Plücker Coordinates for the Rest of Us - Part 1
by Lionel Brits (15 November 2001)
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|When I first happened upon articles dealing with Plücker Coordinates I found it hard to interpret the mathematical expressions geometrically and many of the ideas were intimidating. Since I couldn't find any lighter material on the subject I put off familiarizing myself with the concepts for more than a year. I then revisited the articles but still found them full of big words and unfamiliar notation, so I tried to imitate, by my own means, what Plücker coordinates promised: a way of mathematically describing the relative orientations of lines. After spending a few hours trying to make my scribblings on paper look 3D I ended up with expressions that looked suspiciously like those I had seen in the other articles. After a few more hours of scribbling my arrogance got the best of me and I decided to contribute my own article to the white noise that is the world wide web. Lo and behold, you are reading that very article. Good luck.|
A Word on Convention
For the sake of consistency we will be using the the right-hand coordinate
system throughout this discussion.|
Also, I chose notations to be reasonably consistent with other sources whenever possible.
Defining a Line in 3-Space
All we need to define a line in
are a point
and a direction .|
on the line is then a linear combination of
with a parameter .
We simply refer to this locus of points as the line
through a point
and with a direction
for some parameter .
What we also need is a way to determine the relative orientation
of two lines. Electromagnetic theory gives us a convenient way to express the
orientation of a line in terms of the Biot-Savart law. This law gives us the
magnitude and direction of the magnetic field
due to a current
in a wire some position
away moving in some direction :|
This formula is however too intimidating for our purposes so we will strip from it everything but the bare essentials. The familiar Right-Hand rule gives us a non-mathematical way of finding the direction of the field : Imagine placing your right hand over the wire so that your fingers curl around the wire and your thumb is extended in the direction of the current; your fingers now curl in the direction of .
In the same way we can define a direction around a line. We will follow the convention that when looking along a line in the direction of the line, other lines are either oriented clockwise (CW), counterclockwise (CCW), or intersect or are parallel to the line being viewed. In the following example lines 2 and 3 are moving clockwise around line 1.
This is analogous to the direction of the magnetic field around a wire. Since we aren't interested in any physical interpretation we can get by with only the direction of . Let us now in the same way define a field around our line . For the sake of simplicity we shall locate the origin such that it passes through . Information about the line is not lost however if we simply keep track of the point that locates it. The vector value of the at a point is then given by the cross product of and .
Now we have a powerful way of finding information about the orientation of lines relative to . Consider a line through point in direction . If we are looking down along the direction of, and is in the same direction as , this line would appear to be pointing in a clockwise direction. On the other hand, if is in the opposite direction as , this line would appear to be pointing in a counterclockwise direction.
We can express these cases in terms of the dot product of and . Here receding lines are depicted as and approaching lines as (as darts might appear receding from and approaching you respectively).
There is some ambiguity in the case of . It can happen that and are parallel or antiparallel (they form angles of 0° or 180° respectively) or it might be the case that passes through the origin making zero. In either case, the lines are neither parallel nor antiparallel.
We ought now to generalize these cases given that does not necessarily pass through the origin. Looking back we have two lines and .
The vector value of at the point is now given by:
Which is equivalent to
The orientation we are interested in we can call , and express it as follows:
Which is equivalent to
This expression on its own is sufficient for allowing us to express the relative orientations of lines in and is very important in its own right. To summarize:
Given two lines and , we define the orientation of with respect to as
If , passes in the clockwise direction.
If , passes in the counterclockwise direction.
If , is either parallel or antiparallel to or intersects .
Clocks and Hands and Lines Oh My!
|It doesn't end here, however. If you think you have what it takes, take a stroll through the next installment.|