The Geometry of Quadratic Quaternion Polynomials in Euclidean and Non-euclidean Planes

  • Zijia LiEmail author
  • Josef Schicho
  • Hans-Peter Schröcker
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)


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.


Quaternion Factorization Four-bar linkage Parallelogram Anti-parallelogram Conic Focal point Hyperbolic geometry 



This work was supported by the Austrian Science Fund (FWF): P 31061 (The Algebra of Motions in 3-Space).


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Zijia Li
    • 1
    Email author
  • Josef Schicho
    • 2
  • Hans-Peter Schröcker
    • 3
  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria
  3. 3.Department of Basic Sciences in Engineering SciencesUniversity of InnsbruckInnsbruckAustria

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