flipcode - Pl�cker Coordinates for the Rest of Us

Pl�cker Coordinates for the Rest of Us - Part 2
by Lionel Brits (15 November 2001)

What is Specific to a Line?

The last installment did not venture beyond the familiar realms of coordinate geometry to bring you what Pl�cker coordinates are good for. Don't look too disappointed. We can, however, proceed further in this matter. Again, take a look at the orientation

.

Notice that

is independent of the line

. It is specific to the line

. The direction of

, that is,

, is also specific to

. Since we chose the point

rather arbitrarily as being any point on the line

, we don't regard it as being specific to

. Enter pl�cker coordinates!

Pl�cker Space

Now that we have determined what is specific to a line, we can convert to a system in which we can readily express the orientations of lines in terms of

and

. This system is called 6D Pl�cker Space. A directed line in 3D becomes a 6D coordinate in Pl�cker Space. It is conventional however to refer to it as a homogenous coordinate in 6D in that all lines through the point

with direction being some multiple of

are equivalent. (A homogeneous coordinate in 6D becomes a non-homogeneous point in 5D). We can now define the Pl�cker coordinate

that corresponds to the line through the point

and with the direction

as:

We can also get the Pl�cker coordinate for a line from point

by noting that

. Then,

Which is equivalent to:

The Permuted Inner Product

Let us return to the situation of the two lines

and

. Call their Pl�cker representations

and

respectively.

We gave the orientation of with respect to as

An important identity of vector products is that which allows us to write

It would be sensible to look for symmetry in this expression, and indeed we find that since , we can write

With this in hand we can do away this the negative sign altogether by defining a new product .

This is the permuted inner product of two Pl�cker coordinates. If we rewrite as , then the product of two coordinates representing lines and , that is, and is given by

.

We can now summarize the relative orientations of lines in terms of , recalling that it is the negative of .

	and pass each other counterclockwise
	and pass each other clockwise
	and intersect and/or are parallel or antiparallel to one another

But Where Did My Lines Go?

This is all good and well, I hear you say, "but isn't this a one-way trip?" Lo and behold, only two installments so far and you're already eager to jump the gun. Jump over to the next installment, if you dare.

Article Series:

Pl�cker Coordinates for the Rest of Us - Part 1
Pl�cker Coordinates for the Rest of Us - Part 2
Pl�cker Coordinates for the Rest of Us - Part 3
Pl�cker Coordinates for the Rest of Us - Part 4 - Applications