Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light
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The Osculating Circumference Problem
THE OSCULATING CIRCUMFERENCE PROBLEM School I.S.I.S.S. M. Casagrande Pieve di Soligo, Treviso Italy Students Silvia Micheletto (I) Azzurra Soa Pizzolotto (IV) Luca Barisan (II) Simona Sota (IV) Gaia Barella (IV) Paolo Barisan (V) Marco Casagrande (IV) Anna De Biasi (V) Chiara De Rosso (IV) Silvia Giovani (V) Maddalena Favaro (IV) Klara Metaliu (V) Teachers Fabio Breda Francesco Maria Cardano Davide Palma Francesco Zampieri Researcher Alberto Zanardo, University of Padova Italy Year 2019/2020 The osculating circumference problem ISISS M. Casagrande, Pieve di Soligo, Treviso Italy Abstract The aim of the article is to study the osculating circumference i.e. the circumference which best approximates the graph of a curve at one of its points. We will dene this circumference and we will describe several methods to nd it. Finally we will introduce the notions of round points and crossing points, points of the curve where the circumference has interesting properties. ??? Contents Introduction 3 1 The osculating circumference 5 1.1 Particular cases: the formulas cannot be applied . 9 1.2 Curvature . 14 2 Round points 15 3 Beyond round points 22 3.1 Crossing points . 23 4 Osculating circumference of a conic section 25 4.1 The rst method . 25 4.2 The second method . 26 References 31 2 The osculating circumference problem ISISS M. Casagrande, Pieve di Soligo, Treviso Italy Introduction Our research was born from an analytic geometry problem studied during the third year of high school. The problem. Let P : y = x2 be a parabola and A and B two points of the curve symmetrical about the axis of the parabola. -
Plotting Graphs of Parametric Equations
Parametric Curve Plotter This Java Applet plots the graph of various primary curves described by parametric equations, and the graphs of some of the auxiliary curves associ- ated with the primary curves, such as the evolute, involute, parallel, pedal, and reciprocal curves. It has been tested on Netscape 3.0, 4.04, and 4.05, Internet Explorer 4.04, and on the Hot Java browser. It has not yet been run through the official Java compilers from Sun, but this is imminent. It seems to work best on Netscape 4.04. There are problems with Netscape 4.05 and it should be avoided until fixed. The applet was developed on a 200 MHz Pentium MMX system at 1024×768 resolution. (Cost: $4000 in March, 1997. Replacement cost: $695 in June, 1998, if you can find one this slow on sale!) For various reasons, it runs extremely slowly on Macintoshes, and slowly on Sun Sparc stations. The curves should move as quickly as Roadrunner cartoons: for a benchmark, check out the circular trigonometric diagrams on my Java page. This applet was mainly done as an exercise in Java programming leading up to more serious interactive animations in three-dimensional graphics, so no attempt was made to be comprehensive. Those who wish to see much more comprehensive sets of curves should look at: http : ==www − groups:dcs:st − and:ac:uk= ∼ history=Java= http : ==www:best:com= ∼ xah=SpecialP laneCurve dir=specialP laneCurves:html http : ==www:astro:virginia:edu= ∼ eww6n=math=math0:html Disclaimer: Like all computer graphics systems used to illustrate mathematical con- cepts, it cannot be error-free. -
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. -
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). -
Gottfried Wilhelm Leibnitz (Or Leibniz) Was Born at Leipzig on June 21 (O.S.), 1646, and Died in Hanover on November 14, 1716. H
Gottfried Wilhelm Leibnitz (or Leibniz) was born at Leipzig on June 21 (O.S.), 1646, and died in Hanover on November 14, 1716. His father died before he was six, and the teaching at the school to which he was then sent was inefficient, but his industry triumphed over all difficulties; by the time he was twelve he had taught himself to read Latin easily, and had begun Greek; and before he was twenty he had mastered the ordinary text-books on mathematics, philosophy, theology and law. Refused the degree of doctor of laws at Leipzig by those who were jealous of his youth and learning, he moved to Nuremberg. An essay which there wrote on the study of law was dedicated to the Elector of Mainz, and led to his appointment by the elector on a commission for the revision of some statutes, from which he was subsequently promoted to the diplomatic service. In the latter capacity he supported (unsuccessfully) the claims of the German candidate for the crown of Poland. The violent seizure of various small places in Alsace in 1670 excited universal alarm in Germany as to the designs of Louis XIV.; and Leibnitz drew up a scheme by which it was proposed to offer German co-operation, if France liked to take Egypt, and use the possessions of that country as a basis for attack against Holland in Asia, provided France would agree to leave Germany undisturbed. This bears a curious resemblance to the similar plan by which Napoleon I. proposed to attack England. In 1672 Leibnitz went to Paris on the invitation of the French government to explain the details of the scheme, but nothing came of it. -
Geometry in the Age of Enlightenment
Geometry in the Age of Enlightenment Raymond O. Wells, Jr. ∗ July 2, 2015 Contents 1 Introduction 1 2 Algebraic Geometry 3 2.1 Algebraic Curves of Degree Two: Descartes and Fermat . 5 2.2 Algebraic Curves of Degree Three: Newton and Euler . 11 3 Differential Geometry 13 3.1 Curvature of curves in the plane . 17 3.2 Curvature of curves in space . 26 3.3 Curvature of a surface in space: Euler in 1767 . 28 4 Conclusion 30 1 Introduction The Age of Enlightenment is a term that refers to a time of dramatic changes in western society in the arts, in science, in political thinking, and, in particular, in philosophical discourse. It is generally recognized as being the period from the mid 17th century to the latter part of the 18th century. It was a successor to the renaissance and reformation periods and was followed by what is termed the romanticism of the 19th century. In his book A History of Western Philosophy [25] Bertrand Russell (1872{1970) gives a very lucid description of this time arXiv:1507.00060v1 [math.HO] 30 Jun 2015 period in intellectual history, especially in Book III, Chapter VI{Chapter XVII. He singles out Ren´eDescartes as being the founder of the era of new philosophy in 1637 and continues to describe other philosophers who also made significant contributions to mathematics as well, such as Newton and Leibniz. This time of intellectual fervor included literature (e.g. Voltaire), music and the world of visual arts as well. One of the most significant developments was perhaps in the political world: here the absolutism of the church and of the monarchies ∗Jacobs University Bremen; University of Colorado at Boulder; [email protected] 1 were questioned by the political philosophers of this era, ushering in the Glo- rious Revolution in England (1689), the American Revolution (1776), and the bloody French Revolution (1789). -
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). -
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. -
Differential Geometry of Curves and Surfaces 1
DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 1. Curves in the Plane 1.1. Points, Vectors, and Their Coordinates. Points and vectors are fundamental objects in Geometry. The notion of point is intuitive and clear to everyone. The notion of vector is a bit more delicate. In fact, rather than saying what a vector is, we prefer to say what a vector has, namely: direction, sense, and length (or magnitude). It can be represented by an arrow, and the main idea is that two arrows represent the same vector if they have the same direction, sense, and length. An arrow representing a vector has a tail and a tip. From the (rough) definition above, we deduce that in order to represent (if you want, to draw) a given vector as an arrow, it is necessary and sufficient to prescribe its tail. a c b b a c a a b P Figure 1. We see four copies of the vector a, three of the vector b, and two of the vector c. We also see a point P . An important instrument in handling points, vectors, and (consequently) many other geometric objects is the Cartesian coordinate system in the plane. This consists of a point O, called the origin, and two perpendicular lines going through O, called coordinate axes. Each line has a positive direction, indicated by an arrow (see Figure 2). We denote by R the a P y a y x O O x Figure 2. The point P has coordinates x, y. The vector a has also coordinates x, y. -
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. -
Safe Domain and Elementary Geometry
Safe domain and elementary geometry Jean-Marc Richard Laboratoire de Physique Subatomique et Cosmologie, Universit´eJoseph Fourier– CNRS-IN2P3, 53, avenue des Martyrs, 38026 Grenoble cedex, France November 8, 2018 Abstract A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a “safe” domain. In the simple cases of a constant force, harmonic potential, and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic section which can be derived either by simple calculus or by geometrical considerations. The case of harmonic forces reveals a subtle property of the maximal sum of distances within an ellipse. 1 Introduction A classical problem of classroom mechanics and military academies is the border of the so-called safe domain. A projectile is set off from a point O with an initial velocity v0 whose modulus is fixed by the intrinsic properties of the gun, while its direction can be varied arbitrarily. Its is well-known that, in absence of air friction, each trajectory is a parabola, and that, in any vertical plane containing O, the envelope of all trajectories is another parabola, which separates the points which can be shot from those which are out of reach. This will be shortly reviewed in Sec. 2. Amazingly, the problem of this envelope parabola can be addressed, and solved, in terms of elementary geometrical properties. arXiv:physics/0410034v1 [physics.ed-ph] 6 Oct 2004 In elementary mechanics, there are similar problems, that can be solved explicitly and lead to families of conic trajectories whose envelope is also a conic section. -
On Caustics by Reflexion
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology. Vol. 21. No. 2. pp. 179-199, 1982 W&9383/82/020179-21$03.00/O Printed in Great Britain. Pergamon Press Ltd. ON CAUSTICS BY REFLEXION J. W. BRUCE,t P. J. GIBLIN and C. G. GIBSON (Received 21 May 1980) ONE OF THE fundamental contributions of singularity theory to the physical sciences is the result of Looijenga, discussed in the opening section of this paper, that a generic wavefront gives rise to a caustic whose local structure can be described by a finite number of archetypes. However, when caustics arise in explicit physical situations, there is no reason a priori to suppose that a generic state of the system will necessarily give rise to a generic caustic. In this paper we make a detailed study of the system comprising a mirror M (a smooth hypersurface in R”+‘) and a point source of light L (a point in R”+‘): light rays emanating from L are reflected at M and give rise to a caustic by reflexion, the envelope of the family of reflected rays. And the problem to which we address ourselves is whether a generic state of our system does give rise to a generic caustic. Note that we consider only first reflexions of light from M. Perhaps the most natural question to look at here is that of mirror genericity, i.e. for a fixed position of the source L will a generic mirror M give rise to a generic caustic? We give an affirmative answer to this question, using a classical geometric construction, under the hypotheses that the mirror M is compact and has everywhere positive Gaussian curvature, and the source L lies inside M.