A Quaternionic Product of Lines in the Plane E^2

Abstract

This note introduces a product of plane lines inspired by the product of quaternions. A technical condition is necessary for the existence of this product and some examples (squares, the axes and the bisectrices of the axes) are discussed.

Keywords

• Line
• quaternion
• product
• projective geometry

References

1. Bekta¸s D.B., Aghayev N., On geometric applications of quaternions, Turkish Journal of Mathematics, 44(4), 1289-1303, 2020.
2. Crasmareanu M., Quaternionic product of circles and cycles and octonionic product for pairs of circles, Iranian Journal of Mathematical Sciences and Informatics, 17(1), 227-237, 2022.
3. Crasmareanu M., The Farey sum of Pythagorean and Eisenstein triples, Mathematical Sciences and Applications E-Notes, 12(1), 28-36, 2024.
4. Crasmareanu M., Popescu M., Quaternionic product of equilateral hyperbolas and some extensions, Mathematics, 8(10), 1686, 2020.
5. Crasmareanu M., Pi¸scoran L.-I., A quaternionic product of simple ratios, Carpathian Journal of Mathematics, 41(1), 89-93, 2025.
6. Cushman R.H., Bates L.M., Global Aspects of Classical Integrable Systems, 2nd Edition, Springer, 2015.
7. Giardino S., Differential geometry using quaternions, International Electronic Journal of Geometry, 17(2), 700-711, 2024.
8. Hamilton W.R., On quaternions; or on a new system of imaginaries in algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Supplement to Volume XXV, 1844.

9. Millman R.S., Parker G.D., Geometry. A Metric Approach With Models, 2nd Edition, Springer- Verlag, 1991.