Monday Math 48

Previously, we noted that the hyperboloid of one sheet is one of only three surfaces to be doubly ruled; that is, to be able to be swept out by a moving line in more than one distinct way.
A brief bit of thought should give a second example of a doubly ruled surface, the most trivial of the three: the plane. Not only does the plane admit two distinct rulings (draw a grid on the plane for an example of two rulings), but infinitely many: rotate a ruling of a plane by any angle about an axis normal to the plane and you’ve generated a new, distinct ruling. In fact, the plane is the only n-ruled surface for n>2.

Now, how about the third surface?

Let us consider a line to be swept to form one ruling, and let us choose the line in one particular position to be the y-axis of our coordinate system. Now, let us choose an x-axis, and let us sweep our y-axis line such that all our lines in this first ruling are of constant x coordinate: that is, our lines are of the form x=x0, z=m(x0)y+b(x0), so we have surface z=m(xy+b(x), with continuous functions m(x) and b(x) with m(0)=b(0)=0.

Now, we want m(x) and b(x) such as to admit a second ruling. First, let us consider z=0, which occurs at  (note that the singularity at x=0 is removable). Here, we can choose our origin (and thus x-axis) so that it is the point where this curve crosses the y-axis: namely, so that , and b(x) goes to zero faster than m(x) as x→0.
To make this a line, we need , so that b(x)=-px·m(x), with some constant p.

Thus, our surface is z=m(x)·(ypx). Now, we want a family of lines swept from the line y=px, z=0, which also gives this surface. Let us examine the behavior of z when y=px+q, for q≠0 a constant. Then we have z=m(x)·(px+qpx)=q·m(x). We see immediately that if m(x) is a linear function of x, m(x)=ax (as m(0)=0), then these slices of our surface are automatically lines: z=aqx, y=px+q. Then our surface is just z=ax(ypx)=axyapx2. The p=0 case, z=axy, is a hyperbolic paraboloid. The p≠0 case is just this under a shear transformation, and is thus a hyperbolic paraboloid as well (as a shear can be formed from the composition of rotations and a non-uniform scaling). For each, there is an affine transformation that can map to the general hyperbolic paraboloid , and so the (general) hyperbolic paraboloid is our third doubly ruled surface.