DOCSLIB.ORG

Descriptive Geometry 2 by Pál Ledneczki Ph.D

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Descriptive Geometry 2 by Pál Ledneczki Ph.D Descriptive Geometry 2 By Pál Ledneczki Ph.D. Table of contents 1. Intersection of cone and plane 2. Perspective image of circle 3. Tangent planes and surface normals 4. Intersection of surfaces 5. Ellipsoid of revolution 6. Paraboloid of revolution 7. Torus 8. Ruled surfaces 9. Hyperboloid of one sheet 10. Hyperbolic paraboloid 11. Conoid 12. Developable surfaces 13. Helix and helicoid Intersection of Cone and Plane: Ellipse The intersection of a cone of revolution and a plane is an ellipse if the plane (not passing through the vertex of the cone) intersects all generators. Dandelin spheres: spheres in a cone, tangent to the T2 cone (along a circle) and also tangent to the plane of F2 intersection. Foci of ellipse: F1 and F2, points of contact of plane of intersection and the Dadelin spheres. P P: piercing point of a generator, point of the curve of F 1 intersection. T1 and T2, points of contact of the generator and the Dandelin sphere. T1 PF1 = PT1, PF2 = PT2 (tangents to sphere from an external point). PF1 + PF2 = PT1 + PT2 = T1T2 = constant http://www.clowder.net/hop/Dandelin/Dandelin.html Descriptive Geometry 2 Intersection of cone and plane 2 Construction of Minor Axis α ” Let the plane of intersection α” second projecting A” plane that intersects all generators. L”=C”=D” The endpoints of the major axis are A and B, the β ” K” piercing points of the leftmost and rightmost generators respectively. B” The midpoint L of AB is the centre of ellipse. C’ Horizontal auxiliary plane β” passing through L intersects the cone in a circle with the centre of K. (K is a point of the axis of the cone). A’ K’ L’ B’ The endpoints of the minor axis C and D can be found as the points of intersection of the circle in D’ β and the reference line passing through L”. Descriptive Geometry 2 Intersection of cone and plane 3 Intersection of Cone and Plane: Parabola The intersection of a cone of revolution and a plane is a parabola if the plane (not passing through the vertex d of the cone) is parallel to one generator. Focus of parabola: F, point of contact of the plane of intersection and the Dadelin sphere. E T F 2 T Directrix of parabola: d, line of intersection of the plane of intersection and the plane of the circle on the Dandelin sphere. P: piercing point of a generator, point of the curve of intersection. T: point of contact of the generator and the T1 Dandeli sphere. P PF = PT (tangents to sphere from an external point). PT =T1T2 = PE, dist(P,F) = dist(P,d). http://mathworld.wolfram.com/DandelinSpheres.html Descriptive Geometry 2 Intersection of cone and plane 4 Construction of Point and Tangent α ” Let the plane of intersection α” second projecting V” plane be parallel to the rightmost generator. The vertex of the parabola is V. β ” Horizontal auxiliary plane β” can be used to find P”, P” the second image of a point of the parabola. The tangent t at a point P is the line of intersection of the plane of intersection and the tangent plane of the surface at P. The first tracing point of the tangent N1 is the point of intersection of the first tracing line of the plane of V’ intersection and the firs tracing line of the tangent plane at P, n11 and n12 respectively. t’ n11 n t = | N P| P’ 12 1 N1 Descriptive Geometry 2 Intersection of cone and plane 5 Intersection of Cone and Plane: Hyperbola The intersection of a cone of revolution and a T2 plane is a hyperbola if the plane (not passing through the vertex of the cone) is parallel to two generators. F2 Foci of hyperbola: F1 and F2, points of contact of plane of intersection and the Dadelin spheres. P: piercing point of a generator, point of the curve of intersection. T1 F1 T1 and T2, points of contact of the generator and the Dandelin spheres. P PF1 = PT1, PF2 = PT2 (tangents to sphere from an external point). PF2 – PF1 = PT2 - PT1 = T1T2 = constant. http://thesaurus.maths.org/mmkb/view.html Descriptive Geometry 2 Intersection of cone and plane 6 Construction of Asymptotes β ” α ” Let the plane of intersection α” second projecting plane A” parallel to two generators, that means, parallel to the L” second projecting plane β through the vertex of the K” cone, which intersects the cone in two generators g1 B” and g2. g ”= g” 1 2 The endpoints of the traverse (real) axis are A and B, the piercing points of the two extreme generators. g1’ The midpoint L of AB is the centre of hyperbola. a1’ L’ The asymptotic lines a and a are the lines of B’ 1 2 intersections of the tangent planes along the generators g1 and g2 and the plane of intersection α. a ’ g2’ 2 Descriptive Geometry 2 Intersection of cone and plane 7 Perspective Image of Circle horizon F C axis vanishing line Descriptive Geometry 2 8 Perspective image of circle Construction of Perspective Image of Circle K The distance of the horizon and the (C) is equal to the distance of the axis and the horizon F vanishing line. (C) The type of the perspective image of a circles depends on the number of common points with the vanishing line: axis •no point in common; ellipse vanishing line •one point in common; parabola •two points in common; hyperbola The asymptotes of the hyperbola are the (K) projections of the tangents at the vanishing points. Descriptive Geometry 2 Perspective image of circle 9 Tangent Planes, Surface Normals n n P A A P A P n n A P n P A Descriptive Geometry 2 Tangent planes and surface normals 10 Intersection of Cone and Cylinder 1 Descriptive Geometry 2 Intersection of surfaces 11 Intersection of Cone and Cylinder 2 Descriptive Geometry 2 Intersection of surfaces 12 Intersection of Cone and Cylinder 3 Descriptive Geometry 2 Intersection of surfaces 13 Methods for Construction of Point 1 Auxiliary plane: plane passing through the vertex of the cone and parallel to the axis of cylinder Descriptive Geometry 2 Intersection of surfaces 14 Methods for Construction of Point 2 Auxiliary plane: first principal plane Descriptive Geometry 2 Intersection of surfaces 15 Principal Points S1” K K2 D” 5 K7 K1 K1,2’ S1 S2” K6 S2 K8 K4 K1’ K S ’ 3 D’ 1 S2’ K2’ Double point: D Points in the plane of symmetry: S1, S2 Points On the outline of cylinder: K1, K2, K3, K4 Points On the outline of cone: K5, K6, K7, K8 Descriptive Geometry 2 Intersection of surfaces 16 Surfaces of Revolution: Ellipsoid Prolate ellipsoid Affinity Oblate ellipsoid Axis plane of affinity Capitol Descriptive Geometry 2 Ellipsoid of revolution 17 Ellipsoid of Revolution in Orthogonal Axonometry IV Z Z anti-circle E F1 OiV O Y X XIV =YIV F2 Descriptive Geometry 2 Ellipsoid of revolution 18 Intersection of Ellipsoid and Plane http://www.burgstaller-arch.at/ Descriptive Geometry 2 Ellipsoid of revolution 19 Surfaces of Revolution: Paraboloid Descriptive Geometry 2 Paraboloid of revolution 20 Paraboloid; Shadows Find - the focus of parabola - tangent parallel to f” -self-shadow -cast shadow - projected shadow inside Descriptive Geometry 2 Paraboloid of revolution 21 Torus z (axis of rotation) equatorial circle meridian circle throat circle y x Descriptive Geometry 2 Torus 22 Classification of Toruses http://mathworld.wolfram.com/Torus.html Descriptive Geometry 2 Torus 23 Torus as Envelope of Spheres Descriptive Geometry 2 Torus 24 Outline of Torus as Envelope of Circles Descriptive Geometry 2 Torus 25 Outline of Torus Descriptive Geometry 2 Torus 26 Classification of Points of Surface hyperbolical (yellow) elliptical (blue) parabolical (two circles) Descriptive Geometry 2 Torus 27 Tangent plane at Hyperbolic Point Descriptive Geometry 2 Torus 28 Villarceau Circles Antoine Joseph François Yvon Villarceau (1813-1889) Descriptive Geometry 2 Torus 29 Construction of Contour and Shadow Descriptive Geometry 2 Torus 30 Descriptive Geometry 2 Ruled Surface http://www.amsta.leeds.ac .uk/~khouston/ruled.htm 31 http://www.cs.mtu.edu/~shene/COURSES/cs3621/LAB/surface/ru led.html Ruled surface http://mathworld.wolfram.com/RuledSurface.html http://en.wikipedia.org/wiki/Ruled_surface http://www.geometrie.tuwien.ac.at/havlicek/torse.html http://www.f.waseda.jp/takezawa/mathenglish/geometry/surface 2.htm Hyperboloid of One Sheet http://www.archinform.net/medien/00004570.htm?ID=25c5e478c9a50370 03d984a4e4803ded http://www.jug.net/wt/slscp/slscpa.htm http://www.earth-auroville.com/index.php?nav=menu&pg=vault&id1=18 Descriptive Geometry 2 Hyperboloid of one sheet 32 Hyperboloid of One Sheet Descriptive Geometry 2 Hyperboloid of one sheet 33 Hyperboloid of One Sheet, Surface of Revolution Descriptive Geometry 2 Hyperboloid of one sheet 34 Shadows on Hyperboloid of one Sheet Descriptive Geometry 2 Hyperboloid of one sheet 35 Hyperboloid of One Sheet, Shadow 1 f” The self-shadow outline is the hyperbola h in e” the plane of the self- shadow generators of asymptotic cone. h” f’ e’ The cast-shadow on the hyperboloid itself is an arc of ellipse e inside of the surface. h’ Descriptive Geometry 2 Hyperboloid of one sheet 36 Hyperboloid of One Sheet, Shadow 2 f” The self-shadow outline is the ellipse e1 inside of the surface. The cast-shadow on the hyperboloid itself is an arc e ” 1 of ellipse e2 on the outher side of the surface. e2” f’ e1’ e2’ Descriptive Geometry 2 Hyperboloid of one sheet 37 Hyperboloid of One Sheet, Shadow 3 O2” f” The outline of the self_shadow is a pair of parallel generators g1 and g2.
Load more

Recommended publications
  • Villarceau Circles
    Villarceau Circles by Marcel Berger, Mathematician & former director of IHÉS (Institut des Hautes Études scientifiques) Figure 1: Villarceau Circles on a torus. For me, it is a great pleasure to write a text about Villarceau Circles1, because they are doubly and strongly linked to my personal life. Here is why. My first encounter with these exotic circles came about like this. In my final year of high school in 1943-1944, in addition to the typically recommended maths books, I had at hand the work that my father had had in the very same class, the exact title of these two great volumes being Traité de Géométrie by authors Rouché and de Comberousse. Certainly, they fully treated the geometry programme of final year science, but they were also filled with out of programme appendices (at that time important because they were linked to posterior geometry). Slightly obsessed with geometry, leafing through the second volume, I discovered in an appendix this surprising fact (an understatement), at least for me but probably for you too, dear reader, that a torus (always understood at the time as a torus of revolution), contained many other circles than meridian and parallel circles. I was so surprised that, instead of watching the demonstration, and in great laziness, my first reaction was to try a physical demonstration, namely to saw a ring from a wooden curtain rod. With primitive handsaws, and mainly with a ring that was ultimately too thin, the operation proved inconclusive. Regarding an elementary demonstration, and others at all levels, whose data of this result in preparatory class still remains spectacular, see 1.
    [Show full text]
  • The Hopf Fibration
    THE HOPF FIBRATION The Hopf fibration is an important object in fields of mathematics such as topology and Lie groups and has many physical applications such as rigid body mechanics and magnetic monopoles. This project will introduce the Hopf fibration from the points of view of the quaternions and of the complex numbers. n n+1 Consider the standard unit sphere S ⊂ R to be the set of points (x0; x1; : : : ; xn) that satisfy the equation 2 2 2 x0 + x1 + ··· + xn = 1: One way to define the Hopf fibration is via the mapping h : S3 ! S2 given by (1) h(a; b; c; d) = (a2 + b2 − c2 − d2; 2(ad + bc); 2(bd − ac)): You should check that this is indeed a map from S3 to S2. 3 (1) First, we will use the quaternions to study rotations in R . As a set and as a 4 vector space, the set of quaternions is identical to R . There are 3 distinguished coordinate vectors{(0; 1; 0; 0); (0; 0; 1; 0); (0; 0; 0; 1){which are given the names i; j; k respectively. We write the vector (a; b; c; d) as a + bi + cj + dk. The multiplication rules for quaternions can be summarized via the following: i2 = j2 = k2 = −1; ij = k; jk = i; ki = j: Is quaternion multiplication commutative? Is it associative? We can define several other notions associated with quaternions. The conjugate of a quaternionp r = a + bi + cj + dk isr ¯ = a − bi − cj − dk. The norm of r is jjrjj = a2 + b2 + c2 + d2.
    [Show full text]
  • Surfaces Containing Two Circles Through Each Point
    Surfaces containing two circles through each point M. Skopenkov, R. Krasauskas Abstract We find all analytic surfaces in 3-dimensional Euclidean space such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface (and analyti- cally depending on the point). The search for such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions: 3 - the set f p + q : p 2 α; q 2 β g, where α; β are two circles in R ; p×q 2 - the set f 2 jp+qj2 : p 2 α; q 2 β; p + q 6= 0 g, where α; β are circles in the unit sphere S ; 2 2 2 - the set f (x; y; z): Q(x; y; z; x + y + z ) = 0 g, where Q 2 R[x; y; z; t] has degree 2 or 1. The proof uses a new factorization technique for quaternionic polynomials. Keywords: Moebius geometry, circle, quaternion, Pythagorean n-tuple, polynomials factorization 2010 MSC: 51B10, 13F15, 16H05 1 Introduction Dedicated to the last real scientists, searching only for the truth, not career, not glory, not pushing forward their own field or students For which surfaces in 3-dimensional Euclidean space, through each point of the surface one can draw two transversal circular arcs fully contained in the surface? This is a question which simply must be answered by mathematicians because of a natural statement and obvious architectural moti- vation — recall Shukhov’s hyperboloid structures.
    [Show full text]
  • Single Digits
    ...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library
    [Show full text]
  • How to Make a Torus Laszlo C
    DO THE MATH! How to Make a Torus Laszlo C. Bardos sk a topologist how to make a model of a torus and your instructions will be to tape two edges of a piece of paper together to form a tube and then to tape the ends of the tube together. This process is math- Aematically correct, but the result is either a flat torus or a crinkled mess. A more deformable material such as cloth gives slightly better results, but sewing a torus exposes an unexpected property of tori. Start by sewing the torus The sticky notes inside out so that the stitching will be hidden when illustrate one of three the torus is reversed. Leave a hole in the side for this types of circular cross purpose. Wait a minute! Can you even turn a punctured sections. We find the torus inside out? second circular cross Surprisingly, the answer is yes. However, doing so section by cutting a yields a baffling result. A torus that started with a torus with a plane pleasing doughnut shape transforms into a one with perpendicular to different, almost unrecognizable dimensions when turned the axis of revolu- right side out. (See figure 1.) tion (both such cross So, if the standard model of a torus doesn’t work well, sections are shown we’ll need to take another tack. A torus is generated Figure 2. A torus made from on the green torus in sticky notes. by sweeping a figure 3). circle around But there is third, an axis in the less obvious way to same plane as create a circular cross the circle.
    [Show full text]
  • Conformal Villarceau Rotors
    UvA-DARE (Digital Academic Repository) Conformal Villarceau Rotors Dorst, L. DOI 10.1007/s00006-019-0960-5 Publication date 2019 Document Version Final published version Published in Advances in Applied Clifford Algebras License CC BY Link to publication Citation for published version (APA): Dorst, L. (2019). Conformal Villarceau Rotors. Advances in Applied Clifford Algebras, 29(3), [44]. https://doi.org/10.1007/s00006-019-0960-5 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:29 Sep 2021 Adv. Appl. Clifford Algebras (2019) 29:44 c The Author(s) 2019 0188-7009/030001-20 published online April 30, 2019 Advances in https://doi.org/10.1007/s00006-019-0960-5 Applied Clifford Algebras Conformal Villarceau Rotors Leo Dorst∗ Abstract.
    [Show full text]
  • Hopf Fibration and E8 in Vedic Physics
    Hopf Fibration and E8 in Vedic Physics By John Frederick Sweeney Abstract A parallel construction exists in Vedic Nuclear Physics which appears to be the Exceptional Lie Algebra E8 and the Hopf Fibration. This paper describes the key sphere H7 in Vedic Physics and then attempts to draw isomorphic relationships between the structures. In this way, this paper attempts to explain the relationship between E8 and the Hopf Fibration, which is not well understood generally, in the hope that Vedic Nuclear Physics may provide a heuristic model. Along the way, this paper describes the life cycle of galaxies, from birth to final destruction into Dark Matter. Table of Contents Introduction 3 Hopf Fibration 5 Exceptional Lie Algebra E8 11 H7 Hyper Circle 17 Conclusion 21 Bibliography 24 Introduction In recent years the Exceptional Lie Algebra E8 has taken center stage as somehow central to a Theory of Everything (TOE), first under Garret Lisi, who brought E8 to international fame, and then others who have criticized or supported Lisi. No one really has a definitive answer, and much of what was written about E8 is pure conjecture, based on K Theory or M Theory or Super String or …none of which have much to support them in any event. In contrast, Vedic Physics knows precisely what E8 is and what it does, and what comes after E8 in the sequence of the formation of matter, all included herein. Western mathematical physics knows comparatively little, in fact, since it develops in linear fashion, moving from point A to point B to point C, etc.
    [Show full text]
  • Ortho-Circles of Dupin Cyclides
    Journal for Geometry and Graphics Volume 10 (2006), No. 1, 73{98. Ortho-Circles of Dupin Cyclides Michael Schrott, Boris Odehnal Institute of Discrete Mathematics, Vienna University of Technology Wiedner Hauptstr. 8-10/104, Wien, Austria email: fmschrott,[email protected] Abstract. We study the set of circles which intersect a Dupin cyclide in at least two di®erent points orthogonally. Dupin cyclides can be obtained by inverting a cylinder, or cone of revolution, or by inverting a torus. Since orthogonal intersec- tion is invariant under MÄobius transformations we ¯rst study the ortho-circles of cylinder/cone of revolution and tori and transfer the results afterwards. Key Words: ortho-circle, double normal, dupin cyclide, torus, inversion. MSC: 53A05, 51N20, 51N35 1. Introduction In this paper we investigate the set of circles which intersect Dupin cyclides twice orthogonally. These circles will be called ortho-circles. Dupin cyclides are algebraic surfaces of degree three or four [17]. They are known to be the images of cylinders, or cones of revolution, or tori under inversions. Depending on the location of the center O of the inversion and of the choice of the input surface © we obtain di®erent types of Dupin cyclides (see Fig. 1). Dupin cyclides can be obtained by certain projections from supercyclides [12]. There is also a close relation between Dupin cyclides and line geometry [48] and geometric optics [34]. Dupin cyclides carry at least two one-parameter families of circles. The carrier planes of these circles form pencils with skew axes [13]. These circles comprise the set of lines of curvature on the cyclide.
    [Show full text]
  • Arxiv:1511.08894V2 [Math.AT] 1 Dec 2015  N−P R with Pairwise Skew Affine fibers
    AFFINE HOPF FIBRATIONS VALENTIN OVSIENKO AND SERGE TABACHNIKOV 1. Introduction There exist four fiber bundles, called the Hopf fibrations, whose fibers, total spaces, and bases are spheres: (1) S0 / S1 S1 / S3 S3 / S7 S7 / S15 S1 S2 S4 S8: The vertical arrows denote projections whose fibers, i.e., inverse images of points, are represented by the horizontal arrows. The term “fibration” means that, locally, the total space is the product of the base and the fiber, hence the bigger spheres are filed with smaller ones. For instance, through every point of S3 there passes one circle, S1, and different circles do not intersect. The construction of Hopf fibrations is very simple (we recall it below) and is based on the algebras R; C; H; O; according to a famous theorem of Adams [1], that there are no such fibrations in other dimensions. Note that the \magic" numbers 1; 2; 4; 8 appear twice: as the dimensions of the bases, and also, this time shifted by 1, as the dimensions of the fibers. In a sense, contemporary algebraic topology has grown up with the Hopf fibrations: the development of the theory of characteristic classes, homotopy theory, and K-theory was much influenced by the study of Hopf fibrations; see [4, 5, 8, 13, 15]. This note, based on a recent work [18], concerns a similar problem: for which pairs (p; n), n p is R foliated by pairwise skew affine subspaces R ? Two disjoint affine subspaces are pairwise skew if they do not contain parallel lines.1 In other words, we consider fiber bundles p n (2) R / R arXiv:1511.08894v2 [math.AT] 1 Dec 2015 n−p R with pairwise skew affine fibers.
    [Show full text]
  • Ideal Knots and Other Packing Problems of Tubes
    Ideal Knots and Other Packing Problems of Tubes THÈSE NO 4601 (2010) PRÉSENTÉE LE 22 FÉVRIER 2010 À LA FACULTÉ SCIENCES DE BASE CHAIRE D'ANALYSE APPLIQUÉE PROGRAMME DOCTORAL EN MATHÉMATIQUES ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Henryk GERLACH acceptée sur proposition du jury: Prof. M. Troyanov, président du jury Prof. J. Maddocks, Prof. J. P. Buser, directeurs de thèse Prof. S. Hildebrandt, rapporteur Prof. R. Langevin, rapporteur Prof. N. Monod, rapporteur Suisse 2010 ii Abstract This thesis concerns optimal packing problems of tubes, or thick curves, where thickness is defined as follows. Three points on a closed space curve define a circle. Taking the infimum over all radii of pairwise-distinct point triples defines the thickness ∆. A closed curve with positive thickness has a self-avoiding neighbourhood that consists of a disjoint union of normal disks with radius ∆, which is a tube. The thesis has three main parts. In the first, we study the problem of finding the longest closed tube with prescribed thickness on the unit two-sphere, and show that solutions exist. Furthermore, we give explicit solutions for an infinite sequence π of prescribed thicknesses Θn = sin 2n . Using essentially basic geometric arguments, we show that these are the only solutions for prescribed thickness Θn, and count their multiplicity using algebraic arguments involving Euler's totient function. In the second part we consider tubes on the three-sphere S3. We show that thickness defined by global radius of curvature coincides with the notion of thickness based on normal injectivity radius in S3.
    [Show full text]
  • Darboux Curves on Surfaces R Ronaldo Garcia, R´Emi Langevin and Pawel Walczak
    : Darboux curves on surfaces R Ronaldo Garcia, R´emi Langevin and Pawel Walczak October 21, 2011 Abstract In 1872 G. Darboux defined a family of curves on surfaces of R3 which are preserved by the action of the M¨obius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry, prove some general properties and make them explicit on simple surfaces, retrieving in particular results of Pell (1900) and Santal´o(1941). Contents 1 Preliminaries 2 1.1 The set of spheres in S3 ............................. 2 1.2 Curves in Λ4 andcanalsurfaces......................... 4 2 Spheres and surfaces 6 2.1 Spherestangenttoasurface. ... 6 2.2 Spheres tangenttoasurfacealongacurve . ...... 10 2.3 Localconformalinvariantsofsurfaces . ....... 14 3 Characterization in V (M) and equations of Darboux curves 14 3.1 A geometric relationsatisfied by Darboux curves . ........ 15 3.2 Differential Equation of Darboux curves in a principal chart......... 16 4 A plane-field on V (M) 18 5 Foliations making a constant angle with respect to Principal Foliations 22 5.1 Foliations onDupincyclides.. ... ... ... .. ... ... ... ... 23 Fα 5.2 Foliations onquadrics ............................ 24 Fα 6 Darboux curves in cyclides: a geometric viewpoint 27 7 Darboux curves near a Ridge Point 29 8 Darboux curves on general cylinders, cones and surfaces of revolution 33 8.1 Cylinders ..................................... 34 8.2 Cones ....................................... 34 8.3 Surfacesofrevolution .. .. ... ... ... ... .. ... ... ... .. 35 1 9 Darboux curves on quadrics 37 Introduction Our interest is to understand the extrinsic conformal geometry of a surface M R3 or ⊂ M S3, that is objects invariant by the action of the M¨obius group.
    [Show full text]
  • Hopf Fibrations and Hurwitz-Radon Numbers
    Mathematical Gems and Curiosities Sergei Tabachnikov, Editor opf fibration is a beautiful topological construction Hopf Fibrations and representing a 3-dimensional sphere as a disjoint HHunion of pairwise linked great circles. Judging by its presence on YouTube, it is almost an object of mass Hurwitz-Radon culture; see [23] for a sampler. For visualization, we also strongly recommend the film ‘‘Dimensions’’ [3]. Hopf fibration was defined and studied by the famous Numbers German geometer and topologist Heintz Hopf in 1931 [16]. VALENTIN OVSIENKO AND SERGE TABACHNIKOV In a sense, contemporary algebraic topology has grown up with the Hopf fibration: the development of the theory of characteristic classes, homotopy theory, and K-theory was much influenced by the study of Hopf fibration; see [4, This column is a place for those bits of contagious 5, 8, 13, 14]. Hopf fibration appears in other areas of mathematics and physics, including fluid dynamics, gauge mathematics that travel from person to person in the theories, cosmology, and elementary particles [22]. In fact, there exist four fiber bundles, called the Hopf community, because they are so elegant, surprising, or fibrations, whose fibers, total spaces, and bases are appealing that one has an urge to pass them on. spheres: Contributions are most welcome. ð1Þ The vertical arrows denote projections whose fibers, that is, inverse images of points, are represented by the horizontal arrows. The term ‘‘fibration’’ means that, locally, the total space is the product of the base and the fiber, hence the bigger spheres are filed with smaller ones. For instance, through every point of S3 there passes one circle, S1, and different circles do not intersect.
    [Show full text]