Factorization of quaternionic polynomials of bi-degree (n,1).

We consider polynomials of bi-degree (n, 1) over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot be explained in this way and characterize them geometrically and algebraically. Existence of factorizations is related to the existence of special rulings of two different types (left/right) on the ruled surface parameterized by the bivariate polynomial in the projective space over the quaternions. Special non-uniqueness in above sense can be explained algebraically by commutation properties of factors in suitable factorizations. A necessary geometric condition for this to happen is degeneration to a point of at least one of the left/right rulings.

The Study Variety of Conformal Kinematics.

We introduce the Study variety of conformal kinematics and investigate some of its properties. The Study variety is a projective variety of dimension ten and degree twelve in real projective space of dimension 15, and it generalizes the well-known Study quadric model of rigid body kinematics. Despite its high dimension, co-dimension, and degree it is amenable to concrete calculations via conformal geometric algebra (CGA) associated to three-dimensional Euclidean space. Calculations are facilitated by a four quaternion representation which extends the dual quaternion description of rigid body kinematics. In particular, we study straight lines on the Study variety. It turns out that they are related to a class of one-parametric conformal motions introduced by Dorst in (Math Comput Sci 10:97-113, 2016, 10.1007/s11786-016-0250-8). Similar to rigid body kinematics, straight lines (that is, Dorst's motions) are important for the decomposition of rational conformal motions into lower degree motions via the factorization of certain polynomials with coefficients in CGA.

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.

Factorization of Dual Quaternion Polynomials Without Study's Condition.

In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real polynomial factors in the primal part and factorizability of the norm polynomial over the dual numbers into monic quadratic factors. This obviously necessary condition is also sufficient for existence of factorizations. We present an algorithm to compute factorizations of these polynomials and use it for new constructions of mechanisms which cannot be obtained by existing factorization algorithms for motion polynomials. While they produce mechanisms with rotational or translational joints, our approach yields mechanisms consisting of "vertical Darboux joints". They exhibit mechanical deficiencies so that we explore ways to replace them by cylindrical joints while keeping the overall mechanism sufficiently constrained.

Quadratic Split Quaternion Polynomials: Factorization and Geometry.

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.