Discussions(1) Swept Surfaces: a Euclidean cuckoointhePGnest?Consider this, from the marvellous Wolfram: MathWorld.
The article
describes the hyperboloid as “a surface of revolution.” It is a
fine example
of a socalled “swept surface”, made  somehow  by a plane curve swung bodily around an axis
in the curve's plane. The ‘somehow’ is not discussed.
Such surfaces are disallowed
by pure projective geometry,
as there is no projective way to move (transform) an entire, intact curve in the requisite fashion — — let alone a way to have it leave a trace (a ‘locus’), that somehow ‘builds’ the surface. In particular, in this instance, we see the notion of preserved (constant) radii deployed – perhaps instinctively, or from habit. Now this is a purely Euclidean notion, as size, whether or not constant, is not preserved under projective transformation, so constant radii are simply unavailable.
In consequence, we must conclude for the same reason (viz, that size
is not preserved), that the structure of skew lines between our skew rulers
cannot be a swept surface.
Moreover, the hyperboloid structure (which is actually built of nonincident skew lines), cannot be a continuous surface. In fact, it cannot be a surface at all. See the panel on the right. 
This cannot become a continuous surface,because it is a structure of skews, with each
