An Algorithm for the Factorization of Split Quaternion Polynomials

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.

Spherical kinematics in 3-dimensional generalized space

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space [Formula: see text]. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in [Formula: see text] space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in [Formula: see text] space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for [Formula: see text] in 3-dimensional Eulidean space [Formula: see text] in 3-dimensional Lorentzian space [Formula: see text] respectively.

A Unified Approach: Split Quaternions with Quaternion Coefficients and Quaternions with Dual Coefficients

This paper aims to present, in a unified manner, results which are valid on both split quaternions with quaternion coefficients and quaternions with dual coefficients, simultaneously, calling the attention to the main differences between these two quaternions. Taking into account some results obtained by Karaca, E. et al., 2020, each of these quaternions is studied and some important differences are remarked on.

ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.