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29 pages, 707 KiB

Open AccessArticle

A Novel Approach to Ruled Surfaces Using Adjoint Curve

by Esra Damar

Symmetry 2025, 17(7), 1018; https://doi.org/10.3390/sym17071018 - 28 Jun 2025

Cited by 1 | Viewed by 219

In this study, ruled surfaces are examined where the direction vectors are unit vectors derived from Smarandache curves, and the base curve is taken as an adjoint curve constructed using the integral curve of a Smarandache-type curve generated from the first and second Bishop normal vectors. The newly generated ruled surfaces will be referred to as Bishop adjoint ruled surfaces. Explicit expressions for the Gaussian and mean curvatures of these surfaces have been obtained, and their fundamental geometric properties have been analyzed in detail. Additionally, the conditions for developability, minimality, and singularities have been investigated. The asymptotic and geodesic behaviors of parametric curves have been examined, and the necessary and sufficient conditions for their characterization have been derived. Furthermore, the geometric properties of the surface generated by the Bishop adjoint curve and its relationship with the choice of the original curve have been established. The constructed ruled surfaces exhibit a notable degree of geometric regularity and symmetry, which naturally arise from the structural behavior of the associated adjoint curves and direction fields. This underlying symmetry plays a central role in their formulation and classification within the broader context of differential geometry. Finally, the obtained surfaces are illustrated with figures. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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22 pages, 887 KiB

Open AccessArticle

On the Special Viviani’s Curve and Its Corresponding Smarandache Curves

by Yangke Deng, Yanlin Li, Süleyman Şenyurt, Davut Canlı and İremnur Gürler

Mathematics 2025, 13(9), 1526; https://doi.org/10.3390/math13091526 - 6 May 2025

Cited by 1 | Viewed by 448

Abstract

In the present paper, the special Viviani’s curve is revisited in the context of Smarandache geometry. Accordingly, the paper first defines the special Smarandache curves of Viviani’s curve, including the Darboux vector. Then, it expresses the resulting Frenet apparatus for each Smarandache curve in terms of the Viviani’s curve. The paper is also supported by extensive graphical representations of Viviani’s curve and its Smarandache curves, as well as their respective curvatures. Full article

(This article belongs to the Special Issue Submanifolds in Metric Manifolds, 2nd Edition)

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18 pages, 318 KiB

Open AccessArticle

Spinor Equations of Smarandache Curves in E³

by Zeynep İsabeyoǧlu, Tülay Erişir and Ayşe Zeynep Azak

Mathematics 2024, 12(24), 4022; https://doi.org/10.3390/math12244022 - 22 Dec 2024

Viewed by 751

Abstract

This study examines the spinor representations of

TN

(tangent and normal),

NB

(normal and binormal),

TB

(tangent and binormal) and

TNB

(tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space

E^{3}

. Spinors are complex column vectors and move on Pauli spin [...] Read more.

This study examines the spinor representations of

TN

(tangent and normal),

NB

(normal and binormal),

TB

(tangent and binormal) and

TNB

(tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space

E^{3}

. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the

C^{3}

complex vector space form a two-dimensional surface in the

C^{2}

complex space. Additionally, each isotropic vector in

C^{3}

space corresponds to two vectors in

C^{2}

space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (

TN

NB

TB

and

TNB

)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. Full article

(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)

28 pages, 559 KiB

Open AccessArticle

The Pedal Curves Generated by Alternative Frame Vectors and Their Smarandache Curves

by Davut Canlı, Süleyman Şenyurt, Filiz Ertem Kaya and Luca Grilli

Symmetry 2024, 16(8), 1012; https://doi.org/10.3390/sym16081012 - 8 Aug 2024

Cited by 2 | Viewed by 1942

Abstract

In this paper, pedal-like curves are defined resulting from the orthogonal projection of a fixed point on the alternative frame vectors of a given regular curve. For each pedal curve, the Frenet vectors, the curvature and the torsion functions are found to provide the common relations among the main curve and its pedal curves. Then, Smarandache curves are defined by using the alternative frame vectors of each pedal curve as position vectors. The relations of the Frenet apparatus are also established for the pedal curves and their corresponding Smarandache curves. Finally, the expressions of the alternative frame apparatus of each Smarandache curves are given in terms of the alternative frame elements of the pedal curves. Thus, a set of new symmetric curves are introduced that contribute to the vast curve family. Full article

(This article belongs to the Section Mathematics)

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19 pages, 1700 KiB

Open AccessArticle

Investigation of Special Type-Π Smarandache Ruled Surfaces Due to Rotation Minimizing Darboux Frame in E³

by Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan and Mohamed Abdelkawy

Symmetry 2023, 15(12), 2207; https://doi.org/10.3390/sym15122207 - 17 Dec 2023

Cited by 5 | Viewed by 1496

Abstract

This study begins with the construction of type-

Π

Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in

E^{3}

. The direction vectors of these surfaces are unit vectors that convert Smarandache [...] Read more.

This study begins with the construction of type-

Π

Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in

E^{3}

. The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 2nd Edition)

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15 pages, 280 KiB

Open AccessArticle

On Some Quasi-Curves in Galilean Three-Space

by Ayman Elsharkawy, Yusra Tashkandy, Walid Emam, Clemente Cesarano and Noha Elsharkawy

Axioms 2023, 12(9), 823; https://doi.org/10.3390/axioms12090823 - 27 Aug 2023

Cited by 8 | Viewed by 1486

Abstract

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

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