## Monday Math 47

I imagine that quite a few of you have previously encountered string art, where curves are approximated by a number of straight lines spaced along some pattern (see also here). Most string art is two dimensional, but I’ve also encountered some three dimensional cases. Here, we instead have the lines of string approximating a surface. Such a surface is a ruled surface; through each point on the surface there exists a straight line that lies on the surface. The set of such lines is called a ruling of the surface.

Let us consider a pair of parallel, congruent circles perpendicular to the axis connecting their centers. Let us choose our coordinates so that we have circles of radius R, parallel to the xy-plane with centers on the z axis at z=±h. If we connect corresponding points on the circles with straight lines, then we have a cylinder. Suppose, instead, that we then “twist” the circles, so that we thus have lines connecting points on the circles offset by some angle, which we will call θ. We noted that θ=0 gives a cylinder; one should also see that if θ=π, we have a double cone. So we consider 0<θ<π.

Picking one of the lines (one with a symmetrically placed position), we choose the
line through the points  and  on the circles. Describing this line parametrically, with parameter t such that we have p1 at t=-1 and p2 at t=1:
.
Now, via symmetry, we see the other lines consist of the image of this line under a rotation about the z axis. So, rotating this line by an angle φ about the z axis, we get


So, now, we can use this equation to define a parametric surface, with u (a renamed t) and φ as the parameters:
, with -∞<u<∞ and 0≤φ<2π.

We want to find a non-parametric equation for the surface; to do so, we seek to relate x, y, and z so as to eliminate u and φ. First, we use x(u,φ) and y(u,φ) and solve for cos(φ) and sin(φ):




If we square these and sum them, we obtain




Now, using z=hu, u=z/h, and we have:


This is a one-sheeted circular hyperboloid:

with  and  (as 0<θ<π, a,c>0)

Thus the circular hyperboloid of one sheet is a ruled surface. A suitable (non-uniform) scaling of our y coordinate will show that the general hyperboloid of one sheet x^2/a^2+y^2/b^2-z^2/c^2=1 (the elliptical hyperboloid when a≠b) also has the same property.

Now, note that using a ‘twist’ angle of -θ in our procedure generates the same hyperboloid as θ. Thus, the hyperboloid of one sheet is a doubly ruled surface (not just the circular, but the general hyperboloid of one sheet), a surface with two distinct rulings. It is one of only three such surfaces; does anyone care to guess the other two?

This property of hyperboloids of one sheet leads to them being used in structures, particularly towers, far more than one might realize. For example the classic shape for cooling towers of nuclear power plants is that of a segment of a hyperboloid. For other examples of hyperboloid structures, see here.