[Research Proposal] PERIMETER OF REULEAUX CURVES: ITS RELATION TO THE CIRCUMFERENCE OF A GIVEN CIRCLE

CHAPTER 1

INTRODUCTION

Man is a rational being. Through man’s existence on earth he has been longing and striving for knowledge. He keeps on searching for the things that are good for the body, things that are good for the soul and things that are good for the mind. Man’s hope lies in his rational nature and his determination and courage to reach the success, therefore his quest for the “impossible dream” must go on, and he must continue fighting and searching for the things that are good.

The desire of man is simply to satisfy his needs. Thus, the ability of man to think mathematically existed. According to some mathematicians, everything on earth that man perceives can be interpreted using mathematical expressions. Meaning to say, these things can be explained through numbers and figures. Mathematicians believe that mathematics was related to numbers and figures. Through this, another discipline arose and it is called “Geometry”. Geometry was defined as the branch of mathematics that treat space and its relation, especially as shown in the properties and measurement of points, lines, angles, surfaces and solids.

Studies concerning to the relationship of figures triggers my curiosity which was inspired by different theories. Apparently, Barbier’s theorem indicates that some figures aside from circle possess a constant width. This statement caught my concern pertaining to this figure called “reuleaux polygons”. 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:

Figure 1

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 the 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 squarish) 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, we cannot 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. We will see some interesting contortions in the centroid locus as you change 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.

When we say that the reuleaux is inscribed in a rhombus, we mean that it touches every side, but crosses none of them. It is possible to inscribe and rotate a reuleaux in any polygon other than a rhombus.

He saved the toughest one for last. It is possible to design a special road for the reuleaux. We 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. We tried parametrizing x and y as functions of the rotation angle. What we got was an enormous integral that stumped Derive, Mathematica, and me. Sometimes he just feels 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 Reuleaux curves based on pentagon
5. The Reuleaux 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 particularly to the relationship of the circumference of a given circle to the perimeter of reuleaux curves from the odd number of sides of polygon. This area of research is another interesting topic in Geometry which may later be found significant and applicable in other fields i.e. combustion theory, non-Euclidian geometry (since the sum of the angles of a reuleaux triangle was not equal to 180 degrees but it was less than or greater than 180 degrees) and even to the foundation of modern geometry. Students, professors, scientists, and mathematicians will be benefited to the possible results of the study.

SCOPE AND DELIMITATION

This paper focuses on the construction of reuleaux curves and finding its relationship to the circumference of a given circle. This paper tends to present the proof pertaining to the relationship of a given circle to reuleaux curves (from odd number of sides of a polygon) in a different way. Derivations and expositions of some formulas are presented in this paper. Among the reuleaux curves considered in this paper are equilateral triangle, pentagon as well as n-gon with an odd number of sides.

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).

CHAPTER 2

REVIEW OF RELATED LITERATURE

The Reuleaux triangle Two of the round windows (IV and XII) contain the shape known as the Reuleaux triangle. Starting with an equilateral triangle, we draw three arcs each with centre at a vertex and radius the side-length of the triangle. If the triangle has side-length 1, the area of the Reuleaux triangle is ( – 3)/2 and the perimeter is .

The Reuleaux triangle is a curve which has constant width: it can be rotated between two fixed parallel tangent lines. There are many such curves; the circle is the simplest and best-known. Since a Reuleaux triangle has constant width, it can be rotated inside a square so that it always touches the four edges. This idea has been used to construct a drill which will drill square holes! Two arcs of the Reuleaux triangle give the characteristic Gothic arch; surprisingly, the triangle itself seldom features in Gothic architecture.

Working Model of Rotary Combustion

Some pages for Geometers

Eric W. Weisstein has a comprehensive set of math pages including the Reuleaux Triangle and Constant Width Curves. Alexander Bogomolny has a page on Constant Width Shapes. David Epstein's Geometry Junkyard has a discussion of the Reuleaux Triangles. Ivars Peterson expounds on Reuleaux Polygons. Mathsoft tells us that the Reuleaux Triangle has the largest Magic Geometric Constant.

William Chui has written a few pages about Franz Reuleaux(f) which are mildly interesting to those with a geometrical bent, though there is a slight confusion about what constitutes a Wankel rotary engine (RVICE creeps in).

A Curve of Constant Width constructed by drawing arcs from each Vertex of an Equilateral Triangle between the other two Vertices. It is the basis for the Harry Watt square drill bit. It has the smallest Area for a given width of any Curve of Constant Width.

The Area of each meniscus-shaped portion is

(1)

where we have subtracted the Area of the wedge from that of the Equilateral Triangle. The total Area is then

(2)

When rotated in a square, the fractional Area covered is

(3)

The center does not stay fixed as the Triangle is rotated, but moves along a curve

composed of four arcs of an Ellipse (Wagon 1991).

Kakeya Needle Problem

What is the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also allowed)? When the figure is restricted to be convex, Cunningham and Schoenberg (1965) found there is still no minimum area, although Wells (1991) states that Kakeya discovered that the smallest convex region is an equilateral triangle of unit height. The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by 180° is

(Le Lionnais 1983).

For a general convex shape, Besicovitch (1928) proved that there is no minimum area. This can be seen by rotating a line segment inside a deltoid, star-shaped 5-oid, star-shaped 7-oid, etc. Another iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).

Shapes of Constant Width

Yes - there are shapes of constant width other than the circle. No - you can't drill square holes. But saying this was not just an attention catcher. Now then let us define the subject of our discussion. First we need a notion of width. Let there be a bounded shape. Pick two parallel lines so that the shape lies between the two. Move each line towards the shape all the while keeping it parallel to its original direction. After both lines touched our figure, measure the distance between the two. This will be called the width of the shape in the direction of the two lines. A shape is of constant width if its (directional) width does not depend on the direction. This unique number is called the width of the figure. For the circle, the width and the diameter coincide.

The curvilinear triangle above is built the following way. Start with an equilateral triangle. Draw three arcs with radius equal to the side of the triangle and each centered at one of the vertices. The figure is known as the Reuleaux triangle. Convince yourself that the construction indeed results in a figure of constant width. Starting with this we can create more. Rotating Reuleaux triangle covers most of the area of the enclosing square. For the width=1 the following formula is cited in Eric's Treasure Trove of Mathematics (Oleg Cherevko from Kiev, Ukraine kindly pointed out a misprint in the original quote)

which looks pretty close to 1, the area of the square.

Extend sides of the triangle the same distance beyond its vertices. This will create three 60^o angles external to the triangle. In each of these angles draw an arc with the center at the nearest vertex. All three arcs should be drawn with the same radius. Connect these arcs with each other with circular arcs centered again at the vertices (but now the distant ones) of the triangle.

There are many other shapes of constant width. There are in fact curves of constant width that include no circular arcs however small.

CHAPTER 3

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. This chapter shall discuss the constructions of reuleaux polygons available for the study and what is applicable for it to use. Likewise, the chapter shall present how the research will be implemented and how to come up with pertinent findings. New terms were defined and examples were given. Results formulated were proven and illustrated.

Construction of Reuleaux Triangle

Johnrp (John P. of Middletown, NJ. 2000-10-08) stated that a circle is a two-dimensional object that has constant width; its height remains constant regardless of the orientation.

There are plenty of examples. The simplest is the so-called Reuleaux triangle, pictured at right and named after the German engineer Franz Reuleaux (1829-1905):

Just take the three vertices of an equilateral triangle and connect each pair of vertices with an arc of a circle centered on the third vertex. Joseph Emile Barbier (1839-1889) states that the perimeter of any curve of constant width is p times its width, which was an interesting theorem to be considered.

We can construct figures of constant diameter [constant width] from a regular polygon (with an odd number of vertices) by drawing small circles of radii R around each vertex and then drawing arcs from each vertex as to connect the two opposite circles at a tangent. There is a way to do this with an arbitrary polygon. For example, if I have a triangle that is not equilateral. There are plenty of such irregular curves of constant width and we call them irregrollers.

Note: It's probably better to use the accepted term "width" in this context. Although little confusion is possible with "diameter" here, the standard meaning is different and you may have a need to mention the diameter --the largest width-- in related discussions.

We may build such a shape around a scalene triangle ABC as follows. In this description, we assume, AC is the longest side and BC the shortest (AC>AB>BC).

1. Draw the arc of the circle (of radius AC) centered on C going from A to the intersection B' with the line BC.

2. Draw the arc of the circle (of radius AC-BC) centered on B going from B' to the intersection B" with the line AB.

3. Draw the arc of the circle (of radius AB+AC-BC) centered on A from B" to the intersection C" with the line AC.

4. Draw the arc of the circle (of radius AB-BC) centered on C from C" to the intersection C' with the line BC.

5. Finally draw the arc of the circle (of radius AB) centered on B from C' back to A.

The five arcs you've drawn make up the perimeter of a shape of constant width (W=AB+AC-BC). It has at least one sharp corner (2 in case of an isoceles triangle with a base AC larger than the other sides, and 3 sharp corners in case of an equilateral triangle).

If we want a smooth curve you may increase all of the above radii by the same quantity R. The construction is trivially modified by introducing only two points A' and A" at a distance R from A on AB and AC respectively. The construction starts with A" and ends with an arc of radius R from A' to A", to close the curve. Alternatively, we may describe the new "rounded" shape as the set of all points at a distance R from the (inside of) the previous shape...

We may want to notice that these curves need not involve any circular arcs at all. What we want is to have conjugate arcs (not necessarily circular) on opposite sides of our shape of constant width W so that if the radius of curvature at one point of an arc is R, the radius of curvature at the corresponding point on the other arc is W-R. This means the two arcs have the same evolute.

More precisely, if we roll a segment of length W on any curve we care to choose (without inflexion points) we obtain a pair of conjugate arcs as the trajectories of the segment's endpoints. (Conjugate circular arcs correspond to the degenerate case, where the above "base" curve is reduced to a single point, so the segment just rotates instead of rolling.)

When using such building blocks to make an actual shape of constant width, we only have to make sure the perimeter closes up into a convex shape (we could easily end up with some kind of double spiral).

Remarkably, a few symmetry remarks allow we to find immediately entire families of curves with constant width. Take the deltoid, for example (it does not have to be an exact deltoid; any curve with the same general features and symmetries will do): All its (closed) convex involutes are curves of constant width.

They look very much like the rounded equilateral triangles we mentioned in our question, but without any circular arcs on the perimeters...

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.