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
Authors | Selig, J.M. |
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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. |
Keywords | line geometry; planar rigid body displacements; directed point pairs; Clifford algebra |
Year | 2022 |
Journal | Journal of Geometry |
Publisher | Springer |
ISSN | 1420-8997 |
Digital Object Identifier (DOI) | https://doi.org/10.1007/s00022-022-00661-3 |
Publication dates | |
19 Sep 2022 | |
Publication process dates | |
Accepted | 07 Sep 2022 |
Deposited | 15 Sep 2022 |
Accepted author manuscript | License File Access Level Open |
Additional information | 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 |
https://openresearch.lsbu.ac.uk/item/91x72
Download files
Accepted author manuscript
Points+lines(rev2).pdf | ||
License: Springer Bespoke License | ||
File access level: Open |
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