Rational motions with generic trajectories of low degree

Siegele, Johannes; Scharler, Daniel F.; Schröcker, Hans-Peter

doi:10.1016/j.cagd.2019.101793

“…However, all definitions and results from Sect. 2.1 can be translated to the ring of split quaternions with coefficients in C. This has been done in [16] with the Hamiltonian quaternions. Considering complex numbers as coefficients they are isomorphic to the (complex) split quaternions.…”

Section: Geometry Of the Factorization Algorithmmentioning

confidence: 99%

An Algorithm for the Factorization of Split Quaternion Polynomials

Scharler

¹

,

Schröcker

²

2021

Adv. Appl. Clifford Algebras

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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.

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“…If t 1 = t 2 ∈ C \ R, then the coefficients of R(t 1 ) and R(t 2 ) might be complex numbers. However, all definitions and results from Section 2.1 can be translated to the ring of split quaternions with coefficients in C. This has been done in [13] with the Hamiltonian quaternions. Considering complex numbers as coefficients they are isomorphic to the (complex) split quaternions.…”

Section: Geometry Of the Factorization Algorithmmentioning

confidence: 99%

An Algorithm for the Factorization of Split Quaternion Polynomials

Scharler¹,

Schröcker²

2020

Preprint

0

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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.

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

Lercher

¹

,

Scharler

²

,

Schröcker

³

et al. 2022

Beitr Algebra Geom

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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.

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Rational motions with generic trajectories of low degree

Cited by 6 publications

References 9 publications

An Algorithm for the Factorization of Split Quaternion Polynomials

An Algorithm for the Factorization of Split Quaternion Polynomials

An Algorithm for the Factorization of Split Quaternion Polynomials

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

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