Home Site Map Tutorial Material Of Surfaces and Skews (2)
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 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).The skew lines would, in such a surface,
be required to be
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. You can “spin” the whole thing by clicking and dragging the dashed blue line. You can drag the centres of the circles and adjust their diameters. but it is neither a surface, nor smooth. It is a |
The view below is from ∞ through infinitely powerful binoculars. |
Rigid Rotation is not available to projective geometry.
If, in the physical world,
there is rigid rotation, it does not come from geometry. From what, then? |