It is generally held that a line moving skew to itself sweeps out, or generates, a surface—a hyperboloid, perhaps.

Now, a hyperboloid is widely supposed to be a doubly-ruled surface of revolution—
that is, one allegedly swept out by a line moving skew to itself.

But the axioms of projective geometry categorically disallow surfaces of revolution.

The surface simply cannot be made projectively.

It is instructive to see in some detail why a surface could in any case not be formed by a skew,
even if it were allowed by the axioms to revolve “rigidly”,
meaning that all the figure's radii from the axis of rotation would stay constant.

Consider two skew lines.

Being skew, they have no geometric elements in common (that is, no incidence of any kind with each other).
But they are said to be, while rotating, “sweeping out a surface”,
sometimes called the ‘locus’ of the line.

The skew lines would, in such a surface,
be required to be

1. incident ^{[ * ]} (in order to form a differentiable continuum),


and, simultaneously,


2. non-incident (in order to remain skew).

We have a contradiction. They cannot simultaneously be both.

This is exactly equivalent to stating that a hyperboloid surface cannot be made by integration,
- that is, as the limit of an infinite summation of planar infinitesimals -
because the requisite planes cannot exist.