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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. After the turn of the century it was dis­ covered that L'l-curves (rotors in an equilateral triangle) are closely related to curves of constant width. About the middle of this century renewed interest in these curves was evident as part of the theory of convexity~ In this thesis significant examples of curves of con­ stant width and L'l-curves are described, constructed, and parametrized. A variety of theorems are presented which es­ tablish major characteristics of these curves. The conclu­ sion indicates the present state of research in this field. vi I. Introduction Just before the invention of the wheel primitive people probably used logs as rollers to transport heavy loads. The logs' circular cross sections kept the under­ side of the load always the same distance from the ground. "Ob- viously a wheel must be made in the form of a circle with the hub at the center, since any other form will produce an Fig. 1 up-and-dovm motion." [l,p. 163] If rollers are used, how- ever, the movement of the center is not important. The cross section may be any curve that will serve to keep the load the same distance h above the ground. Such a curve­ which has always the same width h between a supporting line (in this case the ground) and the "top" - is called a "curve of constant width". It can be shown that a curve of constant width h revolves inside a square of sidelength h, and it is therefore called a "rotor in a sguare." Since curves of constant width are rotors in a square, it is natural to ask if there are rotors in an equilateral triangle. Such curves do exist and are called b.-curve~ ("delta-curves"), because the Greek capital letter ~ resembles an equilateral triangle. In this thesis we will study the properties of both types of curves and some of their applications. Chapter II 1 2 gives a brief history, Chapter III gives definitions and introduces some tools from differential geometry for subse­ quent development. In Chapter IV several examples of curves of constant width, including the Reuleaux triangle and in­ volutes of astroids are described and constructed. The next chapter lists and proves several theorems which char­ acterize curves of constant width. In Chapter-VI for the particular case of involutes of astroids parantetrizations are developed, the theorems are verified and some of the in­ volutes are graphed. Chapter VII deals with ~-curves and the analogies to the curves of constant width. II. History Leonhard Euler discovered and studied curves of con- stant width as involutes of tricusped curves (1778). He called them "orbiformes" [2]. Since then, many authors have written about them, yet in 1958 Eggleston was moved to ·1r.rrite 11 it is surprizing, how little is known about them" [3]. Barbier studied the curves in 1860 in connection with Buffon's needle problem and found the theorem named after him: "All curves of constant width h have the same per1meter• II [4,J. In 1875 Franz Reuleaux published a "Text- book of Kinematics" in which a large section is devoted to curves which revolve in a square (namely curves of constant width) and to rotors in an equilateral triangle (~-curves). But the emphasis is on geometrical construction and not on analysis. The well known curvilinear triangle consisting of three congruent circular arcs of radius h is named after Reuleaux [5]. Many papers on curves of constant width were published in the late 19th and the early 20th century, es- pecially in Japanese Mathematics Journals, but written in French, German, and English. Papers treating rotors in the equilateral triangle analytically appeared around 1915, again mostly in Japanese Journals [6],[7],[8],[9]. 3 III. Definitions and General Notions To begin the discussion of curves of constant width, we have to define the width of a closed convex curve in a , given direction d*): Each point of the curve C is project- ed perpendicularly onto a line n also in the given direction. These projections fill the closed line segment AB of the line n. The length of the line segment !AB"J is the width h of the curve C in the direction of n. The line nA, perpendicular to n at A has at least one point in common with the curve C, while C lies completely to one side of nA. Similarly this is the case for the line nB, perpendicular to n at B. These two lines are called a pair of supporting lines in the direct- ion of n. A closed curve has exactly two supporting lines (= one pair of supporting lines) for every direction n. It should be noted that a supporting line does not have to be tangent. We will see later that a Reuleaux triangle has infinitely many supporting lines which are not tangents at each of it's vertices. The definition of curves of constant width may now be stated more precisely: For a curve of constant width the distance between every pair of supporting lines is always *)This discussion follows Rademacher and Toeplitz. pl63ff. 4 5 the same distance h. Now, if one considers any two.pairs of supporting lines of a curve of constant width, they form a rhombus (Fig. 3). If the two pairs are in perpendicu­ lar directions, they form a square of sidelength h. There­ fore all squares circumscribed around a given curve of con­ stant width h are congruent (Fig. 4). This means that a square of sidelength h can be Fig. 3 rotated around a curve of con­ stant width h in such a way, that it always touches the curve at four points. Since the motion is relative, it follows, that a curve of con­ stant width can rotate inside a square of sidelength h. Fig. 4 It is important t.o realize, that only for the circle the center of rotation stays the same. For all other curves of constant width the instantaneous center of rotation changes (Fig. 5). At any point. of contact between the sides of the square and the curve, the center of rotation has to lie on the perpendicular to the square's side at that point. Since this has to be the case for all points of contact, 6 the perpendiculars have to meet in one point M. As the curve rotates, the contact points change, and so does the center of rotation. Definitions and Notions from Differential Geometry In the following we will need some expressions and defi- nitions from Differential Fig. 5 geometry. The function {3.1) X ( t) = (x ( t) , y ( t) ) represents a continuous curve if x (t) and y (t) are defined and continuous functions of a parameter t in some closed interval t1 ~ t ~ t 2 • We can change to another parameter r of t is a monotone function of r in the closed interval r 1 < r < Then the interval r ~ r ~ r is mapped onto t 1 2 1 < t < t in a one-to-one way and we can write 2 X (t (r)) -·(X (t(r)), y (t (r)) or X (r) (x (r), y (r)) for r < r ~ r = 1 2 X (t) can also be looked upon as the path of the end point of the vector X (t) from the origin to the point X with the coordinates (x (t), y (t)). The length of this vector is defined as 2 2 (3.2) lx (t) I = /C (t) + y (t) The linear combination of two vectors x , x is defin- 1 2 7 ed as ( 3. 3) and a, b real numbers. The scalar product of two vectors x1 , x2 is defined a.s (3.4) or (3.5) where x 1 , y 1 , x 21 y 2 are the coordinates of the end points of xll x2, and 8 is the angle between the vectors. In particular we have for any vector ( 3.
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