DOCSLIB.ORG

SMOOTH BODIES of CONSTANT WIDTH in E 2 Sasˇo Bogoev

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SMOOTH BODIES of CONSTANT WIDTH in E 2 Sasˇo Bogoev Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, Vol.8, (2019), Issue 2, pp.74-90 SMOOTH BODIES OF CONSTANT WIDTH IN E2 Sa soˇ Bogoev ABSTRACT . This paper is concerned with another possible construction of two-dimensional bodies of constant width in Euclidean plane. We derive a new parametrization of their boundary curves and give a connection to the standard support function. Considering this parametrization, we also obtain an area formula and show some examples in the sequel. At the end, we take a step back to the Reuleaux polygons in an attempt to represent them holding on to our approach, although they are not smooth and are outside the scope of the title of the paper. 1. I NTRODUCTION Let K E2 be a compact convex body (set), and D its diameter. We say the body K is of constant⊂ width D if it has exactly one pair of points (refer to them as opposite or antipodal points) at distance D in all directions. This definition implies that K is to be strictly convex too. The bodies of constant width have been studied by many authors, but their properties are still subject to analysis. In brief, we only mention a few of their basic features. For instance, the Barbier’s theorem states that any curve of constant width D has length of Dπ. Certainly, the boundary of our body K is a curve of constant width and we denote it by ∂K, as usual. A simple example of a curve of constant width is the circle, but there exist infinitely many others which are commonly named as orbiforms. Another widely known result refers to the area and claims that among all bodies of a given constant width, the disk and the Reuleaux triangle have the largest and (Blaschke–Lebesgue theorem) the smallest area respectively. Further, the only centrally symmetric body of constant width is the disk, and so on. In the sequel, we denote the area of K by A(K) and its perimeter by L(K). The main focus herein is on the bodies of constant width which boundaries are smooth curves of at least class C1. In spite of the fact that our construction is based on a union of two (or more) curves of class C2, the resulting curves might not have C2 continuity at the two (or more) join points. However, regardless of this classification, since C1 continuity is preserved, we simply say K is a smooth body of constant width. 2010 Mathematics Subject Classification: 52A10, 53A04. Keywords : constant width, isoperimetric, curve, body, set, construction, parametrization, area, Reuleaux polygons. 74 SMOOTH BODIES OF CONSTANT WIDTH IN E2 2. P ARAMETRIZATION Now, let’s start speculating. Suppose we have some body K of constant width which diameter is D, and a line segment, let say AB , of the same length. By the definition above, and due to the strict convexity, we are able to fit the whole AB into the interior of K, except the two endpoints A and B which will belong to the boundary of K. What is more important is that we are restricted to do this placement in a unique way for every single direction and hence, we conclude ∂K could be well–defined by the position of AB , taking the direction as a parameter. Moreover, for any plane body of constant width which boundary is a curve of class C2, it is proven that the sum of the radii of curvature at any two opposite points (in our case A and B) equals to the diameter D. This fact leads us to an idea to start a rotation of AB without leaving the interior of K. Although the center of this rotation F will vary, it must lie somewhere on AB in every moment (follows from the mentioned sum of the radii of curvature and the properties of the distance as a metric function). After a rotation of AB by π, we arrive at the same direction which we started from and therefore, its position must match the starting one. The trajectories of the points A and B bring us back the boundary of K. Since we were in the interior of K all the time, and the points A and B were on its boundary, we covered the whole area A(K) as well with this rotation. The conclusion is that we can perform exactly the same movement of AB even if K is not present. All we need to know is how the center of the rotation F changes when we change the direction of AB . Recall that we are dealing with a smooth ∂K, which means its derivative exists everywhere. Finally, let’s fix a coordinate system and place K such that one of its diameters, say AB , lies on the x-axis, and the point A is at the origin, i.e. A = (0, 0) and B = (D, 0) , as shown in Figure 1. FIGURE 1. Rotation of the diameter AB . After a rotation by α, AB arrives at the position A′B′. Denote the points A′ and B′ by their coordinates (xA, yA) and (xB, yB), where all xA, yA, xB, yB are some functions of the parameter α. 75 S. BOGOEV At this position, we are to continue with a rotation which center must be exactly F, as previously explained. Following the point F, we define the function f : [0, π] [0, D] by f (α) = d(A′, F) , → where d is the standard metric in E2 (distance). The smoothness of ∂K (it is of class C1) implies continuity of f almost everywhere on [0, π]. By definition, the function f is also non-negative, bounded by D and therefore, it is measurable. While having this in mind, in order to simplify the analysis, for now we assume f is continuous everywhere on [0, π]. Considering Figure 1, let dl A and dl B be positive lengths, and dα small enough. Then, we may write dl A = f (α) dα , dl = (D f (α)) dα . B − Under this assumptions, the Barbier’s theorem follows immediately, π π L(K) = (dl A + dl B) = D dα = Dπ . Z0 Z0 We track the points A and B separately, and Figure 1 shows that yA(α) is negative, yB(α) is positive on the interval (0, π), while the differential functions dx A and dx B are positive and negative on the same interval respectively. Thus, the differentials of the coordinates of A, B are: π dx = dl cos ( α) = dl sin α = f (α) sin α dα ; A A 2 − A π dy = dl sin ( α) = dl cos α = f (α) cos α dα ; A − A 2 − − A − π (2.1) dx = dl cos ( α) = dl sin α = (D f (α)) sin α dα ; B − B 2 − − B − − π dy = dl sin ( α) = dl cos α = (D f (α)) cos α dα . B B 2 − B − Taking in account the starting values xA(0) = yA(0) = yB(0) = xB(π) = yA(π) = yB(π) = 0, and xA(π) = xB(0) = D, the parametric equations of the coordinates of the points A and B are: α xA(α) = f (t) sin t dt ; Z0 α yA(α) = f (t) cos t dt ; − Z0 α (2.2) xB(α) = D (D f (t)) sin t dt ; − Z0 − α yB(α) = (D f (t)) cos t dt . Z0 − The graph constructed by the union of the points (xA, yA) and (xB, yB) coincide with ∂K or, if we are more formal, ∂K = xA, yA : α [0, π] xB, yB : α [0, π] . Notice that the x-axis splits this union into! two curves such∈ that each S ! of them shows∈ the trajectory of A and B separately. At α = π, the two points interchange their starting coordinates and probably this is the moment when we give up on achieving C2 continuity. Namely, it is easy to observe 76 SMOOTH BODIES OF CONSTANT WIDTH IN E2 = = D that only in case when f (0) f (π) 2 , the starting and the ending curvature match each other at the join points. Having this in mind, since in the start we picked out the diameter AB arbitrarily, if we reparametrize the same curve, we infer that the function f does not even have to be continuous necessarily. Anyway, what is important is that when the points A and B finish their journey, after integrating over [0, π], by (2.1) we conclude π π dx A = xA(π) xA(0) = D f (α) sin α dα = D , Z0 − ⇐⇒ Z0 π π (2.3) dy A = yA(π) yA(0) = 0 f (α) cos α dα = 0 . Z0 − ⇐⇒ Z0 In fact, the last two relations are constraints on f which must hold for any smooth curve of constant width. These constraints obtain the curve to be closed. From this point, if we try to get back step by step aiming to reconstruct ∂K, we will find we have no other obstacle except one which hinder us to guarantee the smoothness of the curve. Let’s suppose we are equipped with a non-negative continuous function f on [0, π], bounded by the length of the segment we rotate, say D, and assume the equations (2.3) hold too. In order to eliminate prospective singular points on the curve (2.2), we are looking for the candidates. By use of (2.1), we set the next conditions: = = x′A(α) f (α) sin α 0 ; y′ (α) = f (α) cos α = 0 ; A − x′ (α) = (D f (α)) sin α = 0 ; B − − y′ (α) = (D f (α)) cos α = 0 .
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]
  • 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.
    [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]