thesis Advisor) Professor of Mathematics Dorothee Blum, Ph.D Associate Professor of Mathematics Roxana Costinescu, Ph.D Assistant Professor of Mathematics 2 Abstract We analyze a conical cup with lid and a circular tin can to find the shortest path connecting two points anywhere on the surface. We show how to transform the problem into an equivalent problem in plane geometry and prove that such paths can be found in most cases. 3 Distance Minimizing Paths on Certain Simple Flat Surfaces with Singularities Contents 1 Introduction 5 2 Simple Flat Surfaces 6 3 Singularities along the Rim 7 3.1 Roulettes Along Lines . 8 3.2 Roulettes Along Circles . 10 4 Distance Minimizing Paths 12 4.1 Derivation of geodesics on cone . 13 4.2 Self Intersecting geodesics on the cone side . 14 5 Geodesics over the Rim 15 6 Tin Can: Lid to Base 18 7 Minimal Geodesics at the Vertex 21 8 Geodesics Minimizing Distance 22 9 Non-Unique Minimal Geodesics 26 9.1 Diametrically Opposed Points . 27 9.2 Open Questions Concerning Non-unique Minimal Paths . 29 References 30 4 1 Introduction In the spring of 2001, I participated in a research seminar directed by Dr. -
The Mechanical Drawing of Cycloids, the Geometric Chuck
The Mechanical Drawing of Cycloids, The Geometric Chuck Robert Craig 8633 E. 96th Terrace Kansas City, MO 64134 E-mail [email protected] Abstract This paper discusses cycloids and their construction using the 19th century mechanical drawing instrument known as the Geometric Chuck. The first part of the paper is a brief history and description of the Geometric Chuck. The last part of the paper is devoted to a discussion the definition of cycloids and examples showing the results that various settings of the Geometric Chuck have on the cycloid patterns produced. This paper is an attempt, in part, to respond to the comment in the Savory book "As this book does not aim at giving a scientific account of the principles on which it works. It might be an exceedingly interesting subject for the scientific person, ….the scientific knowledge required to understand a three-part chuck would be so great that I doubt if there is the person existing who could describe the course of a line that would be produced…."[6] 1. History 1.1 Definition. "The Geometric Chuck is an arrangement of mechanism for producing two or more circular movements in parallel planes. The combination of these movements with different velocity ratios and different radii results in the formation of a great variety of highly interesting curves and geometric figures"[1]. The Geometric Chuck is just one of many devices that were designed to be used as accessories to the Rose Engine Lathe. This lathe, similar to the modern lathe in name only, was designed to produce ornately decorated items. -
Hermite Interpolation with HE-Splines Miroslav L´Aviˇcka, Zbynˇek S´Irˇ and Bohum´Ir Bastl Dept
Geometry and Graphics 2009 205 Hermite interpolation with HE-splines Miroslav L´aviˇcka, Zbynˇek S´ırˇ and Bohum´ır Bastl Dept. of Mathematics, Fac. of Applied Sciences, Univ. of West Bohemia Univerzitn´ı22, 301 00 Plzeˇn, Czech Republic lavicka,zsir,bastl @kma.zcu.cz { } Abstract. Hypo/epicycloids (shortly HE-cycloids) are well-known curves, in detail studied in classical geometry. Hence, one may wonder what new can be said about these traditional geometric objects. It has been proved recently, cf. [12], that all rational HE-cycloids are curves with rational offsets, i.e. they belong to the class of rational Pythagorean hodograph curves. In this paper, we present an algorithm for G1 Hermite interpolation with hypo/epicycloidal arcs which results from their support function representation. Keywords: Hypocycloids; epicycloids; Pythagorean hodograph curves; rational offsets; support function; Hermite interpolation 1 Introduction Hypocycloids and epicycloids are a subclass of the so called roulettes, i.e., planar curves traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. In this case, both curves are circles, cf. [13, 1, 9]. Various sizes of circles generate different hypo/epicycloids, which are closed when the ratio of the radius of the rolling circle and the radius of the other circle is rational. It has been proved recently, cf. [12], that all rational hypo/epicycloids are curves with Pythagorean normals and therefore they provide ratio- nal offsets. This new result was obtained with the help of support function (SF) representation of hypo/epicycloids. In this paper, we present the main idea of the algorithm for G1 Hermite interpolation with hypo/epicycloidal arcs.