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Scrolls, or ruled surfaces, are surfaces generated by straight lines or rulings and have been studied for centuries by geometers such as the Jesuits Roger Boscovich and Andre Tacquet as well as by their famous students such as Gaspar Monge and Phillippe de Lahire. Before the computer age these surfaces were almost impossible to construct because the complicated computations of their formulas placed them out of reach. The trajectories of these six scrolls on cylindrical shells have been computer generated and each of them relate somehow to a conic section. Some of the illustrations result from Mathematica programs written by Dennis Snow of the mathematics department at Notre Dame University.

Warning: html seems to have no Greek symbols so I use Pi for 3.14159265358979323846264338327950288419716939937510 (approx.).

Ruled surfaces include hyperboloids, saddles, helicoids and moebius bands. Of the different kinds of ruled surfaces, three stand out because of their striking shape and relative simplicity of construction; the saddle, the helicoid and the moebius band. The use of spherical, cylindrical and rectangular coordinates all have their advantages and disadvantages, depending on the surface in question. Until computers an insurmountable difficulty has been complicated equations and tedious computations, which are evident from comparing a few surfaces in these three different coordinates. The three forms of the helicoid equation are:

y = x tan(z/k) z =k t R = k t sec A

The three forms of the saddle equations are:

kz = x ^{2}- y^{2}kz =r^{2}cos 2t R = k sec t cot A csc A

On the other hand very dissimilar surfaces may have similar equations if they are compared in one system. An example of this equation similarity is seen in the three different surfaces expressed in polar coordinates.

z = k tis a helicoidz = cos tis a saddlez = (r-a)cot tis a moebius band

#1 The Moebius stripStudy and construction of surfaces like the latter ( z = (r-a)cot t) can be enclosed in a cylinder: in fact it is not difficult to construct such twist surfaces with many half twists. The process starts with the creation of a pattern drawn on a rectangular sheet of paper in (z, t) coordinates. This is pasted around an acrylic cylinder, holes are drilled at the designated spots and the holes are connected with cord, producing the ruled surface. No matter which coordinate system is chosen, however, the object is still to find a pattern from the projections of the surface on a box, a cylinder or a sphere. Then one connects pairs of points on the shell in the proper sequence by straight lines or rulings. For more on this kind of surface see Half Twist Ruled SurfacesThe Moebius strip has always fascinated artists such as Escher, as well as the advertising world, because of the fact that it is unifacial and only has one edge, and therefore forms an endless belt around which a column of ants could march continuously. It was the mechanical devise used in the endless eight-track cassettes tapes The surface is usually formed by giving a half twist to a narrow strip of paper and all their properties are not visible. This scroll intersects itslef along a straight line as the rulings pass between each other.

One half twist

in a cylinderTwo half twists

in a cylinder

#2 The Bifacial SurfaceTwo half twists

in a cylinderis generated in a way similiar to the unifacial surface (#1) 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 scroll highlights symmetries and other properties when suspended from above and twirled around.For more on other p/q twist surfaces see p/q Twist Ruled Surfaces

#3 The right helicoidis a ruled surface swept out by a line which always intersects a fixed axis at right angles and which rotates uniformly as its point of intersection moves uniformly along the axis. It intersects any cylinder concentric with the axis in a helix. The right helicoid has the shape of the thread of a screw. Line L is perpendicular to the z axis, and as its intersection P moves along the z axis, L rotates about the axis, as seen in Figure 1. The right helicoid is a minimalsurface. In fact, it is the only ruled minimal surface apart from the plane and this surface is used in the construction of radio antennae. The intersection of the helicoid with a cylinder produces a helix, which would be equivalent to the railing of a spiral staircase. To construct a simple ruled surface of constant pitch choose for the line of striction a straight line that meets all the generators of the surface at right angles. Let d be the pitch of the surface. If two generators make an angle alpha with each other, and if A and B are the points where they intersect the line of striction, then alpha = AB.d. The surface is transformed into itself by a screw motion of pitch d with the line of striction as the axis. The ruled helicoid obtained when the generating curve is a straight line intersecting the axis at right angles is called a right conoid.

The helicoid has a wide variety of shapes, depending on the pitch, the proximity of the lines and the points, and whether the pairs of points connected by the rulings are in fact not quite diametrically opposite.

In conclusion, we shall construct a particularly simple ruled surface of constant pitch. Here the obvious choice for the line of striction is a straight line that meets all the generators of the surface at right angles. Let d be the (constant) pitch of the surface. If a and b are two generators making an angle a with each other, and if A and B are the points where they intersect the line of striction, thena = ABxd.

Hence the surface, which is called a helicoid, is transformed into itself by a screw motion of the pitch d with the line of striction as axis. The most general helicoid is the surface swept out by an arbitrary space curve performing a uniform screw motion about a fixed axis. Thus our particular ruled helicoid is obtained when the generating curve is astraight lineintersecting the axis at right angles.Boxed HelicoidDouble Helicoid

in a cylinderHelicoid

top view

#4 The Saddle ScrollSaddle

Scroll

is a doubly ruled surface which is generated by two distinct meshes of lines which are skew but appear parallel when viewed from the direction of the top. This surface represents an important point in applied mathematics called the "saddle point" found at its center which is a maximum point in one plane but a minimum point in another plane, but actually is neither for the surface. For more on saddle surfaces see Saddle Surfaces

#5 The Focal Line Scrollis generated by tracing lines along two ellipses whose axes are at right angles. This surface has the unusual property that two lines perpendicular to each other are found where the rays come to a focus in their respective planes. Light rays reflecting off a sphere are represented by this kind of surface. Focal Line

Scroll

#6 The Hyperboloidal ScrollHyperboloidal

Scrollis generated by lines connecting points on parallel ellipses.This surface has the unusual property that it is doubly ruled - there are not one but two families of rulings. Each family covers the surface completely. Every straight line of one family intersects every straight line of the other (or is parallel to it), but any two lines of the same family are mutually skew. Three skew straight lines define a hyperboloid of one sheet.

These two families have a surprising property. If this model were made of lines which intersect in a way which allows rotation but not of sliding, it would seem that straight lines fastened in this way would form a rigid frame. This is not the case, however, the framework is movable. As the distance between the parallel planes and ' increases the angle between the asymptotes of the generating hyperbolas increases. Two such surfaces are used in cogwheel transmission.

An assortment of Ruled surfaces

## References

1 H. M. Cundy and A. P. Rollett,Mathematical Models(London: Oxford University Press,1961)

2 Emch, Arnold,Am. J. Math.,42, 189

3 Emch, Arnold,Mathematical ModelsUn Il. 1920

4 Eves, HowardA Survey of Geometry(1974)

5 D. Hilbert and S . Cohn-Vossen,Geometry and the Imagination(New York: Chelsea, 1952)

6 G. James and R. James,Mathematics Dictionary(New Jersey: Van Nostrand, 1949)

7 Klein, Felix ElementaryMathematics from an Advanced Standpoint.

8 Lipshutz, M. M., 1969,Differential Geometry(New York: McGraw-Hill)

9 MacDonnell, J. F., 1980, Int. J. Math. Educ. Sci. Technol.,11, p. 493.

10 MacDonnell, J. F.,The American Mathematical MonthlyFeb. 198491#2 p 125-

11 Olmsted, J.Solid Analytic Geometry(New York: Appleton Century Crofts, 1947)

12 O'Neill, B. ,Elementary Differential Geometry(New York: Academic Press, 1966)