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

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.

An Algorithm for the Factorization of Split Quaternion Polynomials

“…In the following we recall some basic results on null lines and the null quadric N . For the proofs we refer to [13] and [15].…”

An Algorithm for the Factorization of Split Quaternion Polynomials

“…In the following we recall some basic results on null lines and the null quadric N . For the proofs we refer to [10] and [12].…”

Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions

“…An isomorphism from a Clifford algebra based group to the group SE(3) of rigid body displacements requires a more elaborate construction. An element of C + (3,0,1) is of the shape r = a 0 e 0 + a 3 e 12 − a 2 e 13 + b 1 e 14 + a 1 e 23…”

“…It cannot be visualized in traditional hyperbolic geometry because all rotation centers lie in the exterior of N but is perfectly valid in universal hyperbolic geometry. A more detailed investigation of the underlying geometry of these factorizations is given in [23]. The polynomial of Example 8 parameterizes a circular translation.…”