PERIMETER OF REALEAUX CURVES: ITS RELATION TO THE CIRCUMFERENCE OF A GIVEN CIRCLE
INTRODUCTION
Barbier theorem states that all shapes of constant width D have the same perimeter pi D. The width of a convex figure in a certain direction is the distance between two supporting lines perpendicular to that direction. A straight line is called supporting a convex figure if they have at least one common point and the figure lies in one side from the line. In any direction there are two supporting lines. Shapes of constant width are convex figures that have the same width in any direction. The circle has this property. Barbier theorem also states that all curves of constant width of width w have the same perimeter . This time around I wish to determine if there is a possibility to construct a Reuleaux Curves of constant width from equilateral triangle, extended length of equilateral triangle, irregular triangle and any polygon with an odd number of sides. On the other hand, I wish to present that their perimeter has something to do with the circumference of a given circle.
Background of the Study
The concept of Reuleaux triangle was a very interesting geometric figure, one which was discovered in a roundabout way. The topic was, “Why are manhole covers round?” There were plenty of theories offered, his favorite being, “Because the hole is round.” Kunkel, P. knows a few things about manholes, having spent time crawling into them. He always knew that he would not have to worry about dropping the cover into the hole. Since it is circular, it needs an opening as wide as its diameter. But the hole, also circular, has a seat with a slightly smaller diameter. A regular polygon would not have that property. Someone mentioned this advantage in the newsgroup. In reply, another person described this figure:
Construct an equilateral triangle. On each vertex, center a compass, and draw an arc the short distance between the other two vertices. The perimeter will be three nonconcentric arcs. This is a reuleaux triangle. It is not a circle, but, like a circle, it has constant width, no matter how it is oriented. It is not difficult to see this property, but you should prove it.
Kunkel, P. (1997) made some sketches of the reuleaux and discovered some other interesting properties. It can roll uphill, in a manner of speaking. Notice that as it rolls, its height is constant, but the height of its centroid changes. If it had mass, the centroid would be the center of mass. Imagine that it is standing on one vertex, so that the centroid is at its highest, and imagine that the surface is very slightly inclined. If it moves forward one sixth turn, the centroid will fall. So although the surface rises, the reuleaux is actually falling.
It occurred to me that the figure had constant width and constant height. It should be possible to inscribe one into a square and to turn it freely. I did a sketch to simulate this. Click on the image. As it turned within the square, something looked very familiar about it. It looked a lot like those diagrams for the Wankel rotary engine. I did a Web search, and, sure enough, that is the shape used for the rotor in the engine. That was when I found out what to call it. A guy can learn a lot on the Web.
The square drill bit was not my idea either. Notice that as the reuleaux turns inside the square, its trace nearly fills the entire square. It was noticed that this shape might be used to drill square (actually squares) holes, and in fact someone did manufacture a drill bit base on this concept. It is not just a simple matter of fitting a bit into a drill though. It required a more complex mechanism. Animate that last applet and watch the motion of the centroid. For every rotation of the reuleaux, the centroid makes three revolutions in the opposite direction, and its path is not circular. Your next assignment is to design a mechanism that would make it work.
The reuleaux turns inside the square because the square is formed by two pairs of parallel lines, equally spaced. However, you do not see why they should have to intersect at right angles. Open it again and deform the square. It tilts over into the shape of an oblique rhombus. It is still possible to inscribe a reuleaux and to rotate it. You will see some interesting contortions in the centroid locus as you change the rhombus.
As a reuleaux is inscribed between a pair of parallel lines and rotates between them, is there ever a time when no vertices touch the lines? One vertex? Two Vertices? All three vertices?
If a vertex of the reuleaux can touch a vertex of the rhombus, what can you say about the acute angle of the rhombus?
The shape of the centroid locus is not a single ellipse. It is comprised of parts from four ellipses. To understand this better, see the Sliding Triangle page. Even without understanding the shape of the centroid locus, you should be able to prove that it has four symmetries. Can you prove this and can you define the symmetries?
When I say that the reuleaux is inscribed in a rhombus, I mean that it touches every side, but crosses none of them. Is it possible to inscribe and rotate a reuleaux in any polygon other than a rhombus?
He saved the toughest one for last. Is it possible to design a special road for the reuleaux? I want it to be able to roll along without feeling any bumps, so the centroid would have to trace a level line as the road rises and falls. This one is not easy at all. Remember, it does not have a constant radius, so if the rotational velocity is constant, the horizontal velocity will vary. I tried parametrizing x and y as functions of the rotation angle. What I got was an enormous integral which stumped Derive, Mathematical, and me. Sometimes he just feel like being difficult.
STATEMENT OF THE PROBLEM
This paper intends to show that the circumference of a given circle is equal to the perimeter of a Realeaux curves based on:
1. The equilateral triangle
2.The extended length of equilateral triangle
3.The irregular triangle
4.The Realeaux curves based on pentagon
5.The Realeaux polygon with an odd number of sides
SIGNIFICANCE OF THE STUDY
The result of this study will be a significant contribution to the development of constant width. This area of research is another interesting topic in Geometry which may later be found significant and applicable in other fields.
SCOPE
This paper deals only on finding the perimeter of Reuleaux curves related to circles.
METHODOLOGY
This research is based on the article “Reuleaux Triangle and Constant Width Curves” by Eric W. Wesstien and “Constant Width Shape” by Alexander Bogolmony. Basic concepts and definitions were presented. New terms were defined and examples were given. Results formulated were proven and illustrated.
Concepts and Definitions
Centroid – Center of a Circle
Diameter – The diameter of a circle is the distance from a point on the circle to a point Radians away, and is the maximum distance from one point on a circle to another. The diameter of a sphere is the maximum distance between two antipodal points on the surface of the sphere.
If r is the radius of a circle or sphere, then . The ratio of the circumference C of a circle or great circle of a sphere to the diameter d is pi, Circumference - The perimeter of a circle. For radius r or diameter ,
where is pi.
Curve of Constant Width – Curves which, when rotated in a square, make contact with all four sides. Such curves are sometimes also known as rollers. The “width” of a closed convex curve is defined as the distance between parallel lines bounding it (“supporting lines”). Every curve of constant width is convex. Curves of constant width have the same “width” regardless of their orientation between the parallel lines. In fact, they also share the same perimeter (Barbier’s theorem). Examples include the circle (with largest area), and Reuleaux triangle (with smallest area) but there are an infinite number. A curve of constant width can be used in a special drill chuck to cut square “holes.”
A generalization gives solids of constant width. These do not have the same surface area for a given width, but their shadows are curves of constant width with the same width.
Equilateral Triangle – A triangle with equal sides.
Reuleaux Triangle – A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest area for a given width of any curve of constant width.
Reuleaux Polygon – A curvilinear polygon built up of circular arcs. The Reuleaux polygon is a generalization of the Reuleaux triangle and, for an odd number of sides, is a curve of constant width (Gray 1997).
Analysis
In order to show that the circumference of a given circle is related to the perimeter of a Reuleaux curves in different polygons, we let, the perimeter of a reuleaux triangle be equal to p with one side of a given equilateral triangle is 1. Then, we also let the circumference of a circle be C= 2pr where r=1.
Suppose a given equilateral triangle has s= 1, then the perimeter of a reuleaux triangle becomes p i.e.
P of reuleaux D=sp
P of reuleaux D= (1)p
P of reuleaux D= p
We also try to compute for the perimeter of a circle with r=1 i.e.
C= 2pr
C= 2p(1)
C= 2p
Suppose we have an equilateral triangle placed inside the circle in which one of its vertex is in the center of the given circle. i.e.
Where r= 1
The illustration shows that the computed Circumference is equal to 2p. But pertaining to the perimeter of the minor arc, the perimeter of minor arc was p/2.
In connection to the illustration above, the reuleaux triangle with s= 1 shows that the possible perimeter of the reuleaux triangle is p. However an equilateral triangle inscribed in a circle with s=1 will give a possible minor arc perimeter value of p/3.
The study also aims to identify the relationship of a reuleaux triangle to a given circle. From previous studies, if the triangle has side-length 1, the perimeter of the constructed reuleaux triangle was p.
Consider the illustration below:
Suppose r=1, then the circumference of each circle was C=2p where r=1. The illustration reveals p= C/2 for r=1. However, the equation becomes:
Perimeter of reuleaux D = C/2
Meaning to say, the perimeter of reuleaux D is equivalent to one-half of the circumference of each ⊙ as illustrated by the figure above. Therefore the perimeter of all Reuleaux D is ½ of circumference of the given ⊙“ rÎÂ. Thus, it is also evident to say that “ reuleaux n-gon where n= odd #, the perimeter is always p “ r=1.
The study also discovered that the circumference of a given reuleaux polygon becomes closer to 2p as the # of sides increases up to infinity. The smallest perimeter of a reuleaux is p.
Credit:ivythesis.typepad.com
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