Closed form of inverse Plücker correspondence in line geometry

Abstract

By mapping the homogeneous coordinates of two points in space to the Plücker coordinates of the line they determine, any transformation of type SL(4) upon points in space is mapped to a transformation of type SO0(3, 3), the latter being the connected component containing the identity of the special orthogonal transformation group of the linear space spanned by Plücker coordinates. This is the classical Plücker correspondence, two-to-one and onto. It has important applications in line geometry and projective transformations. While the explicit form of Plücker correspondence is trivial to 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 Plücker correspondence.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11671388) and the CAS Frontier Key Project (Grant No. QYZDJ-SSW-SYS022).

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Correspondence to Hongbo Li.

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Dong, L., Li, H., Huang, L. et al. Closed form of inverse Plücker correspondence in line geometry. Sci. China Math. 62, 2617–2630 (2019). https://doi.org/10.1007/s11425-017-9242-3

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Keywords

  • Plücker correspondence
  • inverse Plücker correspondence
  • line geometry
  • SL(4)
  • SO(3,3)

MSC(2010)

  • 15B10
  • 15A29