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Bertrand partner P-Trajectories in the Euclidean 3-space e3
(Ankara Univ, Fac Sci, 2023) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
The concept of a pair of curves, called as Bertrand partner curves, was introduced by Bertrand in 1850. Bertrand partner curves have been stud-ied widely in the literature from past to present. In this study, we take into account of the concept of Bertrand partner trajectories according to Positional Adapted Frame (PAF) for the particles moving in 3-dimensional Euclidean space. Some characterizations are given for these trajectories with the aid of the PAF elements. Then, we obtain some special cases of these trajectories. Moreover, we provide a numerical example.
Characterization of Tzitzeica Curves Using Positional Adapted Frame
(Mehmet Zeki SARIKAYA, 2022) Özen, Kahraman Esen; İşbilir, Zehra; Tosun, Murat
In this study, Tzitz\'eica curves are taken into consideration in the Euclidean 3-space by using the Positional Adapted Frame (PAF). Such curves are characterized according to PAF elements. Also, some results are obtained on spherical Tzitz\'eica curves. The results obtained in this study are new contributions to the field. It is expected that these results will be useful in various application areas of differential geometry and applied mathematics in the future.
Editorial for special issue IECMSA 2020-International Eurasian Conference on Mathematical Sciences and Applications
(Wiley, 2022) Tosun, Murat; Kara, Emrah Evren; Usta, Fuat
[Bastract Not Available]
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.
Generalized Padovan Pauli Quaternions
(Springer Science and Business Media Deutschland GmbH, 2025) Isbilir, Zehra; Doğan Yazıcı, Bahar; Tosun, Murat
The main purpose of this study is to construct a new type special number system which is defined as generalized Padovan Pauli quaternion with non-negative and negative subscripts. Furthermore, we give some special cases with respect to the initial values and examine them, as well. We obtain not only new equations but also recurrence relations, Binet formulas, generating functions, exponential generating functions, summation formulas, and special determinant equalities with a numerical example regarding this new number system. After all, we construct algorithms for calculating the generalized Padovan Pauli quaternions with non-negative and negative subscripts. Then, we present the R-linear transformation of this new type special Pauli quaternions. © 2025 Elsevier B.V., All rights reserved.
Generalized Rectifying Ruled Surfaces of Special Singular Curves
(Ovidius Univ Press, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
In this study, generalized rectifying ruled surfaces of Frenet-type framed base curves in the three-dimensional Euclidean space are introduced. These surfaces are a generalization of not only the tangent and binormal surfaces of Frenet-type framed base curves, but also the tangent and binormal surfaces of regular curves. Additionally, we present some geometric characterizations and properties of these surfaces. Then, the singular point classes of the surface are scrutinized and the conditions for being a cross-cap surface are stated. Moreover, generalized rectifying surfaces are examined as framed surfaces by using the framed surface theory, and we investigate the basic invariants and curvatures of them. Then, several illustrative examples with figures are given to support the theorems and results.
Generalized ruled surfaces in Myller configuration
(University of Nis, 2025) Isbilir, Zehra; Doğan Yazıcı, Bahar; Tosun, Murat
In this paper, we introduce a quite big ruled surface family, which is called generalized ruled surfaces with Frenet-type frame in Myller configuration for Euclidean 3-space. This paper especially improves the theory of surfaces with respect to ruled surfaces and presents the relationships between the usual theory of curves and the theory of surfaces with Myller configuration. We investigate some special type ruled surfaces, such as rectifying-type ruled surfaces, osculating-type ruled surfaces, tangent-type ruled surfaces and trajectory ruled surfaces with Frenet-type frame in Myller configuration for E3 . We also give some particular cases of these ruled surfaces, as well. Since the geometry of versor fields along a curve with Frenet-type frame in Myller configuration for E3 is a generalization of the usual theory of curves in classical Euclidean space, the surface theory of versor fields along a curve with Frenet-type frame in Myller configuration for E3 is a generalization of the usual theory of surfaces in classical Euclidean space, as well. Then, we establish some numerical examples with some illustrative figures with respect to the ruled surfaces in Myller configuration in order to solidify and concretize the given results. © 2025 Elsevier B.V., All rights reserved.
GENERALIZED SMARANDACHE CURVES WITH FRENET-TYPE FRAME
(Honam Mathematical Soc, 2024) Isbilir, Zehra; Tosun, Murat
In this study, we investigate Smarandache curves with Frenettype frame in Myller configuration for Euclidean 3-space E 3 . Also, we introduce some characterizations and invariants of them. Then, we construct a numerical example with respect to these special Smarandache curves in order to understand the obtained materials.
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.
Mannheim partner trajectories related to pafors
(Univ Nis, 2024) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
In this study, we consider the concept of Mannheim partner trajectories related to the Positional Adapted Frame on Regular Surfaces (PAFORS) for the particles moving on the different regular surfaces in Euclidean 3 -space. We give the relations between the PAFORS elements of these aforementioned trajectories. Also, we obtain the relations between Darboux basis vectors of these trajectories. Furthermore, some special cases of these trajectories are written.
A New Insight on Rectifying-Type Curves in Euclidean 4-Space
(Int Electronic Journal Geometry, 2023) Isbilir, Zehra; Tosun, Murat
In this study, our purpose is to determine the generalized rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4. Also, some characterizations of them are given. We construct some correlations between curvatures and invariants of generalized rectifyingtype curves. Additionally, we obtain an illustrative example with respect to the rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4.
On generalized osculating-type curves in Myller configuration
(Ovidius Univ Press, 2024) Isbilir, Zehra; Tosun, Murat
In this study, we examine osculating-type curves with Frenet-type frame in Myller configuration for Euclidean 3-space E-3. We present the necessary characterizations for a curve to be an osculating-type curve. Characterizations originating from the natural structure of Myller configuration are a generalization of osculating curves according to the Frenet frame. Also, we introduce some new results that are not valid for osculating curves. Then, we give an illustrative numerical example supported by a figure.
On special curves in Lie groups with Myller configuration
(Wiley, 2024) Isbilir, Zehra; Dogan Yazici, Bahar; Tosun, Murat
In this study, we determine a new type comprehensive frame, which is called the generalized Frenet-type frame in three-dimensional Lie groups with Myller configurations, and it includes several special and classical type frames for Euclidean 3-space and three-dimensional Lie groups. After constructing this new comprehensive frame, we obtain derivative formulas with the help of the Lie curvature. In addition, we define some special type curves. The geometry of versor fields along a curve with Frenet-type frame in three-dimensional Lie groups with Myller configurations is a generalization of the usual theory of curves. Since this particular relationship, the osculating-type and rectifying-type curves with Frenet-type frame in three-dimensional Lie groups with Myller configurations include some special cases for osculating and rectifying curves in different spaces.
On the 3-parameter generalized quaternions with generalized tribonacci numbers components
(Univ Nis, Fac Sci Math, 2025) Isbilir, Zehra; Gurses, Nurten; Tosun, Murat
In this paper, we aim to combine 3-parameter generalized quaternions (shortly 3PGQs), which are a general form of the quaternion algebra according to 3-parameters, and generalized Tribonacci number (shortly GTNs), which are also quite a big special number family for third-order recurrence sequences and most general form of all of the third-order recurrence sequences. Namely, we investigate a special new number system called 3-parameter generalized quaternions with generalized Tribonacci numbers components (shortly 3PGQs with GTN components) with both nonnegative and negative subscripts and examine some special cases of them. Then, we construct a Maple code of this special number family. Moreover, we obtain some new and classical well-known equations such as; Binet formulas, generating function, exponential generating function, Poisson generating function, summation formulas, polar representation, and matrix equation. In addition to these, we give also determinant, characteristic polynomial, characteristic equation, eigenvalues, and eigenvectors concerning the matrix representation of 3PGQs with GTN components.
Padovan and Perrin hyperbolic spinors
(Springer Heidelberg, 2025) Isbilir, Zehra; Kosal, Isil Arda; Tosun, Murat
In this study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors, both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Besides, we construct some special equations that give relations between Padovan and Perrin hyperbolic spinors and Padovan and Perrin numbers. Further, we give a short introduction for (s, t)-Padovan and (s, t)-Perrin hyperbolic spinors in order to shed light on future studies.
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.
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.
Spinor representations of framed curves in the three-dimensional lie groups
(Taru Publications, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
In this study, we determine the spinor representations of special singular curves (framed curves) in the three-dimensional Lie groups with a biinvariant metric. Also, we construct spinor framed equations for some special cases and obtain the relations between the spinor representations of the general frame and adapted frame along the framed curves in the three-dimensional Lie groups. Then, we give some geometric properties and results with respect to them.

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