Homogeneous Coordinates - Plücker Coordinates

Plücker Coordinates

Assigning coordinates to lines in projective 3-space is more complicated since it would seem that at total of 8 coordinates, either the coordinates of two points which lie on the line or two planes whose intersection is the line. A useful method, due to Julius Plücker, creates a set of six coordinate as the determinants x_iy_j − x_jy_i (1 ≤ i < j ≤ 4) from the homogeneous coordinates of two points (x₁, x₂, x₃, x₄) and (y₁, y₂, y₃, y₄) on the line. The Plücker embedding is the generalization of this to create homogeneous coordinates of elements of any dimension m in a projective space of dimension n.