Plücker coordinates for lines in 3d ed
Table of Contents
Outer algebra ed
Description ed
Arbitrary (1d) lines in 3d space can be easier handled, when seen as 2d-planes in 4d space. This is the magic of projective geometry. Let's say the line \(L\) goes through the points \(A, B \in \mathbb{R}^3\). Then we can find plane \(L'\) in 4d, passing through the points\[ A' = \begin{pmatrix} A \\ 1 \end{pmatrix} = A + e_4, \quad B' = \begin{pmatrix} B \\ 1 \end{pmatrix}, \quad 0 \; . \]
Since \(L'\) passes through the origin, it is a (2d) linear subspace of \(\mathbb{R}^4\), i.e. it can be associated to the outer product\[ A' \wedge B' = (A + e_4) \wedge (B + e_4) = A \wedge B + (A - B) \wedge e_4 \; . \]This means, we can use the two vectors\[ u = A - B, \quad v = A \times B \]to describe the line \(L\) via\[ L' = u \wedge e_4 + \star v \; . \]