On the Differential Geometric Elements of the Involute $${\tilde{\rm D}}$$ D ~ -Scroll in E 3

Şenyurt, Süleyman; Kilicoglu, Seyda

doi:10.1007/s00006-015-0535-z

Cited by 7 publications

(7 citation statements)

References 3 publications

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“…In this section we will define and work on MDRS, which is known as rectifying developable ruled surface, orD scroll as in [7] where the differential geometric elements of the involuteD − scroll.are examined too. Here first we will give Darboux vector field of the Mannheim partner α * as in the following theorem.…”

Section: Mannheim Darboux Ruled Surfacementioning

confidence: 99%

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On the differential geometric elements of Mannheim Darboux ruled surface in E^3

Kilicoglu¹,

Şenyurt²

2016

ams

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In this paper we consider two special ruled surfaces associated to Mannheim pair {α, α * }. First, Mannheim Darboux Ruled surface (MDRS) of the curve α be defined and examined in terms of the Frenet-Serret apparatus of the Mannheim curve α, in E 3. Further we have examined the differential geometric elements such as, Weingarten map S, Gaus curvature K and mean curvature H of Darboux ruled surface (DRS) and Mannheim Darboux ruled surface (MDRS) relative to each other. Also the first, second and third fundamental forms of Mannheim Darboux ruled surface (MDRS) have been examined in terms of the Mannheim curve α too.

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Section: Mannheim Darboux Ruled Surfacementioning

confidence: 99%

“…Proof. Since the normal vector field N of DR α is [7], and the normal vector field N * of MDRS of the curve α is…”

Section: Definition 22mentioning

confidence: 99%

On the differential geometric elements of Mannheim Darboux ruled surface in E^3

Kilicoglu¹,

Şenyurt²

2016

ams

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“…He also defined the equation for an enveloping curve of the family of normal planes for a space curve. Suleyman and Seyda [3] determined the concept of parallel curves, which means that if the evolute exists, then the evolute of the parallel arc will also exist and the involute will coincide with the evolute. Brewster and David [4] stated that a curve is composed of two arcs with a common evolute, and the common evolute of two arcs must be a curve with only one tangent in each direction.…”

Section: Introductionmentioning

confidence: 99%

On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

2018

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Recently, extensive research has been done on evolute curves in Minkowski space-time. However, the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature. In this study, a special kind of generalized evolute and involute curve is considered in four-dimensional Minkowski space. We consider (1,3)-evolute curves with respect to the casual characteristics of the (1,3)-normal plane that are spanned by the principal normal and the second binormal of the vector fields and the (0,2)-evolute curve that is spanned by the tangent and first binormal of the given curve. We restrict our investigation of (1,3)-evolute curves to the (1,3)-normal plane in four-dimensional Minkowski space. This research contribution obtains a necessary and sufficient condition for the curve possessing the generalized evolute as well as the involute curve. Furthermore, the Cartan null curve is also discussed in detail.

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“…By using the similiar method we produce a new ruled surface based on the other ruled surface. The differential geometric elements of the involuteD scroll are examined in [9]. It is well-known that, if a curve is differentiable in an open interval, at each point, a set of mutually orthogonal unit vectors can be constructed.…”

Section: Introductionmentioning

confidence: 99%

On the singularity and distribution parameters of involutive and Bertrandian Frenet ruled surfaces in $\mathbb{E}^{3}$

Kilicoglu¹,

Şenyurt²

2016

ntmsci

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Abstract:In this paper we consider twelve special ruled surfaces associated to the curve α,the involute curve α * and Bertrand mate α * * , with k 1 = 0. They are called as Frenet ruled, involutive Frenet ruled and Bertrandian Frenet ruled surfaces, cause of their generators are the Frenet vector fields of curve α. First we give all the parametrizations of all Frenet ruled surfaces in terms of the Frenet apparatus of curve α. We examined distribution parameters and Singularity of involutive and Bertrand Frenet ruled surfaces based on the Frenet apparatus of curve α in E 3 .

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On the Differential Geometric Elements of the Involute $${\tilde{\rm D}}$$ D ~ -Scroll in E 3

Cited by 7 publications

References 3 publications

On the differential geometric elements of Mannheim Darboux ruled surface in E^3

On the differential geometric elements of Mannheim Darboux ruled surface in E^3

On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

On the singularity and distribution parameters of involutive and Bertrandian Frenet ruled surfaces in $\mathbb{E}^{3}$

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