Re: 4 points and a plane

Przemyslaw Koprowski <org@xxxxxxxxxxxxxxxxxxx> writes:

> Scott Hemphill wrote:
> > The equation you've written is the "line at infinity" in 2-D projective
> > space. In 3-D homogeneous coordinates, planes are represented by
> > a*x + b*y + c*z + d*w = 0, with (0,0,0,d) representing the plane
> > at infinity. What I did was to project onto 3-D so I wouldn't have to
> > introduce either "d" or "w".
> >
> Yeap, silly mistake. Sorry.
>
> For my excuse, I have been writing a chapter on plane
> projective curves for a last few of weeks. and it seems
> that I have now a total mental brainlock on projective triples.
>
> Anyway, the homogeneous equation of plane at infinity
> w = 0
> is probably more clear, since once you dehomogenize it,
> you take the affine subset of the plane at infinity, which
> is obviously empty. Hence the equation collapses to
> 1 = 0
> and this may confuse the reader who is not familiar with
> projective geometry (like apparently Roger, to whom we
> both answered).

I think you understand my motivation. I was trying to avoid appealing to
projective geometry and let the fact that the only 3-tuple in R^3 that
did not define a real plane was (0,0,0) suggest a reason that there was
only one plane at infinity. But there was a flaw: the equation
a*x + b*y + c*z = 1 omits planes that contain the origin.

Still, I wanted to address the question: if planes can have any orientation,
why isn't there a different plane at infinity for each orientation?

The plane represented by the 3-tuple (a,b,c) has an orientation determined
by its normal vector (a,b,c). This vector is also normal to the plane
represented by (a/n,b/n,c/n), but this plane is n times further from the
origin. As n grows without bound, the plane represented by (a/n,b/n,c/n)
is still normal to (a,b,c) but farther and farther from the origin. In the
limit, the representation goes to (0,0,0). We say this represents the plane
at infinity, and it is normal to any vector.

Scott
--
Scott Hemphill hemphill@xxxxxxxxxxxxxxxxxx
"This isn't flying. This is falling, with style." -- Buzz Lightyear
.

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