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*=*x*_{0}, *z*=*m*(*x*_{0})*y*+*b*(*x*_{0}), so we have surface *z*=*m*(*x*)·*y*+*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*)·(*y*–*px*). 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*+*q*–*px*)=*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*(*y*–*px*)=*axy*–*apx*^{2}. 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.

Tags: Affine Transformation, Doubly Ruled Surface, Geometry, Hyperbolic Paraboloid, Math, Monday Math, Plane

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