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Plücker Coordinates for the Rest of Us - Part 2
by Lionel Brits (15 November 2001)
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What is Specific to a Line?
The last installment did not venture beyond the familiar realms
of coordinate geometry to bring you what Plücker 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 plücker coordinates!
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
This system is called 6D Plücker Space. A directed line in 3D becomes a 6D
coordinate in Plücker 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 Plücker coordinate
that corresponds to the line through the point
and with the direction as:|
We can also get the Plücker coordinate for a line from point to by noting that . Then,
Which is equivalent to:
The Permuted Inner Product
Let us return to the situation of the two lines
Call their Plücker representations
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 Plücker 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 .
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.|