Spherical and Hyperbolic Conics
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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. -
Stereographic Projection Patch Kessler, December 12, 2018
Stereographic Projection Patch Kessler, December 12, 2018 Stereographic projection is a way to flatten out the surface of a ball, such as the celestial sphere, which surrounds the earth and contains all the stars in the sky. The celestial sphere gives order out of chaos. It gives a way to predict star locations at different times, as well as the time of night from the stars. Here is a physical model of the celestial sphere from around 1480. 1480-1481, http://www.artfund.org/artwork/3600/spherical-astrolabe 1 According to legend, Ptolemy (AD 90 - AD 168) thought of stereographic projection after a donkey stepped on his model of the celestial sphere, making it flat. Applying stereographic projection to the celestial sphere results in a planar device called an astrolabe. 16th century islamic astrolabe The tips on the perforated upper plate correspond to stars, while points on the lower plate correspond to viewing directions. For instance, the point on the lower plate that the circles are shrinking towards corresponds to the viewing direction directly overhead (i.e., looking straight up). The stereographic projection of a point X on a sphere S is obtained by drawing a line from the north pole of the sphere through X, and continuing on to the plane G that the sphere is resting on. The projection of X is the point f(X) where this line hits the plane. E X S G f(X) 2 One of the most surprising things about stereographic projection is that circles get mapped to circles. P circle on the sphere E S circle in the plane G Circles that pass through the north pole get mapped to lines, however if you think of a line as an infinite radius circle, then you can say that circles get mapped to circles in all cases. -
Generalization of the Pedal Concept in Bidimensional Spaces. Application to the Limaçon of Pascal• Generalización Del Concep
Generalization of the pedal concept in bidimensional spaces. Application to the limaçon of Pascal• Irene Sánchez-Ramos a, Fernando Meseguer-Garrido a, José Juan Aliaga-Maraver a & Javier Francisco Raposo-Grau b a Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid, España. [email protected], [email protected], [email protected] b Escuela Técnica Superior de Arquitectura, Universidad Politécnica de Madrid, Madrid, España. [email protected] Received: June 22nd, 2020. Received in revised form: February 4th, 2021. Accepted: February 19th, 2021. Abstract The concept of a pedal curve is used in geometry as a generation method for a multitude of curves. The definition of a pedal curve is linked to the concept of minimal distance. However, an interesting distinction can be made for . In this space, the pedal curve of another curve C is defined as the locus of the foot of the perpendicular from the pedal point P to the tangent2 to the curve. This allows the generalization of the definition of the pedal curve for any given angle that is not 90º. ℝ In this paper, we use the generalization of the pedal curve to describe a different method to generate a limaçon of Pascal, which can be seen as a singular case of the locus generation method and is not well described in the literature. Some additional properties that can be deduced from these definitions are also described. Keywords: geometry; pedal curve; distance; angularity; limaçon of Pascal. Generalización del concepto de curva podal en espacios bidimensionales. Aplicación a la Limaçon de Pascal Resumen El concepto de curva podal está extendido en la geometría como un método generativo para multitud de curvas. -
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). -
Geometric Properties of Central Catadioptric Line Images
Geometric Properties of Central Catadioptric Line Images Jo˜aoP. Barreto and Helder Araujo Institute of Systems and Robotics Dept. of Electrical and Computer Engineering University of Coimbra Coimbra, Portugal {jpbar, helder}@isr.uc.pt http://www.isr.uc.pt/˜jpbar Abstract. It is highly desirable that an imaging system has a single effective viewpoint. Central catadioptric systems are imaging systems that use mirrors to enhance the field of view while keeping a unique center of projection. A general model for central catadioptric image formation has already been established. The present paper exploits this model to study the catadioptric projection of lines. The equations and geometric properties of general catadioptric line imaging are derived. We show that it is possible to determine the position of both the effec- tive viewpoint and the absolute conic in the catadioptric image plane from the images of three lines. It is also proved that it is possible to identify the type of catadioptric system and the position of the line at infinity without further infor- mation. A methodology for central catadioptric system calibration is proposed. Reconstruction aspects are discussed. Experimental results are presented. All the results presented are original and completely new. 1 Introduction Many applications in computer vision, such as surveillance and model acquisition for virtual reality, require that a large field of view is imaged. Visual control of motion can also benefit from enhanced fields of view [1,2,4]. One effective way to enhance the field of view of a camera is to use mirrors [5,6,7,8,9]. The general approach of combining mirrors with conventional imaging systems is referred to as catadioptric image formation [3]. -
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 -
Read to the Infinite and Back Again
TO THE INFINITE AND BACK AGAIN Part I Henrike Holdrege A Workbook in Projective Geometry To the Infinite and Back Again A Workbook in Projective Geometry Part I HENRIKE HOLDREGE EVOLVING SCIENCE ASSOCIATION ————————————————— A collaboration of The Nature Institute and The Myrin Institute 2019 Copyright 2019 Henrike Holdrege All rights reserved. Second printing, 2019 Published by the Evolving Science Association, a collaboration of The Nature Institute and The Myrin Institute The Nature Institute: 20 May Hill Road, Ghent, NY 12075 natureinstitute.org The Myrin Institute: 187 Main Street, Great Barrington, MA 01230 myrin.org ISBN 978-0-9744906-5-6 Cover painting by Angelo Andes, Desargues 10 Lines 10 Points This book, and other Nature Institute publications, can be purchased from The Nature Institute’s online bookstore: natureinstitute.org/store or contact The Nature Institute: [email protected] or 518-672-0116. Table of Contents INTRODUCTION 1 PREPARATIONS 4 TEN BASIC ENTITIES 6 PRELUDE Form and Forming 7 CHAPTER 1 The Harmonic Net and the Harmonic Four Points 11 INTERLUDE The Infinitely Distant Point of a Line 23 CHAPTER 2 The Theorem of Pappus 27 INTERLUDE A Triangle Transformation 36 CHAPTER 3 Sections of the Point Field 38 INTERLUDE The Projective versus the Euclidean Point Field 47 CHAPTER 4 The Theorem of Desargues 49 INTERLUDE The Line at Infinity 61 CHAPTER 5 Desargues’ Theorem in Three-dimensional Space 64 CHAPTER 6 Shadows, Projections, and Linear Perspective 78 CHAPTER 7 Homologies 89 INTERLUDE The Plane at Infinity 96 CLOSING 99 ACKNOWLEDGMENTS 101 BIBLIOGRAPHY 103 I n t r o d u c t i o n My intent in this book is to allow you, the reader, to engage with the fundamental concepts of projective geometry. -
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. -
A Gallery of Conics by Five Elements
Forum Geometricorum Volume 14 (2014) 295–348. FORUM GEOM ISSN 1534-1178 A Gallery of Conics by Five Elements Paris Pamfilos Abstract. This paper is a review on conics defined by five elements, i.e., either lines to which the conic is tangent or points through which the conic passes. The gallery contains all cases combining a number (n) of points and a number (5 − n) of lines and also allowing coincidences of some points with some lines. Points and/or lines at infinity are handled as particular cases. 1. Introduction In the following we review the construction of conics by five elements: points and lines, briefly denoted by (αP βT ), with α + β =5. In these it is required to construct a conic passing through α given points and tangent to β given lines. The six constructions, resulting by giving α, β the values 0 to 5 and considering the data to be in general position, are considered the most important ([18, p. 387]), and are to be found almost on every book about conics. It seems that constructions for which some coincidences are allowed have attracted less attention, though they are related to many interesting theorems of the geometry of conics. Adding to the six main cases those with the projectively different possible coincidences we land to 12 main constructions figuring on the first column of the classifying table below. The six added cases can be considered as limiting cases of the others, in which a point tends to coincide with another or with a line. The twelve main cases are the projectively inequivalent to which every other case can be reduced by means of a projectivity of the plane. -
Pencils of Conics
Chaos\ Solitons + Fractals Vol[ 8\ No[ 6\ pp[ 0960Ð0975\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Pergamon Printed in Great Britain \ 9859Ð9668:87 ,08[99 ¦ 9[99 PII] S9859!9668"86#99978!0 Pencils of Conics] a Means Towards a Deeper Understanding of the Arrow of Time< METOD SANIGA Astronomical Institute\ Slovak Academy of Sciences\ SK!948 59 Tatranska Lomnica\ The Slovak Republic "Accepted 7 April 0886# Abstract*The paper aims at giving a su.ciently complex description of the theory of pencil!generated temporal dimensions in a projective plane over reals[ The exposition starts with a succinct outline of the mathematical formalism and goes on with introducing the de_nitions of pencil!time and pencil!space\ both at the abstract "projective# and concrete "a.ne# levels[ The structural properties of all possible types of temporal arrows are analyzed and\ based on symmetry principles\ the uniqueness of that mimicking best the reality is justi_ed[ A profound connection between the character of di}erent {ordinary| arrows and the number of spatial dimensions is revealed[ Þ 0887 Elsevier Science Ltd[ All rights reserved 0[ INTRODUCTION The depth of our penetration into the problem is measured not by the state of things on the frontier\ but rather how far away the frontier lies[ One of the most pronounced examples of such a situation concerns\ beyond any doubt\ the nature of time[ What we have precisely in mind here is a {{puzzling discrepancy between our perception of time and what modern physical theory tells us to believe|| ð0Ł\ see -
THE HESSE PENCIL of PLANE CUBIC CURVES 1. Introduction In
THE HESSE PENCIL OF PLANE CUBIC CURVES MICHELA ARTEBANI AND IGOR DOLGACHEV Abstract. This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3surfaceswithPicardnumber20whicharise naturally in connection with the configuration. 1. Introduction In this paper we discuss some old and new results about the widely known Hesse configuration of 9 points and 12 lines in the projective plane P2(k): each point lies on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94). Through most of the paper we will assume that k is the field of complex numbers C although the configuration can be defined over any field containing three cubic roots of unity. The Hesse configuration can be realized by the 9 inflection points of a nonsingular projective plane curve of degree 3. This discovery is attributed to Colin Maclaurin (1698-1746) (see [46], p. 384), however the configuration is named after Otto Hesse who was the first to study its properties in [24], [25]1. In particular, he proved that the nine inflection points of a plane cubic curve form one orbit with respect to the projective group of the plane and can be taken as common inflection points of a pencil of cubic curves generated by the curve and its Hessian curve. In appropriate projective coordinates the Hesse pencil is given by the equation (1) λ(x3 + y3 + z3)+µxyz =0. The pencil was classically known as the syzygetic pencil 2 of cubic curves (see [9], p. -
The Synthetic Treatment of Conics at the Present Time
248 SYNTHETIC TEEATMENT OF CONICS. [Feb., THE SYNTHETIC TREATMENT OF CONICS AT THE PRESENT TIME. BY DR. E. B. WILSON. ( Read before the American Mathematical Society, December 29, 1902. ) IN the August-September number of the Jahresbericht der Deutschen Mathernatiker-Vereinigung for the year 1902 Pro fessor Reye has printed an address, delivered on entering upon the rectorship of the university at Strassburg, on u Syn thetic geometry in ancient and modern times," in which he sketches in a general manner, as the nature of his audience necessitated, the differences between the methods and points of view in vogue in geometry some two thousand years ago and at the present day. The range of the present essay is by no means so extended. It covers merely the different methods which are available at the present time for the development of the elements of synthetic geometry — that is the theory of the conic sections and its relation to poles and polars. There is one method of treatment which was much in use, the world over, half a century ago. At the present time this method is perhaps best represented by Cremona's Projective Geometry, translated into English by Leudesdorf and published by the Oxford University Press, or by RusselPs Pure Geom etry from the same press. This treatment depends essentially upon the conception and numerical measure of the double or cross ratio. It starts from the knowledge the student already possesses of Euclid's Elements. This was the method of Cre mona, of Steiner, and of Chasles, as set forth in their books.