Points in the Plane, Lines in Space

Journal article

Selig, J.M. (2022). Points in the Plane, Lines in Space. Journal of Geometry. https://doi.org/10.1007/s00022-022-00661-3
AuthorsSelig, J.M.
Abstract

The 1911 Grunwald - 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.

Keywordsline geometry; planar rigid body displacements; directed point pairs; Clifford algebra
Year2022
JournalJournal of Geometry
PublisherSpringer
ISSN1420-8997
Digital Object Identifier (DOI)https://doi.org/10.1007/s00022-022-00661-3
Publication dates
Print19 Sep 2022
Publication process dates
Accepted07 Sep 2022
Deposited15 Sep 2022
Accepted author manuscript
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Open
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This is a post-peer-review, pre-copyedit version of an article published in Journal of Geometry. The final authenticated version is available online: https://link.springer.com/article/10.1007/s00022-022-00661-3

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