The geometry of quadratic quaternion polynomials in Euclidean and non-Euclidean planes. (English) Zbl 1400.51009

Cocchiarella, Luigi (ed.), ICGG 2018 – Proceedings of the 18th international conference on geometry and graphics. 40th anniversary – Milan, Italy, August 3–7, 2018. In 2 volumes. Cham: Springer; Milan: Politecnico de Milano (ISBN 978-3-319-95587-2/pbk; 978-3-319-95588-9/ebook). Advances in Intelligent Systems and Computing 809, 298-309 (2019).
Summary: 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.
For the entire collection see [Zbl 1403.00028].


51M05 Euclidean geometries (general) and generalizations
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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[1] Gallet, M., Koutschan, C., Li, Z., Georg Regensburger, G., Schicho Josef Villamizar, N.: Planar linkages following a prescribed motion. Math. Comp. 86 (2017) · Zbl 1404.70007 · doi:10.1090/mcom/3120
[2] Hegedüs, G., Schicho, J., Schröcker, H.P.: Factorization of rational curves in the Study quadric and revolute linkages. Mech. Machine Theory 69(1), 142-152 (2013) · doi:10.1016/j.mechmachtheory.2013.05.010
[3] Inalcik, A., Ersoy, S., Stachel, H.: On instantaneous invariants of hyperbolic planes. Math. Mech. Solids 22(5), 1047-1057 (2017) · Zbl 1371.70008 · doi:10.1177/1081286515616283
[4] Li, Z., Rad, T.D., Schicho, J., Schröcker, H.P.: Factorization of rational motions: a survey with examples and applications. In: S.H. Chang (ed.) Proceedings of the 14th IFToMM World Congress (2015)
[5] Li, Z., Schicho, J., Schröcker, H.P.: Factorization of motion polynomials. Accepted for publication in J, Symbolic Comp (2018)
[6] Li, Z., Schröcker, H.P.: Factorization of left polynomials in Clifford algebras: state of the art, applications, and open questions. Submitted for publication (2018)
[7] Schoger, A.U.: Koppelkurven und Mittelpunktskegelschnitte in der hyperbolischen Ebene. J. Geom. 57(1-2), 160-176 (1996) · Zbl 0864.51016 · doi:10.1007/BF01229260
[8] Wildberger, N.: Universal hyperbolic geometry II: a pictorial overview. KoG 14, 3-24 (2010) · Zbl 1217.51009
[9] Wildberger, N.: Universal hyperbolic geometry III: first steps in projective triangle geometry. KoG 15, 25-49 (2011) · Zbl 1262.51016
[10] Wildberger, N.: Universal hyperbolic geometry I: trigonometry. Geom. Dedicata 163, 215-274 (2013) · Zbl 1277.51018 · doi:10.1007/s10711-012-9746-9
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