Plucker Coodinates provide a convenient representation
for directed lines in affine 3-Dimensional space, **R**3[1,2]. This
section reviews relevant notation for, and some useful properties of, Plucker
coordinates.

Each ordered pair of distinct points p = (px, py, pz) and q = (qx, qy, qz) defines a directed line in 3-Dimensions. This line corresponds to the following set of six coefficients called the Plucker coordinates of the line:

**L = (L[0], L[1], L[2], L[3], L[4], L[5])**

Each Plucker coordinate is the determinant of a 2x2 minor of the matrix:

**p _{x} p_{y}
p_{z} 1**

**q _{x} q_{y}
q_{z} 1**

We adopt the following convention relating these minors and the Plucker coordinates L[i]:

**
L[0] = p _{x} q_{y} - q_{x}
p_{y}**

**Side Operator:**

If a and b are two directed lines, and a[i] and
b[i] their corresponding Plucker mappings, a relation side(a,b) can be
defined as the permuted inner product

**side(a,b) = a[0]*b[4] + a[1]*b[5] + a[2]*b[3]
+ a[3]*b[2] + a[4]*b[0] + a[5]*b[1]**

which is **zero whenever a and b intersect or
are parallel, and non-zero otherwise.**

However, although every directed line in **R**3
maps to a point in the Plucker coordinates, not every six-tuple X of **P**5
corresponds to a real line. Only the points satisfying the quadratic relation:

**side(X,X) = 0 and X[2], X[4] and X[5] not all
equal to zero**

correspond to real lines in **R**3. If

then it corresponds to a line at infinity. The remaining points do not correspond to lines in P3.

The plucker coordinates are
determined only to within a scale factor. That is if points p, q and r
lie on a line L, plucker(p,q) and plucker(p,r) are identical up to a scale,
that is there is some constant c (not zero) such that plucker(p,q) = c*plucker(p.r).
Note that this property is made use of in converting
back
from the plucker coordinates to the 3-Dimensions.

References:

[1] Jorge Stolfi. Primitives for computational
geometry. Technical Report 36, DEC SRC, 1989

[2] Seth Teller & Michael Hohmeyer, Determining
the Lines Through Four Lines, Journal of Graphic Tools.