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Points in the plane, lines in space

Journal of Geometry volume 113, Article number: 46 (2022) Cite this article

Abstract

The 1911 Grünwald–Blaschke mapping is reviewed from the point of view of a particular Clifford algebra. This is a mapping between the group of proper Euclidean displacements of the plane and an open set in 3-dimensional real projective space. The image of the set of group elements which displace an arbitrary point to another fixed point is a line in the projective space. In this way, a correspondence is established between point-pairs in the plane and lines in 3-dimensional projective space. The space of lines in 3 dimensions is an object of classical study usually called the Klein quadric. The action of the group of planar rigid-body displacements on the Klein quadric is different from the usually considered action of the spatial group. The quadratic invariants with respect to this representation are found and interpretations in terms of point-pairs are given. Some subspaces of lines, including line complexes and congruences, are investigated and their interpretation as sets of point-pairs in the plane are given.

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References

  1. Bottema, O., Roth, B.: Theoretical Kinematics. Dover Publications, New York (1990)

    MATH  Google Scholar 

  2. Coolidge, J.L.: A Treatise on the Circle and the Sphere. Clarendon Press, Oxford (1916)

    MATH  Google Scholar 

  3. Grünwald, J.: Ein Abbildungsprinzip, welches die ebene Geometrie und Kinematik mit der räumlichen Geometrie verknüpft. Sitzungsberichten der kaiserl. Akademie der Wissenschaften in Wein. Mathem.-naturw. Klasse; Bd. CXX. Abt. II a, 120(2a), 677–741 (1911). (English translation: D. H. Delphenich, http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/grunwald_-_kinematic_mapping.pdf)

  4. Guth, L., Katz, N.H.: On the Erdős distinct distance problem in the plane. Ann. Math. 181(1), 155–190 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ivory, J.: On the attractions of homogeneous ellipsoids. Philos. Trans. R. Soc. Lond. 99, 345–372 (1809)

    Google Scholar 

  6. Jessop, C.M.: A Treatise on the Line Complex. Cambridge University Press, Cambridge (1903)

    MATH  Google Scholar 

  7. Klawitter, D., Hagemann, M.: Kinematic mappings for Cayley–Klein geometries via Clifford algebras. Beiträge Algebra Geom. Contrib. Algebra Geom. 54(2), 737–761 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Murphy, B., Petridis, G., Pham, T., Rudnev, M., Stevens, S.: On the pinned distances problem in positive characteristic. J. Lond. Math. Soc. 105(1), 469–499 (2022). https://doi.org/10.1112/jlms.12524.

    Article  MathSciNet  Google Scholar 

  9. Picard, E.: Notice sur la vie et les travaux de Georges–Henri Halphen, Membre de la Section de Géométrie. C. R. hebd. séances l’Acad. Sci. 110(N10), 489–497 (1890)

    MATH  Google Scholar 

  10. Pottmann, H., Wallner, J.: Computational Line Geometry Mathematics and Visualization. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  11. Rudnev, M., Selig, J.M.: On the use of Klein quadric for geometric incidence problems in two dimensions. SIAM J. Discrete Math. 30(2), 934–954 (2016). https://doi.org/10.1137/16M1059412.

    Article  MathSciNet  MATH  Google Scholar 

  12. Selig, J.M.: Geometrical Fundamentals of Robotics, 2nd edn Springer, New York (2005)

    MATH  Google Scholar 

  13. Semple, J.G., Roth, L.: Introduction to Algebraic Geometry. Clarendon Press, Oxford (1985).(First published 1949)

  14. Tao, T.: Lines in the Euclidean Group SE(2) (2011). https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/

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  1. School of Engineering, London South Bank University, London, SE1 OAA, UK

    J. M. Selig

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Selig, J.M. Points in the plane, lines in space. J. Geom. 113, 46 (2022). https://doi.org/10.1007/s00022-022-00661-3

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  • DOI: https://doi.org/10.1007/s00022-022-00661-3

Mathematics Subject Classification

  • Primary 53A20
  • Secondary 51A10, 53A25