Dmitry Sustretov wrote: > On May 16, 4:16 pm, Timothy Murphy <gayle...@eircom.net> wrote: >> Dmitry Sustretov wrote: >> > is there a projective quadratic surface that contains two non- >> > intrersecting curves?
>> Suppose you take the surface x^2 - y^2 + z^2 - t^2 = 0. >> Then the lines x = y, z = t = 0 and x = y = 0, z = t >> don't meet.
> The loci of these equations are points, not lines.
You're quite right, of course. I should have said something like
x = y, z = t and x = -y, z = -t.
Basically, I was thinking of a ruled surface, as has been pointed out.
In article <f4b27853-16cf-45fb-8559-3df2021bf...@d1g2000hsg.googlegroups.com>, Dmitry Sustretov <dmitry.sustre...@gmail.com> wrote:
> .... By the way, are hyperboloid and hyperbolic paraboloid (that > have been mentioned) the only quadric surfaces that are ruled?
Your geometry here is presumably affine or Euclidean, not purely projective.
The hyperboloid of one sheet and the hyperbolic paraboloid are the only proper (non-degenerate) quadrics with real rulings, yes. Cones, cylinders and plane-pairs also have real rulings, and if you allow complex coordinates then even ellipsoids are ruled.
On May 18, 7:04 pm, Ken Pledger <ken.pled...@mcs.vuw.ac.nz> wrote:
> In article > <f4b27853-16cf-45fb-8559-3df2021bf...@d1g2000hsg.googlegroups.com>, > Dmitry Sustretov <dmitry.sustre...@gmail.com> wrote:
> > .... By the way, are hyperboloid and hyperbolic paraboloid (that > > have been mentioned) the only quadric surfaces that are ruled?
> Your geometry here is presumably affine or Euclidean, not purely > projective.
Why?
> The hyperboloid of one sheet and the hyperbolic paraboloid are the > only proper (non-degenerate) quadrics with real rulings, yes. Cones, > cylinders and plane-pairs also have real rulings, and if you allow > complex coordinates then even ellipsoids are ruled.
Mostly because when working over the complex numbers there is no difference between an ellipsoid and an hyperboloid! ;-)
In article <92887ab2-0d8a-4296-8034-114c86aae...@56g2000hsm.googlegroups.com>, Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com> wrote:
> On May 18, 7:04 pm, Ken Pledger <ken.pled...@mcs.vuw.ac.nz> wrote: > > In article > > <f4b27853-16cf-45fb-8559-3df2021bf...@d1g2000hsg.googlegroups.com>, > > Dmitry Sustretov <dmitry.sustre...@gmail.com> wrote:
> > > .... By the way, are hyperboloid and hyperbolic paraboloid (that > > > have been mentioned) the only quadric surfaces that are ruled?
> > Your geometry here is presumably affine or Euclidean, not purely > > projective.
> Why? ....
In projective geometry there's only one kind of proper quadric. To switch to affine geometry you choose an absolute (plane at infinity); then a hyperboloid meets that plane in a real proper conic and a hyperbolic paraboloid meets it in a real line-pair.
