Abstract
We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split quaternion polynomials of degree two are related to the coupler motion of “four-bar linkages” with equal opposite sides in universal hyperbolic geometry. A factorization corresponds to a leg of the four-bar linkage and during the motion the legs intersect in points of a conic whose focal points are the fixed revolute joints. The number of factorizations is related by the number of real focal points which can, indeed, be six in universal hyperbolic geometry.
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Acknowledgements
This work was supported by the Austrian Science Fund (FWF): P 31061 (The Algebra of Motions in 3-Space).
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Li, Z., Schicho, J., Schröcker, HP. (2019). The Geometry of Quadratic Quaternion Polynomials in Euclidean and Non-euclidean Planes. In: Cocchiarella, L. (eds) ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, vol 809. Springer, Cham. https://doi.org/10.1007/978-3-319-95588-9_24
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DOI: https://doi.org/10.1007/978-3-319-95588-9_24
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