We are interested in converting back the plucker
coordinates
to the real affine space **R**3. 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 **R**3 corresponding
to the plucker coordinates.

**Code:**

The code can be found here