Generalization of the Pedal Concept in Bidimensional Spaces. Application to the Limaçon of Pascal• Generalización Del Concep
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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. -
Orthocorrespondence and Orthopivotal Cubics
Forum Geometricorum b Volume 3 (2003) 1–27. bbb FORUM GEOM ISSN 1534-1178 Orthocorrespondence and Orthopivotal Cubics Bernard Gibert Abstract. We define and study a transformation in the triangle plane called the orthocorrespondence. This transformation leads to the consideration of a fam- ily of circular circumcubics containing the Neuberg cubic and several hitherto unknown ones. 1. The orthocorrespondence Let P be a point in the plane of triangle ABC with barycentric coordinates (u : v : w). The perpendicular lines at P to AP , BP, CP intersect BC, CA, AB respectively at Pa, Pb, Pc, which we call the orthotraces of P . These orthotraces 1 lie on a line LP , which we call the orthotransversal of P . We denote the trilinear ⊥ pole of LP by P , and call it the orthocorrespondent of P . A P P ∗ P ⊥ B C Pa Pc LP H/P Pb Figure 1. The orthotransversal and orthocorrespondent In barycentric coordinates, 2 ⊥ 2 P =(u(−uSA + vSB + wSC )+a vw : ··· : ···), (1) Publication Date: January 21, 2003. Communicating Editor: Paul Yiu. We sincerely thank Edward Brisse, Jean-Pierre Ehrmann, and Paul Yiu for their friendly and valuable helps. 1The homography on the pencil of lines through P which swaps a line and its perpendicular at P is an involution. According to a Desargues theorem, the points are collinear. 2All coordinates in this paper are homogeneous barycentric coordinates. Often for triangle cen- ters, we list only the first coordinate. The remaining two can be easily obtained by cyclically permut- ing a, b, c, and corresponding quantities. Thus, for example, in (1), the second and third coordinates 2 2 are v(−vSB + wSC + uSA)+b wu and w(−wSC + uSA + vSB )+c uv respectively. -
Automated Study of Isoptic Curves: a New Approach with Geogebra∗
Automated study of isoptic curves: a new approach with GeoGebra∗ Thierry Dana-Picard1 and Zoltan Kov´acs2 1 Jerusalem College of Technology, Israel [email protected] 2 Private University College of Education of the Diocese of Linz, Austria [email protected] March 7, 2018 Let C be a plane curve. For a given angle θ such that 0 ≤ θ ≤ 180◦, a θ-isoptic curve of C is the geometric locus of points in the plane through which pass a pair of tangents making an angle of θ.If θ = 90◦, the isoptic curve is called the orthoptic curve of C. For example, the orthoptic curves of conic sections are respectively the directrix of a parabola, the director circle of an ellipse, and the director circle of a hyperbola (in this case, its existence depends on the eccentricity of the hyperbola). Orthoptics and θ-isoptics can be studied for other curves than conics, in particular for closed smooth strictly convex curves,as in [1]. An example of the study of isoptics of a non-convex non-smooth curve, namely the isoptics of an astroid are studied in [2] and of Fermat curves in [3]. The Isoptics of an astroid present special properties: • There exist points through which pass 3 tangents to C, and two of them are perpendicular. • The isoptic curve are actually union of disjoint arcs. In Figure 1, the loops correspond to θ = 120◦ and the arcs connecting them correspond to θ = 60◦. Such a situation has been encountered already for isoptics of hyperbolas. These works combine geometrical experimentation with a Dynamical Geometry System (DGS) GeoGebra and algebraic computations with a Computer Algebra System (CAS). -
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. -
Elementary Euclidean Geometry an Introduction
Elementary Euclidean Geometry An Introduction C. G. GIBSON PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011–4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon´ 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org C Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2003 Printed in the United Kingdom at the University Press, Cambridge Typeface Times 10/13 pt. System LATEX2ε [TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data ISBN 0 521 83448 1 hardback Contents List of Figures page viii List of Tables x Preface xi Acknowledgements xvi 1 Points and Lines 1 1.1 The Vector Structure 1 1.2 Lines and Zero Sets 2 1.3 Uniqueness of Equations 3 1.4 Practical Techniques 4 1.5 Parametrized Lines 7 1.6 Pencils of Lines 9 2 The Euclidean Plane 12 2.1 The Scalar Product 12 2.2 Length and Distance 13 2.3 The Concept of Angle 15 2.4 Distance from a Point to a Line 18 3 Circles 22 3.1 Circles as Conics 22 3.2 General Circles 23 3.3 Uniqueness of Equations 24 3.4 Intersections with -
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). -
CATALAN's CURVE from a 1856 Paper by E. Catalan Part
CATALAN'S CURVE from a 1856 paper by E. Catalan Part - VI C. Masurel 13/04/2014 Abstract We use theorems on roulettes and Gregory's transformation to study Catalan's curve in the class C2(n; p) with angle V = π=2 − 2u and curves related to the circle and the catenary. Catalan's curve is defined in a 1856 paper of E. Catalan "Note sur la theorie des roulettes" and gives examples of couples of curves wheel and ground. 1 The class C2(n; p) and Catalan's curve The curves C2(n; p) shares many properties and the common angle V = π=2−2u (see part III). The polar parametric equations are: cos2n(u) ρ = θ = n tan(u) + 2:p:u cosn+p(2u) At the end of his 1856 paper Catalan looked for a curve such that rolling on a line the pole describe a circle tangent to the line. So by Steiner-Habich theorem this curve is the anti pedal of the wheel (see part I) associated to the circle when the base-line is the tangent. The equations of the wheel C2(1; −1) are : ρ = cos2(u) θ = tan(u) − 2:u then its anti-pedal (set p ! p + 1) is C2(1; 0) : cos2 u 1 ρ = θ = tan u −! ρ = cos 2u 1 − θ2 this is Catalan's curve (figure 1). 1 Figure 1: Catalan's curve : ρ = 1=(1 − θ2) 2 Curves such that the arc length s = ρ.θ with initial condition ρ = 1 when θ = 0 This relation implies by derivating : r dρ ds = ρdθ + dρθ = ρ2 + ( )2dθ dθ Squaring and simplifying give : 2ρθdθ dρ + (dρ)2θ2 = (dρ)2 So we have the evident solution dρ = 0 this the traditionnal circle : ρ = 1 But there is another one more exotic given by the other part of equation after we simplify by dρ 6= 0. -
Generating Negative Pedal Curve Through Inverse Function – an Overview *Ramesha
© 2017 JETIR March 2017, Volume 4, Issue 3 www.jetir.org (ISSN-2349-5162) Generating Negative pedal curve through Inverse function – An Overview *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur. Abstract This paper attempts to study the negative pedal of a curve with fixed point O is therefore the envelope of the lines perpendicular at the point M to the lines. In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k2. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the center of inversion, the circle the circle of inversion, and k the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself. is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. The inverse function of f is also denoted. -
Deltoid* the Deltoid Curve Was Conceived by Euler in 1745 in Con
Deltoid* The Deltoid curve was conceived by Euler in 1745 in con- nection with his study of caustics. Formulas in 3D-XplorMath: x = 2 cos(t) + cos(2t); y = 2 sin(t) − sin(2t); 0 < t ≤ 2π; and its implicit equation is: (x2 + y2)2 − 8x(x2 − 3y2) + 18(x2 + y2) − 27 = 0: The Deltoid or Tricuspid The Deltoid is also known as the Tricuspid, and can be defined as the trace of a point on one circle that rolls inside * This file is from the 3D-XplorMath project. Please see: http://3D-XplorMath.org/ 1 2 another circle of 3 or 3=2 times as large a radius. The latter is called double generation. The figure below shows both of these methods. O is the center of the fixed circle of radius a, C the center of the rolling circle of radius a=3, and P the tracing point. OHCJ, JPT and TAOGE are colinear, where G and A are distant a=3 from O, and A is the center of the rolling circle with radius 2a=3. PHG is colinear and gives the tangent at P. Triangles TEJ, TGP, and JHP are all similar and T P=JP = 2 . Angle JCP = 3∗Angle BOJ. Let the point Q (not shown) be the intersection of JE and the circle centered on C. Points Q, P are symmetric with respect to point C. The intersection of OQ, PJ forms the center of osculating circle at P. 3 The Deltoid has numerous interesting properties. Properties Tangent Let A be the center of the curve, B be one of the cusp points,and P be any point on the curve. -
The Cycloid Scott Morrison
The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on. -
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. -
PHD THESIS ISOPTIC CURVES and SURFACES Csima Géza Supervisor
PHD THESIS ISOPTIC CURVES AND SURFACES Csima Géza Supervisor: Szirmai Jenő associate professor BUTE Mathematical Institute, Department of Geometry BUTE 2017 Contents 0.1 Introduction . V 1 Isoptic curves on the Euclidean plane 1 1.1 Isoptics of closed strictly convex curves . 1 1.1.1 Example . 3 1.2 Isoptic curves of Euclidean conic sections . 4 1.2.1 Ellipse . 5 1.2.2 Hyperbola . 7 1.2.3 Parabola . 10 1.3 Isoptic curves to finite point sets . 12 1.4 Inverse problem . 15 2 Isoptic surfaces in the Euclidean space 18 2.1 Isoptic hypersurfaces in En ........................ 18 2.1.1 Isoptic hypersurface of (n−1)−dimensional compact hypersur- faces in En ............................. 19 2.1.2 Isoptic surface of rectangles . 20 2.1.3 Isoptic surface of ellipsoids . 23 2.2 Isoptic surfaces of polyhedra . 25 2.2.1 Computations of the isoptic surface of a given regular tetrahedron 27 2.2.2 Some isoptic surfaces to Platonic and Archimedean polyhedra 29 3 Isoptic curves of conics in non-Euclidean constant curvature geome- tries 31 3.1 Projective model . 31 3.1.1 Example: Isoptic curve of the line segment on the hyperbolic and elliptic plane . 32 3.1.2 The general method . 33 3.2 Elliptic conics and their isoptics . 34 3.2.1 Equation of the elliptic ellipse and hyperbola . 34 3.2.2 Equation of the elliptic parabola . 36 II 3.2.3 Isoptic curves of elliptic ellipse and hyperbola . 36 3.2.4 Isoptic curves of elliptic parabola . 37 3.3 Hyperbolic conic sections and their isoptics .