Re: Strange problem about double points (difficult?)

"computer" <computer@xxxxxxxxxxxxxxxxx> writes:

>In real projective plane I call a point {t(x,y,1)} PROPER if it comes from a
>real point (x,y) and IMPROPER (or a direction) if it comes from a direction
>(m,n) (a vector in R^2) and is represented as {t(m,n,0)}.

Okay.

>Suppose to have the following degenerate conic:
>
>(x-y)(x-y+1)=0
>
>that in the affine plane R^2 represents two parallel lines.
>In projective (pluckerian) coordinates:
>
>(x_1 - x_2)(x_1-x_2+x_3)=0.
>
>The matrix related to this conic is
>
> 1 -1 1/2
>-1 1 -1/2 = A
>1/2 -1/2 0
>
>and has rank 2. The problem is that the solution of
> y_1
>A* y_2
> y_3
>
>yields to the solution (1,1,0) (improper point, a direction) which should be
>a PROPER double point.

The "problem" is not that the intersection of the projective completions
of two parallel lines is a point at infinity (what you call an improper
point), it is that you think it "should be" otherwise.

>I always thought

, for some reason you have not given us (if you would like to
give your reason, I would be happy to hear it),

>that degenerate projective conics
>should have at least one proper double point, like these:
>
>\ /
> \ /
> \ /
> \ /
> X ---> this is the double point,
> / \ CASE 1: degeneration in two distinct real lines.
> / \
> / \

This case is in fact the case you are studying, except
that the double point is on the line at infinity (which,
since you specified above that you are thinking about
"projective conics", is not privileged above any other
line).

> \
> \ ----> CASE 2: the conic degenerate in two parallel lines.
> \ It has infinite proper duoble points.
> \
> \

This case you seem to have described incoherently. What
you drew (given that it is meant to be a degenerate conic)
does, indeed, have "infinite[ly many] proper double points";
but it is *not* "two parallel lines", it is *one* line counted
with multiplicity 2.

> X -----> CASE 3: the conic splits in two immaginary conjugate lines.
> These lines meet in a real point, that is a double point.

Lee Rudolph
.

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