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Of Surfaces and Skews (1)

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.

Please enable Java for an interactive construction (with Cinderella).

Consider two mutually skew lines–any two, say, of the four above.

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

They should therefore have that surface in common

The skew lines would, in such a surface,
be required to be
incident [ projection ] (in order to form a differentiable continuum), and non-incident (in order to remain skew).

We have a contradiction. They cannot 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.


This is a “much-ruled” net of skews.
(i.e., Lines join many “nodes”- those joining distinct node-pairs are skew. The lines are artificially truncated for clarity - imagine one circle behind the other.)

You can “spin” the whole thing by clicking and dragging the dashed blue line, or by using the animation controls. You can drag the centres of the circles and adjust their diameters.

It may look like a continuous, smooth surface - it is commonly taken to be one -
but it is neither a surface, nor smooth.


The view is from ∞ through infinitely powerful binoculars.

Please enable Java for an interactive construction (with Cinderella).


Rigid Rotation is not available to projective geometry.

The rotation imparted to the figure is ‘illusory’, in that it is achieved by calculation based on
assumed equality and isotropy,
not by projection.

Under true projection, radii out from the “spin axis”, such as MP, could not be constant.

If, in the physical world,
there is rigid rotation,
it does not come from geometry.

From what, then?
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