Re: Families of Straight Lines
- From: Timothy Little <tim-usenet@xxxxxxxxxxxxxxxxxx>
- Date: Sun, 14 Aug 2005 09:58:27 +0000 (UTC)
Ulysse Keller wrote:
><mauriziobarbato@xxxxxxxx> wrote:
>> I'm thinking about some particular families of straight lines, which
>> I called "closed under intersection".
>
> do you have a non trivial example of such a family ?
One example would be a family of lines passing through a point, with
another line intersecting them. Another would be a square with lines
through each diagonal.
The latter example is closed under intersection in the Euclidean
plane, but not under the extended Euclidean plane with parallel lines
intersecting at a directional "point at infinity".
In the extended plane, the "closure under intersection" of a unit
square at the origin consists of the set of lines passing through
every pair of rational points.
Closure under intersection is preserved by projective transformations,
and any convex quadrilateral can be mapped to a square by such a
transformation. Hence the closure of any convex quadrilateral will be
a projective transformation of such a diagram. For concave
quadrilaterals, there is always a convex quadrilateral
Adding extra points to the diagram will add more lines to the set, but
a countable set of initial points will always have a countable set of
lines in the closure which will therefore never cover the whole plane.
- Tim
.
- References:
- Families of Straight Lines
- From: Maury Barbato
- Re: Families of Straight Lines
- From: Ulysse Keller
- Families of Straight Lines
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