 Plücker coordinates

A line in real projective space ${\mathbb{P}}^3$ is the span of two independent homogeneous vectors $A,B$. Its Plücker coordinates are the entries of the displayed rank-two antisymmetric matrix, determined up to a nonzero scalar. They satisfy $L^{01}{L}^{23}-L^{02}{L}^{13}+L^{03}{L}^{12}=0$. With a chosen volume orientation, its dual plane form is $L_{ij}=\frac{1}{2}{\epsilon }_{ijkl}{L}^{kl}$. This is projective duality using the volume form, not lowering indices with a Euclidean metric.

Plücker coordinates are a system of homogeneous coordinates used to represent lines in projective space, particularly in three-dimensional projective space \( \mathbb{P}^3 \). They are named after the mathematician Julius Plücker.