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Half Twist Ruled Surfaces




A familiar object in elementary Differential Geometry is the Moebius band which is useful in illustrating a non-orientable surface. One can easily make a Moebius band out of a flat oblong strip of paper by giving it a half twist and gluing its ends together. A square piece, however, would refuse to take shape unless it were cut because such a surface self-intersects. Any horizontal plane will cut this surface in the pattern of Figure 2 showing a double point D. This paper examines these double points for each arbitrary horizontal plane. It is seen that all these double points of this Moebius surface fall on a straight line.

The line PP' generates the surface as it passes along the red circle. The double points are evident where the red dotted line passes the self-intersection of the curve in a fixed z-plane.

The Moebius band described above is constructed as a ruled surface by using the straight line generator PQP' shown in Figure 1. A point Q moves along the circumference of a unit circle centered at O in the xy plane making angle t with the x-axis. A straight line PQP' makes an angle A = t/2 with the z axis. As Q traverses one complete revolution around the circumference, t has changed 360o while A has changed only 180o , so that line PQP' reverses direction to P'QP. The ruled surface generated by the ruling PQP' experiences a half twist resulting in a Moebius strip. Such a surface is one-sided since the normal at any point changes its sense during a complete revolution of Q: i.e., the inside of the surface has become the outside.


For a point P on the surface thus generated, let P' denote the projection of P onto the xy-plane and r = |OP'l. Angle QPP' equals angle A = t/2 so that the point P is located in cylindrical coordinates by the equation tan t/2 = (r-1)/z which is the equation of the surface. In Cartesian coordinates it is

(x2 + y2 + z2)y--2z(x2 + y2) + 2xz - y = 0.

An arbitrary horizontal plane z = b cuts the surface in a cross sectional curve as seen in Fig. 2.


A double point D occurs at the non-zero value of r1 = - r2 and the curve along which the surface self intersects is clearly the collection of these double points. By using the usual polar-Cartesian transformations as well as some elementary algebraic and trigonometric relations, it is now shown that this curve is the straight line y = z and x = 1.


The computations do not fit HTML consistently so are omitted, but they can be seen in my article on the subject which appeared in The American Mathematical Monthly,: "A Ruled Moebius Band Which Self-intersects in a Straight Line" and is cited below. Some of the illustrations here result from Mathematica programs written by Dennis Snow of the mathematics department at Notre Dame University.The conclusion is that all double points of this Moebius surface fall on the line y = z and x = 1, which is to say that this Moebius band self intersects along a straight line.
Further study is profitable for other fractions of the angle t besides t/2 by 3t/2, 5t/2, 7t/2 - all of which are unifacial. An even numerator, however, such as 2t/2, 4t/2, . . . results in a bifacial surface.

The Moebius band has always fascinated artists such as Escher, as well as today's advertising world, because of its peculiar unifacial property possessing only one edge. It forms an endless belt around which a column of ants could march continuously and in the past provided a mechanical devise for an endless eight-track cassettes tapes.

Escher's moebius strip frustrating a small colony of ants



The surface is usually formed by giving a half twist to a narrow strip of paper but all its properties are not obvious. When the strip width increases only a ruled surface will help visualize the one-sided property because intersects itself along a straight line. We find that this process will not work for a square sheet of paper as it refuses to take shape unless it is cut because the moebius surface must self-intersect or it will self-destruct. This square sheet of paper is a useless model but rulings pass comfortably and unhindered between each other.

The formulas needed for making the ruled moebius strip

Two mathematicians, Gaspard Monge (a graduate of the Jesuit school in Lyons, France) and Arnold Emch worked out the necessary mathematical formulas for the fractional (p/q) twist surfaces. In particular I start with p/q = 1/2 for the half twist or "moebius" surface. Monge showed that ruled surface equation satisfies the partial differential equation
x2zxx + 2xyzxy + y2zyy = 0

and also that the solution to this PD equation must take on the algebraic form
z = xF(y/x) + G(y/x)


Then, Emch described such a surface using the z-axis as C1, the first generator and C2, a circle of radius a lying in the xy-plane as the second generator. A ruling passing through the z-axis at an anglea meets C2 at point m. A constant angle alpha would form a cone, but if the angle a varied as the point m moved around the circumference C2 so that a = pt/q, a p/q twist surface results. The equation for such a surface is given in cylindrical coordinates (r, t, z).

r = a + z tan pt/q (or z = (r-a) cot pt/q)


Using tedious but cunning manipulation of algebra and trigonometry formulas Emch derives formulas needed to prove his theorems and determine the order of the surface (the number of times a straight line intersects the surface) and the degree of the surfaces' equations. He also studies the curves of self-intersection made by all the p/q twist surfaces. To verify the theorems and illustrate the interesting properties he describes, however, the algebra and calculus required is daunting and heuristic geometrical models are not available. Computer programs to study the p/q twist surfaces such as MATHEMATICA has the power not only to display these elusive properties, thereby making these interesting surfaces accessible, but also to display the surface and the curves of self- intersection on a screen. MATHEMATICA can also illustrate any number of rulings for the cylindrical and cartesian coordinates, thereby facilitating the actual construction of these models enclosed in a box or in a cylinder.

The moebius surface enclosed in a box

A Moebius surface is generated by a line PQP', which pivots about point Q, making an angle t/2 with the vertical z axis as Q moves around the circumference of circle C of radius a in the xy plane. Angle xOQ = t. This means that Q moves about the circle one complete revolution, while line PQP' makes only half a rotation about the point Q.

After one trip around the circle, line PQP' takes a new position, P'QP, so the surface thus generated has undergone one-half twist. In so doing, it has intersected itself along a straight line. Using elementary analytic geometry, we arrive at the following equations for the three coordinate systems:
rectangular: y(x2+y2+z2)-2z(x2+y2)+2azx - a2y=0

cylindrical: z = (r-a) cot t/2

spherical: R cos f= (cot t/2)(R sin f - a)


The rectangular model can be constructed between two pairs of planes x = + (or-) xo and z = +(or-)zo, but the computation is more easily done using the cylindrical coordinate on the plane faces. The projection of the surface at these planes using parameters r and t results in the equations:

Plane x=xo: z=(xo sec t-a)cot (t/2): y=xotan t

Plane x=-xo: z=-(xo sec t+ a)cot(t/2): y=-xotan t

Plane z= zo: r= zo tan (t/2)+ a

The curves projected on planes zo are mirror images of one another, while the curves in planes xo differ slightly. Points are marked along the curve for every 10 degrees difference in t.
After the four planes are assembled, pairs of points are connected by a ruling starting from the point (a, 0, zo) in the z = zo plane to the point (a, 0, -zo) in the z = -zo plane. Then the sequence of points in the upper z=zo plane moving counterclockwise are connected to the sequence of points in the lower z=-zo plane, also moving counter-clockwise, until the surface is completely constructed. The three patterns for the corresponding points for the planes x=xo, x=-xo and z= zo are seen in the figures below.

The position of the points for constructing a rectangular model
of the half twist surface.


The moebius surface enclosed in a cylinder

The key to success using a cylindrical frame is the orientation of the cylinder. If the axis of the cylinder were along the z axis too many points would be missing since the tangent of angles around q = + k90o would be too large for the cylinder of height 4b. So the axis of symmetry chosen for the cylindrical model is the y axis. Then x = b cos t and z = b sin t where phi is the polar angle in the xz plane. So the coordinates for the drilling pattern are (bf,y) and the values of the 36 points in these coordinates must be put in terms of the variable q. To accomplish this, some rather complicated variable shifting is required, but since these equations (4), (5) and (6) can be fed into a computer, tedious computations are avoided. The major steps in developing these equations are stated below.

Use of the polar-Cartesian relation r = x sec t and the circle in the xz plane

x2 + z2 = b2 produce the equation:
tan t/2= (x sect -a)/{b2-a2}.5
Squaring this, we have

(b2-a2)tan2t/2 = x2sec2t - 2axsec t + a2

Rearranging the terms would produce the equation

x=(a sec t + tan t/2{b2 sec q+b2tan t/2 - a2}.5)/(sec2 t+tan2 t/2)
which indicates the location of the points for the pattern to determine the position of the pairs of points to be joined.

The position of the points for constructing a cylindrical model
of the half twist surface with 36 lines





The moebius surface enclosed in a sphere

In some ways the construction of a moebius surface in spherical coordinates is easier. Since the sphere has a fixed radius Ro, the first requirement is a set of points in the other two variables (t, f). By applying trigonometric identities to the third form of the moebius equation given above, the function f = f(t) is found:
t = { arctan Ro2 tan t/2 ±a [Ro2(tan2 t/2 +1)-a2].5 }/ (R2 - a2)
The resulting curve on the spherical surface indicates the location of the pairs of points to be joined, using each corresponding phi-value for 36 values of t between 0o and 360o. These points are plotted on 2 hemispheres. Because of trigonometric symmetries, the curves on the upper and lower hemispheres are identical, but 180o out of phase. So the curve can be plotted on the two hemispheres, which are then fastened together after a 180o rotation so that the curves form spherical mirror images of each other. The 72 holes are paired off by taking the first set of connecting points (Ro, 0, p/2) and (Ro, p, p/2), which will be a diameter, the only horizontal line, and in fact the only line passing through the center of the sphere. Adjacent pairs of points are then connected by a line so that each ruling has only one point on each hemisphere. This is repeated until the rulings are completed.

The position of the intersections with the surface in constructing
a spherical model of the half twist surface



Half twist surfaces


References


[1] A. Emch, "On a certain class of rational ruled surfaces",
American Journal of Mathematics, 42 (1920) 189-210

[2] J. MacDonnell, S.J., "A ruled Moebius band which self-intersects in a straight line",
American Mathematical Monthly, 91 (1984) 125-127




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