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    Generalized osculating-type ruled surfaces of singular curves
    (Wiley, 2023) Yazıcı, Bahar Doğan; İşbilir, Zehra; Tosun, Murat
    In this study, we introduce generalized osculating-type ruled surfaces of special singular curves. We give some theories and results about the geometric structure of the surface. In addition, the singular point classes of the surface are examined, and the conditions for being a cross-cap surface are expressed. Generalized osculating-type ruled surface is considered as a framed surface and its basic invariants are found and some results are given. Finally, we give some examples and figures to support the theories.
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    MANNHEIM PARTNER P-TRAJECTORIES IN THE EUCLIDEAN 3-SPACE E-3
    (Honam Mathematical Soc, 2022) İşbilir, Zehra; Özen, Kahraman Esen; Tosun, Murat
    Mannheim introduced the concept of a pair of curves, called as Mannheim partner curves, in 1878. Until now, Mannheim partner curves have been studied widely in the literature. In this study, we take into account of this concept according to Positional Adapted Frame (PAF) for the particles moving in the 3-dimensional Euclidean space. We introduce a new type special trajectory pairs which are called Mannheim partner P-trajectories in the Euclidean 3-space. The relationships between the PAF elements of this pair are investigated. Also, the relations between the Serret-Frenet basis vectors of Mannheim partner P-trajectories are given. Afterwards, we obtain the necessary conditions for one of these trajectories to be an osculating curve and for other to be a rectifying curve. Moreover, we provide an example including an illustrative figure.
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    On the combined Jacobsthal-Padovan generalized quaternions
    (Tbilisi Centre Math Sci, 2022) Gürses, Nurten; İşbilir, Zehra
    In this article, we examine the combined Jacobsthal-Padovan (CJP) generalized quaternions with four special cases: Jacobsthal-Padovan, Jacobsthal-Perrin, adjusted Jacobsthal-Padovan and modified Jacobsthal-Padovan generalized quaternions. Then, recurrence relation, generating function, Binet-like formula and exponential generating function of these quaternions are examined. In addition to this, some new properties, special determinant equations, matrix formulas and summation formulas are discussed.
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    Padovan, Perrin and Pell-Padovan Dual Quaternions
    (2023) İşbilir, Zehra; Gürses, Nurten
    In this present study, we intend to determine the Padovan, Perrin and Pell-Padovan dual quaternions with nonnegative and negative subscripts. In line with this purpose, we construct some new properties such as; special determinant equalities, new recurrence relations, matrix formulas, Binet-like formulas, generating functions, exponential generating functions, summation formulas, and binomial properties for these special dual quaternions.
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    Spinor representation of framed Mannheim curves
    (Scientific and Technological Research Council Turkey, 2022) Doğan Yazıcı, Bahar; İşbilir, Zehra; Tosun, Murat
    In this paper, we obtain spinor with two complex components representations of Mannheim curves of framed curves. Firstly, we give the spinor formulas of the frame corresponding to framed Mannheim curve. Later, we obtain the spinor formulas of the frame corresponding to framed Mannheim partner curve. Moreover, we explain the relationships between spinors corresponding to framed Mannheim pairs and their geometric interpretations. Finally, we present some geometrical results of spinor representations of framed Mannheim curves.
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    The spinor representations of framed Bertrand curves
    (Univ Nis, Fac Sci Math, 2023) İşbilir, Zehra; Yazıcı, Bahar Doğan; Tosun, Murat
    In this study, we intend to examine the framed Bertrand curves in three-dimensional Euclidean space E3 by using the spinors, which have a fundamental place and importance in different disciplines from mathematics to physics. For this purpose, we investigate the spinor representations of framed Bertrand mates in E3. Additionally, we present some geometric results and interpretations. Then, we construct numerical examples with illustrative figures in order to support the given materials.