Re: Question about Pluecker coordinates

In article <5b5ca8a6-c625-4f9e-a4da-
3a58844a1f3b@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, anonimo.passante@xxxxxxxxx
says...
You can specify a line in 3D space by a point (3 numbers) and a
direction (2 numbers), but then you have one degree of redundancy
because any point on the line is equivalent.

Good point, there's clearly redundancy.

To see intuitively that 4 numbers are sufficients, think so:
- direction of the line: 2 numbers -- you get a doubly-infinite pencil
of parallel lines;
- distance from the origin: 1 number -- you get a single-infinite
pencil of parallel lines tangent to a sphere;
- angle in the plane containing the origin and normal to the lines: 1
number -- you get one single line.

Hmm.. Something still worries me. The problem is the last one: angle
with respect to what?

Usually an arbitrary dimensional treatment shows off structure better
than some specific dimension. So, assume an n-dimensional space. To
specify a direction you need (n - 1) numbers. A vector N with that
direction has (n - 1) - dimensional orthogonal complement. Thus it would
seem that you need (n - 1) + (n - 1) = 2n - 2 numbers:

1d -> 0
2d -> 2
3d -> 4
4d -> 6
etc.

0 makes sense for 1d, because there is only one line. 2 is ok for 2d,
use an angle and a distance from the origin.

But in 3d and higher... It seems like there are always singularities. If
you construct an orthonormal basis with N as one vector, you still have
the freedom to rotate the basis around N. To fix this, you could choose
another vector U from the orthogonal complement, but no single vector U
works for all N (when N = kU). How to fix this?
.

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