Closed form of inverse Plücker correspondence in line geometry

Abstract

By mapping the homogeneous coordinates of two points in space to the Plucker coordinates of theline they determine, any transformation of type $SL(4)$ upon points in space is mapped to a transformationof type $SO_0(3,3)$, the latter being the connected componentcontaining the identity of the special orthogonal transformation group of the linear spacespanned by Plucker coordinates. This is the classical Plucker correspondence, two-to-one and onto. It has importantapplications in line geometry and projective transformations. While the explicit form of Plucker correspondence is trivialto present, its inverse in explicit form, which is also important in application, is not found in the literature.In this paper, we present a simple and unified formula for the inverse of the Plucker correspondence.