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, L or L 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.
Let us suppose that L 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' = x y' - x' y
L' = -x - x'
L' = x - x'
L' = -y - y'
L' = 2
L' = y' - y
Thus by scaling the given plucker coordinates so that L is scaled to L' ( =2) we easily get that:
x = (L - L)/L
y = -(L + L)/L
x'= -(L + L)/L
y'= (L - L)/L
And that gives us the line in R3 corresponding to the plucker coordinates.
The code can be found here