Closed form of inverse Plücker correspondence in line geometry

  • Lei Dong
  • Hongbo LiEmail author
  • Lei Huang
  • Changpeng Shao


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.


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


15B10 15A29 


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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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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Lei Dong
    • 1
  • Hongbo Li
    • 1
    Email author
  • Lei Huang
    • 1
  • Changpeng Shao
    • 1
  1. 1.Key Laboratory of Mathematics MechanizationAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

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