curves on quadrics
Dmitry Sustretov
is there a projective quadratic surface that contains two non-
intrersecting curves?
--
Dmitry
Timothy Murphy
> 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.
Caveat: I know almost nothing about algebraic geometry.
Dmitry Sustretov
> 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.
Mariano Suárez-Alvarez
wrote:
> Hello,
>
> is there a projective quadratic surface that contains two non-
> intrersecting curves?
The surface given by
x y - z w = 0
in P^3 is ruled, so it has lots and lots of
non intersecting curves.
-- m
Timothy Murphy
> 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.
Dmitry Sustretov
Oh, right, thanks. Now I know that this kind of surfaces are called
ruled.
Dmitry Sustretov
Right. By the way, are hyperboloid and hyperbolic paraboloid (that
have been mentioned) the only quadric surfaces that are ruled?
Ken Pledger
<f4b27853-16cf-45fb...@d1g2000hsg.googlegroups.com>,
Dmitry Sustretov <dmitry.s...@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.
Ken Pledger.
Mariano Suárez-Alvarez
> 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! ;-)
-- m
Ken Pledger
<92887ab2-0d8a-4296...@56g2000hsm.googlegroups.com>,
Mariano Suárez-Alvarez <mariano.su...@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.
Ken Pledger.