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

TL;DR

This paper explores the geometric reasons behind the differing factorizations of quadratic polynomials over split and Hamiltonian quaternions, linking algebraic properties to hyperbolic geometry and four-bar linkages.

Contribution

It provides a geometric explanation for the maximum number of factorizations of quadratic split quaternion polynomials, connecting algebraic factorizations to hyperbolic geometric configurations.

Findings

Quadratic split quaternion polynomials can have up to six factorizations.

The number of factorizations relates to the number of real focal points in hyperbolic geometry.

A geometric model involving four-bar linkages explains the factorization behavior.

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.