However, it is more common and simpler algebraically to use coordinates (l, m) where the equation of the line is lx + my + 1 = 0.
In this equation, only the ratios between l, m and n are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same.
If points in the real projective plane are represented by homogeneous coordinates (x, y, z), the equation of the line is lx + my + nz = 0, provided (l, m, n) ≠ (0,0,0) .
A linear equation in line coordinates has the form al + bm + c = 0, where a, b and c are constants.
Six numbers in a set of coordinates only represent a line when they satisfy an additional equation.
More generally, the lines in n-dimensional projective space are determined by a system of n(n − 1)/2 homogeneous coordinates that satisfy a set of (n − 2)(n − 3)/2 conditions, resulting in a manifold of dimension 2n− 2.
Isaak Yaglom has shown[1] how dual numbers provide coordinates for oriented lines in the Euclidean plane, and split-complex numbers form line coordinates for the hyperbolic plane.
The motions of the line geometry are described with linear fractional transformations on the appropriate complex planes.