Actually, Adam's parametrization makes it clear that the surface is doubly-ruled, since by symmetry the expressions for x and y can be interchanged to get the same surface. --Dan Adam wrote: <<

It occurs to me to ask: Is there a parametrization of the one-sheeted hyperboloid that makes it manifest that it's a ruled surface?

I believe the following is a valid parametrisation, for much the same reason as the hyperbolic slot demonstration works: x = p cos(theta) + sin(theta), y = p sin(theta) - cos(theta), z = p, where p is a real number; theta is in the interval (-pi,pi]. When theta is constant, varying p corresponds to motion along a straight line. Substituting these expressions into x^2 + y^2 = z^2 + 1 confirms that this parametrisation does indeed describe the hyperboloid.

And: Is there a parametrization that makes it manifest that it's a doubly-ruled surface?

Yes, if there are twelve constants, {a,b,c,d,e,f,g,h,i,j,k,l}, . . . . . .

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