New Tools in Geogebra Offering Novel Opportunities to Teach Loci and Envelopes
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Section 1.7B the Four-Vertex Theorem
Section 1.7B The Four-Vertex Theorem Matthew Hendrix Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 A vertex of a regular plane curve α :[a; b] ! R is a point t 2 [a; b] where k0(t) = 0. -
Evolutes of Curves in the Lorentz-Minkowski Plane
Evolutes of curves in the Lorentz-Minkowski plane 著者 Izumiya Shuichi, Fuster M. C. Romero, Takahashi Masatomo journal or Advanced Studies in Pure Mathematics publication title volume 78 page range 313-330 year 2018-10-04 URL http://hdl.handle.net/10258/00009702 doi: info:doi/10.2969/aspm/07810313 Evolutes of curves in the Lorentz-Minkowski plane S. Izumiya, M. C. Romero Fuster, M. Takahashi Abstract. We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz- Minkowski plane. This leads naturally to a well defined evolute asso- ciated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz- Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points. x1. Introduction The evolute of a regular plane curve is a classical subject of differen- tial geometry on Euclidean plane which is defined to be the locus of the centres of the osculating circles of the curve (cf. [3, 7, 8]). -
The Illinois Mathematics Teacher
THE ILLINOIS MATHEMATICS TEACHER EDITORS Marilyn Hasty and Tammy Voepel Southern Illinois University Edwardsville Edwardsville, IL 62026 REVIEWERS Kris Adams Chip Day Karen Meyer Jean Smith Edna Bazik Lesley Ebel Jackie Murawska Clare Staudacher Susan Beal Dianna Galante Carol Nenne Joe Stickles, Jr. William Beggs Linda Gilmore Jacqueline Palmquist Mary Thomas Carol Benson Linda Hankey James Pelech Bob Urbain Patty Bruzek Pat Herrington Randy Pippen Darlene Whitkanack Dane Camp Alan Holverson Sue Pippen Sue Younker Bill Carroll John Johnson Anne Marie Sherry Mike Carton Robin Levine-Wissing Aurelia Skiba ICTM GOVERNING BOARD 2012 Don Porzio, President Rich Wyllie, Treasurer Fern Tribbey, Past President Natalie Jakucyn, Board Chair Lannette Jennings, Secretary Ann Hanson, Executive Director Directors: Mary Modene, Catherine Moushon, Early Childhood (2009-2012) Com. College/Univ. (2010-2013) Polly Hill, Marshall Lassak, Early Childhood (2011-2014) Com. College/Univ. (2011-2014) Cathy Kaduk, George Reese, 5-8 (2010-2013) Director-At-Large (2009-2012) Anita Reid, Natalie Jakucyn, 5-8 (2011-2014) Director-At-Large (2010-2013) John Benson, Gwen Zimmermann, 9-12 (2009-2012) Director-At-Large (2011-2014) Jerrine Roderique, 9-12 (2010-2013) The Illinois Mathematics Teacher is devoted to teachers of mathematics at ALL levels. The contents of The Illinois Mathematics Teacher may be quoted or reprinted with formal acknowledgement of the source and a letter to the editor that a quote has been used. The activity pages in this issue may be reprinted for classroom use without written permission from The Illinois Mathematics Teacher. (Note: Occasionally copyrighted material is used with permission. Such material is clearly marked and may not be reprinted.) THE ILLINOIS MATHEMATICS TEACHER Volume 61, No. -
The Implementation of Ict on Various Curves Through
THE IMPLEMENTATION OF ICT ON VARIOUS CURVES THROUGH WINPLOT J.Sengamala Selvi,1,* Dr.T.Venugopal 2 1Asst.Professor, Dept.of Mathematics, SCSVMV University, Enathur, Tamilnadu, Kanchipuram, India. 2Professor of Mathematics, Director, Research and Publications, SCSVMV University, Enathur, Kanchipuram, Tamilnadu, India. E-mail: [email protected]. ABSTRACT: ml This paper aims to take a step further into the 2. Click on the link: Winplot for Windows development and innovation of teaching, 95/98/ME/2K/XP (558K) using contemporary instruments, which 3. When prompted, click save to save the develops the necessary skills of the student file to the desktop. community. To introduce the Win plot 4. Double click the icon wp32z.exe on the software in making teaching mathematical desktop. problems into an easier one so as to cheer up 5. WinZip will prompt to extract the files to the students to develop their application C:\peanut. Click “Unzip”. oriented skills. This paper deals with the hope 6. Win Plot is now installed in C:\peanut that the mathematics classroom will soon be directory. converted into modern classroom where software oriented learning will take place in the near future. KEYWORDS: Higher education Mathematics, ICT, Mathematics pedagogy.Win plot software 1. Introduction: ICT is one of the factors to shaping and changing the world rapidly.ICT has also brought a paradigm shift in education. It touches every aspect of Education.ICT has brought a new networks of teachers and learners connecting them throughout the globe.ICT is a means of accessing, processing, sharing, editing, selecting, B.BASICS OF WINPLOT presenting, communicating through the The first time Win Plot is used, the screen information in a variety of media. -
Strophoids, a Family of Cubic Curves with Remarkable Properties
Hellmuth STACHEL STROPHOIDS, A FAMILY OF CUBIC CURVES WITH REMARKABLE PROPERTIES Abstract: Strophoids are circular cubic curves which have a node with orthogonal tangents. These rational curves are characterized by a series or properties, and they show up as locus of points at various geometric problems in the Euclidean plane: Strophoids are pedal curves of parabolas if the corresponding pole lies on the parabola’s directrix, and they are inverse to equilateral hyperbolas. Strophoids are focal curves of particular pencils of conics. Moreover, the locus of points where tangents through a given point contact the conics of a confocal family is a strophoid. In descriptive geometry, strophoids appear as perspective views of particular curves of intersection, e.g., of Viviani’s curve. Bricard’s flexible octahedra of type 3 admit two flat poses; and here, after fixing two opposite vertices, strophoids are the locus for the four remaining vertices. In plane kinematics they are the circle-point curves, i.e., the locus of points whose trajectories have instantaneously a stationary curvature. Moreover, they are projections of the spherical and hyperbolic analogues. For any given triangle ABC, the equicevian cubics are strophoids, i.e., the locus of points for which two of the three cevians have the same lengths. On each strophoid there is a symmetric relation of points, so-called ‘associated’ points, with a series of properties: The lines connecting associated points P and P’ are tangent of the negative pedal curve. Tangents at associated points intersect at a point which again lies on the cubic. For all pairs (P, P’) of associated points, the midpoints lie on a line through the node N. -
An Investigation of the Four Vertex Theorem and Its Converse Rebeka Kelmar Union College - Schenectady, NY
Union College Union | Digital Works Honors Theses Student Work 6-2017 An Investigation of the Four Vertex Theorem and its Converse Rebeka Kelmar Union College - Schenectady, NY Follow this and additional works at: https://digitalworks.union.edu/theses Part of the Geometry and Topology Commons Recommended Citation Kelmar, Rebeka, "An Investigation of the Four Vertex Theorem and its Converse" (2017). Honors Theses. 52. https://digitalworks.union.edu/theses/52 This Open Access is brought to you for free and open access by the Student Work at Union | Digital Works. It has been accepted for inclusion in Honors Theses by an authorized administrator of Union | Digital Works. For more information, please contact [email protected]. An Investigation of the Four Vertex Theorem and its Converse By Rebeka Kelmar *************** Submitted in partial fulfillment of the requirements for Honors in the Department of Mathematics UNION COLLEGE March, 2017 Abstract In the study of curves there are many interesting theorems. One such theorem is the four vertex theorem and its converse. The four vertex theorem says that any simple closed curve, other than a circle, must have four vertices. This means that the curvature of the curve must have at least four local maxima/minima. In my project I explore different proofs of the four vertex theorem and its history. I also look at a modified converse of the four vertex theorem which says that any continuous real- valued function on the circle that has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. -
A Special Conic Associated with the Reuleaux Negative Pedal Curve
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 10 (2021), No. 2, 33–49 A SPECIAL CONIC ASSOCIATED WITH THE REULEAUX NEGATIVE PEDAL CURVE LILIANA GABRIELA GHEORGHE and DAN REZNIK Abstract. The Negative Pedal Curve of the Reuleaux Triangle w.r. to a pedal point M located on its boundary consists of two elliptic arcs and a point P0. Intriguingly, the conic passing through the four arc endpoints and by P0 has one focus at M. We provide a synthetic proof for this fact using Poncelet’s porism, polar duality and inversive techniques. Additional interesting properties of the Reuleaux negative pedal w.r. to pedal point M are also included. 1. Introduction Figure 1. The sides of the Reuleaux Triangle R are three circular arcs of circles centered at each Reuleaux vertex Vi; i = 1; 2; 3. of an equilateral triangle. The Reuleaux triangle R is the convex curve formed by the arcs of three circles of equal radii r centered on the vertices V1;V2;V3 of an equilateral triangle and that mutually intercepts in these vertices; see Figure1. This triangle is mostly known due to its constant width property [4]. Keywords and phrases: conic, inversion, pole, polar, dual curve, negative pedal curve. (2020)Mathematics Subject Classification: 51M04,51M15, 51A05. Received: 19.08.2020. In revised form: 26.01.2021. Accepted: 08.10.2020 34 Liliana Gabriela Gheorghe and Dan Reznik Figure 2. The negative pedal curve N of the Reuleaux Triangle R w.r. to a point M on its boundary consist on an point P0 (the antipedal of M through V3) and two elliptic arcs A1A2 and B1B2 (green and blue). -
Final Projects 1 Envelopes, Evolutes, and Euclidean Distance Degree
Final Projects Directions: Work together in groups of 1-5 on the following four projects. We suggest, but do not insist, that you work in a group of > 1 people. For projects 1,2, and 3, your group should write a text le with (working!) code which answers the questions. Please provide documentation for your code so that we can understand what the code is doing. For project 4, your group should turn in a summary of observations and/or conjectures, but not necessarily code. Send your les via e-mail to [email protected] and [email protected]. It is preferable that the work is turned in by Thursday night. However, at the very latest, please send your work by 12:00 p.m. (noon) on Friday, August 16. 1 Envelopes, Evolutes, and Euclidean Distance Degree Read the section in Cox-Little-O'Shea's Ideals, Varieties, and Algorithms on envelopes of families of curves - this is in Chapter 3, section 4. We will expand slightly on the setup in Chapter 3 as follows. A rational family of lines is dened as a polynomial of the form Lt(x; y) 2 Q(t)[x; y] 1 L (x; y) = (A(t)x + B(t)y + C(t)); t G(t) where A(t);B(t);C(t); and G(t) are polynomials in t. The envelope of Lt(x; y) is the locus of and so that there is some satisfying and @Lt(x;y) . As usual, x y t Lt(x; y) = 0 @t = 0 we can handle this by introducing a new variable s which plays the role of 1=G(t). -
Moon in a Puddle and the Four-Vertex Theorem
Moon in a puddle and the four-vertex theorem Anton Petrunin and Sergio Zamora Barrera with artwork by Ana Cristina Chavez´ Caliz´ Abstract. We present a proof of the moon in a puddle theorem, and use its key lemma to prove a generalization of the four-vertex theorem. arXiv:2107.08455v1 [math.DG] 18 Jul 2021 INTRODUCTION. The theorem about the Moon in a puddle provides the simplest meaningful example of a local-to-global theorem which is mainly what differential geometry is about. Yet, the theorem is surprisingly not well-known. This paper aims to redress this omission by calling attention to the result and applying it to a well-known theorem MOON IN A PUDDLE. The following question was initially asked by Abram Fet and solved by Vladimir Ionin and German Pestov [10]. Theorem 1. Assume γ is a simple closed smooth regular plane curve with curvature bounded in absolute value by 1. Then the region surrounded by γ contains a unit disc. We present the proof from our textbook [12] which is a slight improvement of the original proof. Both proofs work under the weaker assumption that the signed curva- ture is at most one, assuming that the sign is chosen suitably. A more general statement for a barrier-type bound on the curvature was given by Anders Aamand, Mikkel Abra- hamsen, and Mikkel Thorup [1]. There are other proofs. One is based on the curve- shortening flow; it is given by Konstantin Pankrashkin [8]. Another one uses cut locus; it is sketched by Victor Toponogov [13, Problem 1.7.19]; see also [11]. -
Unified Higher Order Path and Envelope Curvature Theories in Kinematics
UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS By MALLADI NARASH1HA SIDDHANTY I; Bachelor of Engineering Osmania University Hyderabad, India 1965 Master of Technology Indian Institute of Technology Madras, India 1969 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1979 UNIFIED HIGHER ORDER PATH AND ENVELOPE CURVATURE THEORIES IN KINEMATICS Thesis Approved: Thesis Adviser J ~~(}W>p~ Dean of the Graduate College To Mukunda, Ananta, and the Mothers ACKNOWLEDGMENTS I take this opportunity to express my deep sense of gratitude and thanks to Professor A. H. Soni, chairman of my doctoral committee, for his keen interest in introducing me to various aspects of research in kinematics and his continued guidance with proper financial support. I thank the "Soni Family" for many warm dinners and hospitality. I am thankful to Professors L. J. Fila, P. F. Duvall, J. P. Chandler, and T. E. Blejwas for serving on my committee and for their advice and encouragement. Thanks are also due to Professor F. Freudenstein of Columbia University for serving on my committee and for many encouraging discussions on the subject. I thank Magnetic Peripherals, Inc., Oklahoma City, Oklahoma, my em ployer since the fall of 1977, for the computer facilities and the tui tion fee reimbursement. I thank all my fellow graduate students and friends at Stillwater and Oklahoma City for their warm friendship and discussions that made my studies in Oklahoma really rewarding. I remember and thank my former professors, B. -
FOUR VERTEX THEOREM and ITS CONVERSE Dennis Deturck, Herman Gluck, Daniel Pomerleano, David Shea Vick
THE FOUR VERTEX THEOREM AND ITS CONVERSE Dennis DeTurck, Herman Gluck, Daniel Pomerleano, David Shea Vick Dedicated to the memory of Björn Dahlberg Abstract The Four Vertex Theorem, one of the earliest results in global differential geometry, says that a simple closed curve in the plane, other than a circle, must have at least four "vertices", that is, at least four points where the curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya proved this for strictly convex curves in the plane, and in 1912 Adolf Kneser proved it for all simple closed curves in the plane, not just the strictly convex ones. The Converse to the Four Vertex Theorem says that any continuous real- valued function on the circle which has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. In 1971 Herman Gluck proved this for strictly positive preassigned curvature, and in 1997 Björn Dahlberg proved the full converse, without the restriction that the curvature be strictly positive. Publication was delayed by Dahlberg's untimely death in January 1998, but his paper was edited afterwards by Vilhelm Adolfsson and Peter Kumlin, and finally appeared in 2005. The work of Dahlberg completes the almost hundred-year-long thread of ideas begun by Mukhopadhyaya, and we take this opportunity to provide a self-contained exposition. August, 2006 1 I. Why is the Four Vertex Theorem true? 1. A simple construction. A counter-example would be a simple closed curve in the plane whose curvature is nonconstant, has one minimum and one maximum, and is weakly monotonic on the two arcs between them. -
Generating the Envelope of a Swept Trivariate Solid
GENERATING THE ENVELOPE OF A SWEPT TRIVARIATE SOLID Claudia Madrigal1 Center for Image Processing and Integrated Computing, and Department of Mathematics University of California, Davis Kenneth I. Joy2 Center for Image Processing and Integrated Computing Department of Computer Science University of California, Davis Abstract We present a method for calculating the envelope of the swept surface of a solid along a path in three-dimensional space. The generator of the swept surface is a trivariate tensor- product Bezier` solid and the path is a non-uniform rational B-spline curve. The boundary surface of the solid is the combination of parametric surfaces and an implicit surface where the determinant of the Jacobian of the defining function is zero. We define methods to calculate the envelope for each type of boundary surface, defining characteristic curves on the envelope and connecting them with triangle strips. The envelope of the swept solid is then calculated by taking the union of the envelopes of the family of boundary surfaces, defined by the surface of the solid in motion along the path. Keywords: trivariate splines; sweeping; envelopes; solid modeling. 1Department of Mathematics University of California, Davis, CA 95616-8562, USA; e-mail: madri- [email protected] 2Corresponding Author, Department of Computer Science, University of California, Davis, CA 95616-8562, USA; e-mail: [email protected] 1 1. Introduction Geometric modeling systems construct complex models from objects based on simple geo- metric primitives. As the needs of these systems become more complex, it is necessary to expand the inventory of geometric primitives and design operations.