Converting  Plucker Coordinates to 3-Dimensional Space

We are interested in converting back the plucker coordinates to the real affine space R3. We will make use of the fact that the Plucker coordinates of a line are not independent and are determined to only within a scale factor.

We consider plucker coordinates that correspond only to a real line in R3, i.e, side(L,L) = 0.

We make use of one more observation that the line will intersect at least one of the following pair of planes:

X = 1 and X = -1
Y = 1 and Y = -1
Z = 1 and Z = -1

Or, we can alternatively say that,  for any real line L at least one out of L[2], L[4] or L[5] will be non zero, and hence we can choose any non zero plucker coordinate out of these three and consider the corresponding pair of planes.

An Example:
Let us suppose that L[4] is not zero. Hence this line will definetely intersect the pair of planes Z = 1 and Z = -1. Hence, we consider two points on this plane : p(x, y, 1) and q(x', y',-1). The plucker coordinates for the line through these two points are:

                    L'[0] =  x y' - x' y
                    L'[1] = -x - x'
                    L'[2] =   x - x'
                    L'[3] =  -y - y'
                    L'[4] =   2
                    L'[5] =  y' - y

Thus by scaling the given plucker coordinates so that  L[4] is scaled to L'[4] ( =2) we easily get that:

x =    (L[2] - L[1])/L[4]
y = -(L[3] + L[5])/L[4]
x'= -(L[1] + L[2])/L[4]
y'=    (L[5] - L[3])/L[4]

And that gives us the line in R3 corresponding to the plucker coordinates.

The code can be found here