A Handbook on Curves and Their Properties
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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. -
1915-16.] the " Geometria Organica" of Colin Maclaurin. 87 V.—-The
1915-16.] The " Geometria Organica" of Colin Maclaurin. 87 V.—-The " Geometria Organica " of Colin Maclaurin: A Historical and Critical Survey. By Charles Tweedie, M.A., B.Sc, Lecturer in Mathematics, Edinburgh University. (MS. received October 15, 1915. Bead December 6,1915.) INTRODUCTION. COLIN MACLATJBIN, the celebrated mathematician, was born in 1698 at Kilmodan in Argyllshire, where his father was minister of the parish. In 1709 he entered Glasgow University, where his mathematical talent rapidly developed under the fostering care of Professor Robert Simson. In 171*7 he successfully competed for the Chair of Mathematics in the Marischal College of Aberdeen University. In 1719 he came directly under the personal influence of Newton, when on a visit to London, bearing with him the manuscript of the Geometria Organica, published in quarto in 1720. The publication of this work immediately brought him into promi- nence in the scientific world. In 1725 he was, on the recommendation of Newton, elected to the Chair of Mathematics in Edinburgh University, which he occupied until his death in 1746. As a lecturer Maclaurin was a conspicuous success. He took great pains to make his subject as clear and attractive as possible, so much so that he made mathematics " a fashionable study." The labour of teaching his numerous students seriously curtailed the time he could spare for original research. In quantity his works do not bulk largely, but what he did produce was, in the main, of superlative quality, presented clearly and concisely. The Geometria Organica and the Geometrical Appendix to his Treatise on Algebra give him a place in the first rank of great geometers, forming as they do the basis of the theory of the Higher Plane Curves; while his Treatise of Fluxions (1742) furnished an unassailable bulwark and text-book for the study of the Calculus. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
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. -
Using Elimination to Describe Maxwell Curves Lucas P
Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2006 Using elimination to describe Maxwell curves Lucas P. Beverlin Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Applied Mathematics Commons Recommended Citation Beverlin, Lucas P., "Using elimination to describe Maxwell curves" (2006). LSU Master's Theses. 2879. https://digitalcommons.lsu.edu/gradschool_theses/2879 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. USING ELIMINATION TO DESCRIBE MAXWELL CURVES A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Master of Science in The Department of Mathematics by Lucas Paul Beverlin B.S., Rose-Hulman Institute of Technology, 2002 August 2006 Acknowledgments This dissertation would not be possible without several contributions. I would like to thank my thesis advisor Dr. James Madden for his many suggestions and his patience. I would also like to thank my other committee members Dr. Robert Perlis and Dr. Stephen Shipman for their help. I would like to thank my committee in the Experimental Statistics department for their understanding while I completed this project. And ¯nally I would like to thank my family for giving me the chance to achieve a higher education. -
Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-Space E3
Fundamentals of Contemporary Mathematical Sciences (2020) 1(2) 63 { 70 Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-space E3 Abdullah Yıldırım 1,∗ Feryat Kaya 2 1 Harran University, Faculty of Arts and Sciences, Department of Mathematics S¸anlıurfa, T¨urkiye 2 S¸ehit Abdulkadir O˘guzAnatolian Imam Hatip High School S¸anlıurfa, T¨urkiye, [email protected] Received: 29 February 2020 Accepted: 29 June 2020 Abstract: In this study, evolute-involute curves are researched. Characterization of evolute-involute curves lying on the surface are examined according to Darboux frame and some curves are obtained. Keywords: Curve, surface, geodesic, curvature, frame. 1. Introduction The interest of special curves has increased recently. Some of these are associated curves. They are curves where one of the Frenet vectors at opposite points is linearly dependent to the other curve. One of the best examples of these curves is the evolute-involute partner curves. An involute thought known to have been used in his optical work came up in 1658 by C. Huygens. C. Huygens discovered involute curves while trying to make more accurate measurement studies [5]. Many researches have been conducted about evolute-involute partner curves. Some of them conducted recently are Bilici and C¸alı¸skan [4], Ozyılmaz¨ and Yılmaz [9], As and Sarıo˘glugil[2]. Bekta¸sand Y¨uceconsider the notion of the involute-evolute curves lying on the surfaces for a special situation. They determine the special involute-evolute partner D−curves in E3: By using the Darboux frame of the curves they obtain the necessary and sufficient conditions between κg , ∗ − ∗ ∗ τg; κn and κn for a curve to be the special involute partner D curve. -
Ioan-Ioviţ Popescu Optics. I. Geometrical Optics
Edited by Translated into English by Lidia Vianu Bogdan Voiculescu Wednesday 31 May 2017 Online Publication Press Release Ioan-Ioviţ Popescu Optics. I. Geometrical Optics Translated into English by Bogdan Voiculescu ISBN 978-606-760-105-3 Edited by Lidia Vianu Ioan-Ioviţ Popescu’s Optica geometrică de Ioan-Ioviţ Geometrical Optics is essentially Popescu se ocupă de buna a book about light rays: about aproximaţie a luminii sub forma de what can be seen in our raze (traiectorii). Numele autorului universe. The name of the le este cunoscut fizicienilor din author is well-known to lumea întreagă. Ioviţ este acela care physicists all over the world. He a prezis existenţa Etheronului încă is the scientist who predicted din anul 1982. Editura noastră a the Etheron in the year 1982. publicat nu demult cartea acelei We have already published the descoperiri absolut unice: Ether and book of that unique discovery: Etherons. A Possible Reappraisal of the Ether and Etherons. A Possible Concept of Ether, Reappraisal of the Concept of http://editura.mttlc.ro/iovit Ether, z-etherons.html . http://editura.mttlc.ro/i Studenţii facultăţii de fizică ovitz-etherons.html . din Bucureşti ştiau în anul 1988— What students have to aşa cum ştiam şi eu de la autorul know about the book we are însuşi—că această carte a fost publishing now is that it was îndelung scrisă de mână de către handwritten by its author over autor cu scopul de a fi publicată în and over again, dozens of facsimil: ea a fost rescrisă de la capăt times—formulas, drawings and la fiecare nouă corectură, şi au fost everything. -
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. -
Coverrailway Curves Book.Cdr
RAILWAY CURVES March 2010 (Corrected & Reprinted : November 2018) INDIAN RAILWAYS INSTITUTE OF CIVIL ENGINEERING PUNE - 411 001 i ii Foreword to the corrected and updated version The book on Railway Curves was originally published in March 2010 by Shri V B Sood, the then professor, IRICEN and reprinted in September 2013. The book has been again now corrected and updated as per latest correction slips on various provisions of IRPWM and IRTMM by Shri V B Sood, Chief General Manager (Civil) IRSDC, Delhi, Shri R K Bajpai, Sr Professor, Track-2, and Shri Anil Choudhary, Sr Professor, Track, IRICEN. I hope that the book will be found useful by the field engineers involved in laying and maintenance of curves. Pune Ajay Goyal November 2018 Director IRICEN, Pune iii PREFACE In an attempt to reach out to all the railway engineers including supervisors, IRICEN has been endeavouring to bring out technical books and monograms. This book “Railway Curves” is an attempt in that direction. The earlier two books on this subject, viz. “Speed on Curves” and “Improving Running on Curves” were very well received and several editions of the same have been published. The “Railway Curves” compiles updated material of the above two publications and additional new topics on Setting out of Curves, Computer Program for Realignment of Curves, Curves with Obligatory Points and Turnouts on Curves, with several solved examples to make the book much more useful to the field and design engineer. It is hoped that all the P.way men will find this book a useful source of design, laying out, maintenance, upgradation of the railway curves and tackling various problems of general and specific nature. -
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. -
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.