DOCSLIB.ORG

The Shortest Planar Arc of Width 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Shortest Planar Arc of Width 1 The Shortest Planar Arc of Width 1 Ani Adhikari Department of Statistics Sequoia Hall Stanford University Stanford, CA 94305 and Jim Pitman Department of Statistics University of California Berkeley, CA 94720 Technical Report No. 113 September 1987 Department of Statistics University of California Berkeley, California 1.1 THE SHORTEST PLANAR ARC OF WIDTH 1 ANI ADHIKARI AND JIM PITMAN Departrnent of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305 and Department of Statistics, University of California, Berkeley, CA 94720 1. Introduction. A floor is ruled with parallel lines spaced unit distance apart. You are given a piece of wire of length I which you are free to bend but not stretch. Can you bend the wire so that if dropped on the floor the bent piece of wire is certain to cross at least one of the lines, no matter how it falls? In this article we find the least l such that this can be done, and show how the wire should be bent. More formally, let a: [0, 1] -4 R2 be a continuous rectifiable arc, of length 1(a). For 0.9.< r, define the width of a between parallels at angle 9 by w0(a) = distance between supporting parallel lines at an angle 9 to the x-axis. FIG. 1. The width between parallels at angle 0 And define the width of a by 1.2 w (a) = inf wQ(x) Our problem is to identify an arc of minimal length among all arcs of width at least 1. A cir- cular arc of diameter 1 has length it= 3.14* h.Tree sides of the unit square reduces the length to 3. Two sides of an equilateral triangle of altitude I further reduces the length to 4/t3= 2.309401 . The minimal length tums out only slightly less, namely 2.27829 * - *, and is achieved by bending the wire into a shape which we call the calliper, shown in Fig. 2. FIG. 2. The calliper. In Section 2 we identify the calliper of length 2.27829 - as the shortest convex arc of width 1. In Sections 3 and 4 we show that the calliper is in fact shortest among all arcs, con- vex and non-convex. For the most part, our arguments are geometric, in the spirit of Kazar- inoff (1961), Yaglom and Boltyanski (1961) and Niven (1982). Our problem is a deterministic variation of Buffon's famous needle problem: in case you do not bend the wire at all, but leave it as a straight needle of length 1, what is the chance that the needle crosses at least one line if dropped at random on the floor? As observed by Bertrand in the last century, and argued in detail by Barbier (1860), under natural assumptions of random- ness the expected number of crossings of the grid by a needle bent into a planar curve is a constant c times the length 1, regardless of the shape of the curve. A closed convex curve of constant width 1 must cross the lines exactly twice, no matter how it falls. Since a circle of diameter I has constant width 1 and length it, it follows that 1.3 (i) the constant is c = 2/xt, (ii) every closed convex curve of constant width 1 has the same length it, and (iii) for I . 1 the probability that a randomly dropped straight needle crosses the lines is (2/i)l. For other questions about randomly dropped curves, and further references, see DeTemple and Robertson (1980). The problem solved in this article is a special case of a stochastic geometry problem which we do not know how to solve for all l. Given a wire of length l, how should it be bent to max- imize the probability that it crosses at least one line when tossed at random on the ruled floor? Here we just find the least length L such that this maximum probability is one. For 0 < I < 1 the best strategy is easily shown to be leaving the wire straight, with crossing probability (2/7i)1. But for 1 <1 < L we do not know how to bend the wire to maximize the crossing probability. Some remarks on this and other problems in the same vein are mentioned in the final section. Notation. Since rotation, translation, and reflection of an arc in the plane affect neither its length nor its width, we will often use these operations to reduce calculations to arcs with some convenient orientation. A generic arc a will often be denoted instead by A-B to indi- A cate that it starts at A and ends at B. To -indicate the arc is convex we write A B instead of A-B. For points A and B in the plane we use the following notation: AB for the straight line through A and B, AB for the segment of this line between A and B, I AB I for the length of this line segment. Constructions of arcs may be indicated by notation such as this: if A B is some arc under consideration, then A BA is the closed arc obtained by following A-B from A to B, then returning to A along a line segment. In such constructions the precise paramaterization of the arc will be irrelevant, though the order in which points are visited may be important 2.1 2. The Shortest Convex Arc of Width One. Suppose now that the arc A B is convex, A meaning that the closed arc A BA is the boundary of a convex subset of the plane: A B A B A FIG. 3. Some convex arcs. A A convex arc A B must lie on one side of the line AB. Without loss of generality, we take AB to be horizontal, and suppose A B lies above AB. We propose to find the shortest such A arc of width one. In case A B extends to the left of A or to the right of B, a shorter arc with greater width is obtained by dropping a perpendicular to AB: A 3 FIG. 4. Reduction to arcs over an interval. So we may assume that d = AB 2 1, and confine our search to arcs A B lying in the strip above AB, such as the leftmost arc in Fig. 3. In this case the width condition can be reformu- lated more simply as follows, with reference to the horizontal segment A'B' at unit height above AB, and open discs Disc(A) and Disc(B) centered at A and B, with radii 1, as shown in the next figure: Al . ;; _-- Ds(B)"C I,A A JB FIG. 5. An arc of width greatera an 1. LEMMA. For a convex arc A B above AB with l AB > 1 width (A B) > 1 if and only if the following three conditions all hold: A (i) A B intersects A'B' at some point F A (ii) A F does not intersect Disc (B) A (iii) F B does not intersect Disc (A) Proof of Sufficiency. Suppose the three conditions hold. Width at least 1 in the horizontal and vertical directions is clear. For the width between down-sloping parallels, consider two paral- lels distance 1 apart, one through A, the other above it tangent to Disc (A ). The upper paral- lel must hit the arc between F and B by (i) and (iii). Similarly, width at least 1 between up- sloping parallels is implied by (i) and (ii). Proof of Necessity. If (i) fails, the vertical width is less than 1. The only remaining case is if A (i) holds but either (ii) or (iii) fails. If say (ii) fails, there exist points C and D on A B as in the following diagram such that C and D lie on Circle (B) bounding Disc (B), and apart from these points arc C D lies inside the open Disc (B). defined in tems of CD and isc (B ). Th upe-uprigl'e fABprle oC A'.A By convexity of A B, the arc lies inside the shaded region in the diagram, with boundary d d e f n c . ppersupportinglineofA 2.3 must therefore intersect Disc (B ), giving a width less than 1. According to the lemma, for fixed A and B at distance d . 1, the shortest convex arc A B of width 1 lying above AB is the shortest path from A to B via some F on A'B', not entering Disc (B ) on the way from A to F, and not entering Disc (A ) on the way from F to B. In case d > 2/43, this shortest path is obvious. For F the midpoint of A'B' the straight lines AF and FB do not meet the discs, so AFB is the shortest, with length V1Id. For d =2h13, the length is 2.30940... .. FIG. 7. Two sides of an equilateral triangle: d =P3 If d < 2/43, the constraint of the discs is felt. For any F on A'B', it is easy to see that the shortest way to get from A to F without entering Disc (B) is to go along AF, if this can be done without entering Disc (B); else to go from A to Circle (B ) along a tangent, stay on the circle awhile, and leave for F along another tangent. And the shortest way to return from F to B without entering Disc (A) can be described similarly, with Circle (A) replacing Circle (B). A ' .;.F A.:.
Load more

Recommended publications
  • Representation of Curves of Constant Width in the Hyperbolic Plane
    PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 3 { 1998 REPRESENTATION OF CURVES OF CONSTANT WIDTH IN THE HYPERBOLIC PLANE P.V. Araujo¶ * Abstract: If γ is a curve of constant width in the hyperbolic plane H2, and l is a diameter of γ, the track function x(θ) gives the coordinate of the point of intersection l(x(θ)) of l with the diameter of γ that makes an angle θ with l. We show that x(θ) determines the shape of γ up to the choice of a constant; this provides a representation of all curves of constant width in H2. The track function is locally Lipschitz on (0; ¼), satis¯es jx0(θ) sin θj < 1 ¡ ² for some ² > 0, and, if l is appropriately chosen, has a continuous extension to [0; ¼] such that x(0) = x(¼); conversely, any function satisfying these three conditions is the track function of some curve of constant width. As a by-product of the representation thus obtained, we prove that each curve of constant width in H2 can be uniformly approximated by real analytic curves of constant width, and extend to all curves of constant width some results previously established under restrictive smoothness assumptions. 1 { Introduction A closed convex curve γ in the Euclidean plane is said to have constant width W if the distance between every two distinct parallel lines of support of ­ is equal to W; equivalently, γ has constant width W if, for each p 2 γ, the maxi- mum distance from p to other points of γ is equal to W. This latter condition can be taken as the de¯nition of constant width for simple closed curves in ar- bitrary metric spaces: here we are concerned with such curves in the hyperbolic Received: February 26, 1997; Revised: July 5, 1997.
    [Show full text]
  • Mizel's Problem on a Circle References
    Mizel`s problem on a circle Maxim V. Tkachuk Institute of Mathematics of the National Academy of Sciences of Ukraine [email protected] The problem is known in literature as Mizel’s problem(A characterization of the circle): A closed convex curve such that, if three vertices of any rectangle lie on it, so does the fourth, must be a circle. In 1961 A.S. Besicovitch [1] solved this problem. Later, a modified proof of this statement was presented by L.W. Danzer, W.H. Koenen, C. St. J. A. Nash-Williams, A.G.D. Watson [2, 3, 4, 5]. In 1989 T. Zamfirescu [6] proved a similar result for a Jordan curve (not convex a priory) and for a rectangle with the infinitesimal relation between its sides: ¯ ¯ ¯a¯ ¯ ¯ · " > 0; b where a and b are sidelengths of a rectangle. In 2006 M. Tkachuk [7] obtained the most general result in this area for any arbitrary compact set C ½ R2, where the complement R2nC is not connected. T. Zamfirescu proved that every analytic curve of constant width satisfying the infinitesimal rectangle property is a circle. Our aim is to prove the theorem for a convex curve without the analyticity condition(Theorem 1 [6]) and to discuss new unsolved problems concerning Mizel’s problem. Theorem.[9] Any convex curve of constant width satisfying the infinitesimal rectangular condition is a circle. References [1] Besicovich A. S. A problem on a Circle. - J. London Math. Soc.,1961 - 36.- p.241-244. [2] Watson A. G. D., On Mizel’s problem, J. London Math.
    [Show full text]
  • J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2013 Intro: Smooth
    J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2013 Intro: smooth curves and surfaces in Euclidean space (usu- We instead focus on regular smooth curves α. Then if ally R3). Usually not as level sets (like x2 + y2 = 1) as in ϕ: J → I is a diffeomorphism (smooth with nonvanishing algebraic geometry, but parametrized (like (cos t, sin t)). derivative, so ϕ−1 is also smooth) then α ◦ ϕ is again smooth Euclidean space Rn 3 x = (x ,..., x ) with scalar product and regular. We are interested in properties invariant under P 1 n √ (inner prod.) ha, bi = a · b = aibi and norm |a| = ha, ai). such smooth reparametrization. This is an equivalence rela- tion. An unparametrized (smooth) curve can be defined as an equivalence class. We study these, but implicitly. A. CURVES For a fixed t0 ∈ I we define the arclength function s(t):= R t |α˙(t)| dt. Here s maps I to an interval J of length len(α). t0 I ⊂ R interval, continuous α: I → Rn is a parametrized If α is a regular smooth curve, then s(t) is smooth, with pos- n curve in R . We write α(t) = α1(t), . , αn(t) . itive derivatives ˙ = |α˙| > 0 equal to the speed. Thus it has a The curve is Ck if it has continuous derivs of order up to smooth inverse function ϕ: J → I. We say β = α ◦ ϕ is the ∞ k.(C0=contin, C1,..., C =smooth). Recall: if I not open, arclength parametrization (or unit-speed parametrization) of differentiability at an end point means one-sided derivatives α.
    [Show full text]
  • Heinz Hopf, Selected Chapters of Geometry
    Selected Chapters of Geometry ETH Zurich,¨ summer semester 1940 Heinz Hopf (This is the text of a course that Heinz Hopf gave in the summer of 1940 at the ETH in Zurich.¨ It has been reconstructed and translated by Hans Samelson from the notes that he took as student during the course and that resurfaced in 2002.) Contents Chapter I. Euler’s Formula I.1. The Euclidean and the Spherical Triangle .............1 I.2. The n Dimensional Simplex, Euclidean and Spherical .......2 I.3. Euler’s Theorem on Polyhedra, other Proofs and Consequences .4 I.4. Convex Polyhedra ..........................6 I.5. More Applications ..........................8 I.6. Extension of Legendre’s Proof to n Dimensions ..........9 I.7. Steiner’s Proof of Euler’s Theorem ................ 10 I.8. The Euler Characteristic of a (Bounded Convex) 3 Cell .... 11 Chapter II. Graphs II.1. The Euler Formula for the Characteristic ........... 15 II.2. Graphs in the Plane (and on the Sphere) ............. 16 II.3. Comments and Applications .................... 18 II.4. A Result of Cauchy’s ........................ 20 Chapter III. The Four Vertex Theorem and Related Matters III.1. The Theorem of Friedrich Schur (Erhard Schmidt’s Proof) .. 22 III.2. The Theorem of W. Vogt ...................... 25 III.3. Mukhopadaya’s Four Vertex Theorem .............. 26 III.4. Cauchy’s Congruence Theorem for Convex Polyhedra ..... 27 Chapter IV. The Isoperimetric Inequality IV.1. Proofs of H.A. Schwarz (1884), A. Hurwitz, Erhard Schmidt .. 30 IV.2. The Isoperimetric Inequality in Rn ............... 33 selected chapters of geometry 1 Chapter I. Euler’s Formula I.1. The Euclidean and the Spherical Triangle For the Euclidean triangle we have the well known P fact “Sum of the angles = π ”or α−π=0.
    [Show full text]
  • Reinventing the Wheel: an Experiment in Evolutionary Geometry
    Reinventing the Wheel: An Experiment in Evolutionary Geometry Josh C. Bongard Hod Lipson Computational Synthesis Laboratory Computational Synthesis Laboratory Sibley School of Mechanical and Aerospace Sibley School of Mechanical and Aerospace Engineering Engineering Cornell University Cornell University Ithaca, NY 14850 USA Ithaca, NY 14850 USA [email protected] [email protected] ABSTRACT 1 1 0.8 In the domain of design, there are two ways of viewing the competi- 0.8 0.6 0.6 tiveness of evolved structures: they either improve in some manner 0.4 0.4 on previous solutions; they produce alternative designs that were 0.2 0.2 not previously considered; or they achieve both. In this paper we 0 0 show that the way in which designs are genetically encoded influ- −0.2 −0.2 ences which alternative structures are discovered, for problems in −0.4 −0.4 which a set of more than one optimal solution exists. The prob- −0.6 −0.6 lem considered is one of the most ancient known to humanity: de- −0.8 −0.8 sign a two-dimensional shape that, when rolled across flat ground, −1 −1 a −1 −0.5 0 0.5 1 b −1 −0.5 0 0.5 1 maintains a constant height. It was not until the late 19th century— 1 1 roughly 7000 years after the discovery of the wheel—that Franz 0.8 0.8 Reuleaux showed that a circle is not the only optimal solution. Here 0.6 0.6 we demonstrate that artificial evolution repeats this discovery in un- 0.4 0.4 der one hour.
    [Show full text]
  • Radio Frequency Identification Based Smart
    ISSN (Online) 2456-1290 International Journal of Engineering Research in Mechanical and Civil Engineering (IJERMCE) Vol 2, Issue 5, May 2017 Design and Fabrication of Precession Square Hole Drilling [1]Abijeth kc ,[2]Avinash s , [3]Avinash mp , [4]Balu n , [6]Dr C Anilkumar [1],[2],[3],[4]UG Scholars [5]- Professor Department of Mechanical Engineering Sri Sai Ram College of Engineering, Anekal, Bengaluru Abstract: -- Hole serves various purposes in all machine elements. These holes may be round, square, rectangular or any other shape depending on the requirement or design. This paper discusses the mechanical design and simulation of a square hole producing tool based on Reuleaux Triangle. The main aim of this paper is to investigate how a circular motion can be converted into a square motion by purely a mechanical linkage; an application of which is to construct a special tool that drills exact square holes. A geometrical construction that fulfils the laid objective is Reuleaux Triangle. Additionally, for this geometry to work from a rotating drive (such as a drill press) one must force the Reuleaux triangle to rotate inside a square, and that requires a square template to constrain the Reuleaux triangle as well as a special coupling to address the fact that the centre of rotation also moves. The practical importance of this enhancement is that the driving end can be placed in a standard drill press; the other end is restricted to stay inside the fixed square, will yield a perfectly square locus and this can be turned into a working square-to drill hole.
    [Show full text]
  • The Rosenthal-Szasz Inequality for Normed Planes
    The Rosenthal-Szasz inequality for normed planes Vitor Balestro CEFET/RJ Campus Nova Friburgo 28635000 Nova Friburgo Brazil [email protected] Horst Martini Fakult¨at f¨ur Mathematik Technische Universit¨at Chemnitz 09107 Chemnitz Germany [email protected] Abstract We aim to study the classical Rosenthal-Szasz inequality for a plane whose ge- ometry is given by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterization of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes. Keywords: anti-norm, Rosenthal-Szasz inequality, normed plane, constant width, Radon plane, support function. MSC 2010: 52A10, 52A21, 52A40, 53A35. arXiv:1804.00215v3 [math.MG] 27 May 2018 1 Introduction The classical Rosenthal-Szasz theorem (see [14], [4, Section 44], [5, p. 143], and [12, p. 386]) says that for a compact, convex figure K in the Euclidean plane with perimeter p(K) and diameter D(K) the inequality p(K) ≤ πD(K) holds, with equality if and only if K is a planar set of constant width D(K). That any figure of the same constant width satisfies the equality case is clear by Barbier’s theorem (cf.
    [Show full text]
  • Namic Eom Etry ®
    for D y n a m i c G e o m e t r y ® A c t i v i t i e s ISBN 1-55953-462-1 90000 Key Curriculum Press Key Curriculum Press Key College Publishing Key College Publishing 9 781559 534628 Contents Project Numbers Art/Animation 1–9 Calculus 10–14 Circles 15–20 Conic Sections 21–25 Fractals 26–28 The Golden Ratio 29–31 Graphing/Coordinate Geometry 32–36 Lines and Angles 37–39 Miscellaneous 40–45 Polygons 46–49 Quadrilaterals 50–54 Real World Modeling 55–69 Special Curves 70–74 TechnoSketchpad 75–76 Transformations and Tessellations 77–86 Triangles 87–98 Trigonometry 99–101 Appendix 1: Plotting in Sketchpad Appendix 2: Sliders Bibliography Editor and Art Designer: Steven Chanan Co-editor: Dan Bennett Contributors: Masha Albrecht, Dan Bennett, Steven Chanan, Bill Finzer, Dan Lufkin, John Olive, John Owens, Ralph Pantozzi, James M. Parks, Cathi Sanders, Daniel Scher, Audrey Weeks, Janet Zahumeny Sketchpad Design and Implementation: Nick Jackiw Production Editor: Jennifer Strada Art and Design Coordinator: Caroline Ayres Production and Manufacturing Manager: Diana Jean Parks Layout and Cover Designer: Kirk Mills Publisher: Steven Rasmussen © 2001 by Key Curriculum Press. All rights reserved. ®Key Curriculum Press, The Geometer’s Sketchpad, and Dynamic Geometry are registered trademarks of Key Curriculum Press. All other registered trademarks and trademarks in this book are the property of their respective holders. Key Curriculum Press, 1150 65th Street, Emeryville, CA 94608 http://www.keypress.com Printed in the United States of America 10 9 8 7 04 03 ISBN 1-55953-462-1 101 Project Ideas for The Geometer’s Sketchpad® There are a million and one things you can do with The Geometer’s Sketchpad, and this little book covers just a few.
    [Show full text]
  • Shapes of Constant Width and Beyond
    Shapes of Constant Width and Beyond Sept. 18, 2019 — Math Club, CU Boulder Yuhao Hu University of Colorado Boulder A. 50 Pence A. 50 Pence A. 50 Pence A. 50 Pence B. Manhole Cover B. Manhole Cover C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) C. Ruler-Compass Construction (Starting with a regular (2n + 1)-gon . ) D. Ruler-Compass Construction (Removing corners . ) D. Ruler-Compass Construction (Removing corners . ) D. Ruler-Compass Construction (Removing corners . ) D. Ruler-Compass Construction (Removing corners . ) D. Ruler-Compass Construction (Removing corners . ) D. Ruler-Compass Construction (Removing corners . ) E. Ruler-Compass Construction (From any triangle with sides a > b > c ...) E. Ruler-Compass Construction (From any triangle with sides a > b > c ...) E. Ruler-Compass Construction (From any triangle with sides a > b > c ...) E. Ruler-Compass Construction (From any triangle with sides a > b > c ...) E. Ruler-Compass Construction (From any triangle with sides a > b > c ...) E.
    [Show full text]
  • June, 1978 the Thesis of Hedwig Gertrud Knauer Is Approved
    CALIFORNIA STATE UNIVERSITY, NORTHRIDGE CURVES OF CONSTANT HIDTH AND A-CURVES ti .A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics by Hedwig Gertrud Knc-:J.cr- June, 1978 The Thesis of Hedwig Gertrud Knauer is approved: (William Karush) Advisor (Joel L. Zeitlin) (Date · Committee Chairman . \. California State University, Northridge ii Dedication To my husband Wolfgang and my sons Tom, Stephen, and Peter iii Acknowledgement: I wish to express my gratitude to Dr. Huriel Wright and to Dr. Joel Zeitlin for their constant en­ couragement and help with this project.. iv TABLE OF CONTENTS Dedication p. iii Acknowledgment p. iv •ra.ble of Contents p. v Abstract p. vi I. Introduction p. 1 II. History p. 3 III. Definitions artd General Notions p. 4 IV. Curves of Constant Width or Rotors in the Square A. Circular Arc Constructions (l) The Reuleaux Triangle and its Applications p. 12 (2) Reuleaux Polygons and Their Construction p. 18 B. Non-Circular Arc Constructions: Involutes of Astroids p. 20 V. General Theorems p. 24 VI. Involutes of Astroids as Curves of p. 36 Const.ant Width VII. ~-Curves or Rotors in the Equilateral p. 49 Triangle VIII.Conclusion p. 61 Footnotes and Bibliography p. 63 v ABSTRACT CURVES OF CONSTANT WIDTH AND L'l-CURVES by Hedwig Gertrud Knauer Master of Science in Mathematics This thesis deals with curves of constant width (rotors in a square), a topic which first appeared in 1780 in a tre~ atise by Leonard Euler and reappeared interrnittently in the late 19·th and early 20th century in widely different mathe­ matical contexts.
    [Show full text]
  • On the Geometry of Piecewise Circular Curves Author(S): Thomas Banchoff and Peter Giblin Source: the American Mathematical Monthly, Vol
    On the Geometry of Piecewise Circular Curves Author(s): Thomas Banchoff and Peter Giblin Source: The American Mathematical Monthly, Vol. 101, No. 5 (May, 1994), pp. 403-416 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2974900 . Accessed: 01/03/2011 07:56 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=maa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org On the Geometlg of Piecewise Circular Curves Thomas Banchoff and Peter Giblin In this articlewe would like to promote a class of plane curvesthat have a number of special and attractiveproperties, the piecewise circularcurves, or PC curves.
    [Show full text]
  • Tuesday 17 January 2012 – Morning A2 GCE MATHEMATICS (MEI) 4754B Applications of Advanced Mathematics (C4) Paper B: Comprehension
    Tuesday 17 January 2012 – Morning A2 GCE MATHEMATICS (MEI) 4754B Applications of Advanced Mathematics (C4) Paper B: Comprehension INSERT *4733160112* Duration: Up to 1 hour INFORMATION FOR CANDIDATES • This insert contains the text for use with the questions. • This document consists of 8 pages. Any blank pages are indicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR • Do not send this insert for marking; it should be retained in the centre or recycled. Please contact OCR Copyright should you wish to re-use this document. © OCR 2012 [T/102/2653] OCR is an exempt Charity DC (NH/JG) 44283/3 Turn over 2 Curves of Constant Width Introduction Imagine you need to move a heavy object over level ground. You could rest your object on a horizontal platform and use cylindrical rollers, all with the same radius, as shown in Fig. 1. direction of motion object platform Fig. 1 Using cylindrical rollers in this way would ensure that the motion is smooth and horizontal; the object 5 remains at the same height above the ground at all times. If the cross-section of each roller was an ellipse, rather than a circle, then the motion would not be smooth. Fig. 2 shows different orientations of an elliptical roller. Fig. 2 You might be surprised to learn that there are infinitely many shapes which, like the circle, form the cross-section of a roller that would produce smooth horizontal motion of the object. These shapes are said 10 to have constant width and the boundary of such a shape is called a curve of constant width.
    [Show full text]