arXiv subjects

Zülal Derin Yaqub

Publications and source records attributed to Zülal Derin Yaqub.

3 recordsLinked to original sources

arXiv2026

Hypersurfaces with Non-diagonalizable Shape Operators in Minkowski Spaces

In this paper, we study hypersurfaces $M^4$ in the Minkowski space $\mathbb{E}^5_1$ whose shape operator $S$ is non-diagonalizable, having a triple real eigenvalue $λ$ with a one-dimensional eigenspace together with a simple eigenvalue $μ$ (Case~(iii) in the classification of Magid). Assuming that $λ$ and $μ$ satisfy a functional relation $λ=f(μ)$ and that the eigenvector field of $μ$ is geodesic, we show that $λ$ and $μ$, as well as the pseudo-orthonormal frame, are intrinsic invariants of $M$, determined by the induced metric up to sign. As a consequence, we prove that such hypersurfaces are unique up to isometries of $\mathbb{E}^5_1$. We also obtain a classification of the warped product hypersurfaces of this type.

math.DG↗

arXiv2025

Rational Motions of Minimal Quaternionic Degree with Prescribed Line Trajectories

In this paper, we study how to find rational motions that move a line along a given rational ruled surface. Our goal is to find motions with the lowest possible degree using dual quaternions. While similar problems for point trajectories are well known, the case of line trajectories is more complicated and has not been studied. We explain when such motions exist and how to compute them. Our method gives explicit formulas for constructing these motions and shows that, in many cases, the solution is unique. We also show examples and explain how to use these results to design simple mechanisms that move a line in the desired way. This work helps to better understand the relationship between rational motions and ruled surfaces and may be useful for future research in mechanism design.

math.RA↗

arXiv2025

Rational Motions of Minimal Quaternionic Degree with Prescribed Plane Trajectories

This paper investigates the construction of rational motions of a minimal quaternionic degree that generate a prescribed plane trajectory (a ``rational torse''). Using the algebraic framework of dual quaternions, we formulate the problem as a system of polynomial equations. We derive necessary and sufficient conditions for the existence of such motions, establish a method to compute solutions and characterize solutions of minimal degree. Our findings reveal that a rational torse is realizable as a trajectory of a rational motion if and only if its Gauss map is rational. Furthermore, we demonstrate that the minimal degree of a motion polynomial is geometrically related to a drop of degree of the Gauss and algebraically determined by the structure of the torse's associated plane polynomial and the real greatest common divisor of its vector part. The developed theoretical framework has potential applications in robotics, computer-aided design, and computational kinematic, offering a systematic approach to constructing rational motions of small algebraic complexity.

math.RA↗