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p/q Twist Ruled Surfaces



Figure 1

Figure 2

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.


This is a study concerning other p/q fractions of the angle t besides t/2 such as 3t/2, 5t/2, 7t/2, t/5, etc.



The Case: 2 half twist surfaces


A bifacial surface is generated in a way similar to the unifacial (moebius) surface but has two edges and two sides. This surface has the unusual property that although each point of the generating circle has one unique generator, each line has another line parallel to it. This surface highlights symmetries and other properties visible when suspended from above and rotated. Here is shown the pattern needed for such a 2 half twist surface to be enclosed in a cylinder.

Figure 3 The pattern for two half twists



The case of 1 to 8 half twist surfaces


Surfaces with the equation
z = (r-a) cot pt/2
where the numerator p takes on the values: 1,2,3,4,5,6,7,8 have been constructed and enclosed in a sphere. The theory is an extension of the moebius strip in the first case p="1". This Moebius surface of one half twist occurs when the angle between generator PQP and the vertical is t/2. If the angle a was changed to kt/2 where k is an odd integer, there would be k half twists and the above formulas would apply if t/2 were replaced by kt/2. A longer cylinder is required: e.g. for k= 3, a cylinder of radius b would have to be 5b long instead of 4b. But if k were even, the surface would be two sided and not a moebius surface and instead of self-intersecting, it would swing around and become tangent to itself. It is possible to create this model in the same way but because of the formula, many points would be missing near t=90 degrees. This can be avoided by aligning the axis of the cylinder with the x axis instead of the y axis. The development of the equations is similar but the points on the surface of the cylinder would have coordinates (t, x) instead of (f, y) and the 36 values of t would be fed into the formulas:

f=arccos (y/b)
x=b cos f cot t and
y=(a csc t + (or -) tan t/2{b2 csc t+ b2 tan t/2 - a2}.5)/(csc2 t + tan2 t/2)

The construction of such a surface follows the method described for the surfaces with an odd number of half twists. Here is the set of all eight half twist patterns needed for a sphere.

Figure 4 The patterns for 1, 2, 3, 4, 5, 6, 7 and 8 half twists for a spherical frame





The case of other p/q twist surfaces enclosed in a cylinder

Other p/q twist surface, e.g. 1/5, 1/7, 3/4, 2/5 are made by using the patterns shown below, wrapped around a cylinder as was done previously, to provide the connecting points for the rulings. From Mathematica programs, some of the patterns used for case 1/5, 1/7, 2/5, 3/4 are shown here

Figure 5 The patterns for 1/7 and 5/2 twists



Some p/q twist models are shown below in
rectangular, spherical and cylindrical frames.




p/q twist surfaces

1/2 twist 2 half twists in a sphere 3/2 twists in a cylinder
1/4 twist in a cylinder 4/2 twists in a cylinder 1/5 twist in a cylinder
2/3 twists in a cylinder2/2 twists in a cylinder 2 half twists in a box


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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