scholarly journals Sweeping surface of center curve on surface in Euclidean 3-space E³

2021 ◽  
Vol 20 ◽  
pp. 235-243
Author(s):  
Rashad A. Abdel-Baky ◽  
Fatemah Mofarreh
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Moving Frame ◽  
Regular Surface ◽  
Geometrical Properties ◽  
Darboux Frame ◽  
Curve On Surface ◽  
Necessary And Sufficient

For the curve on the regular surface, there is moving frame with this thatis named Darboux frame. Sweeping surfaces through the curve associated with Darboux frame are introduced and their geometrical properties are investigated. Moreover, we obtain the necessary and sufficient conditions of this kind of surfaces to be developable ruled surfaces. Finally, an example to illustrate the application of the results is introduced.

scholarly journals Smarandache Ruled Surfaces according to Darboux Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Soukaina Ouarab
Keyword(s):  
Architectural Design ◽  
Sufficient Conditions ◽  
Surface Modeling ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Regular Surface ◽  
New Approach ◽  
Darboux Frame ◽  
Necessary And Sufficient

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. The principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface in E 3 . It concerns T g -Smarandache ruled surface, T n -smarandache ruled surface, and g n -Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an example with illustrations is presented.

scholarly journals Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nidal Echabbi ◽  
Amina Ouazzani Chahdi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Formula Unit ◽  
Regular Surface ◽  
Darboux Vector ◽  
Spherical Curve ◽  
Constant Torsion ◽  
Darboux Frame ◽  
Necessary And Sufficient ◽  
Unit Normal

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.

scholarly journals Characteristic Properties of Type-2 Smarandache Ruled Surfaces According to the Type-2 Bishop Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ibrahim Al-Dayel ◽  
E. M. Solouma
Keyword(s):  
Sufficient Conditions ◽  
Special Kind ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Minimal Amount ◽  
Bishop Frame ◽  
Necessary And Sufficient

In this paper, we define and investigate a special kind of ruled surfaces called type-2 Smarandache ruled surfaces related to the type-2 Bishop frame in E 3 . From this point and depending on the type-2 Bishop curvature, we provide the necessary and sufficient conditions that allow these surfaces to be developable in a minimal amount of time. Furthermore, an example is given to clear the results.

scholarly journals Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Soukaina Ouarab
Keyword(s):  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Regular Curve ◽  
Necessary And Sufficient

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E 3 . It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

scholarly journals K-type slant helices on spacelike and timelike surfaces

2021 ◽  
Vol 25 (2) ◽  
pp. 201-220
Author(s):  
Santosh Kumar ◽  
Buddhadev Pal
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Timelike Surface ◽  
Spacelike Curve ◽  
Slant Helix ◽  
Darboux Frame ◽  
Necessary And Sufficient

We have derived a necessary and sufficient condition for a non-null normal spacelike curve lying in a spacelike or a timelike surface M ⊂ E13, so that the curve becomes a K-type spacelike slant helix with K ∈ {1,2,3}. We have used Darboux frame to define necessary and sufficient conditions. An example is given for a 1-type spacelike slant helix having a spacelike normal and a timelike binormal.

scholarly journals Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Developable Surface ◽  
Bishop Frame ◽  
Tangential Surface ◽  
Necessary And Sufficient

In this paper, we express timelike sweeping surfaces using rotation minimizing frames in Minkowski 3–Space E3 1 . Necessary and sufficient conditions for timelike sweeping surfaces to be developable ruled surfaces are derived. Using these, we analyze the conditions when the resulting timelike developable surface is a cylinder, cone or tangential surface.

Surface family with a common involute asymptotic curve

2016 ◽  
Vol 13 (05) ◽  
pp. 1650062 ◽  
Author(s):  
Ergi̇n Bayram ◽  
Mustafa Bi̇li̇ci̇
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Necessary And Sufficient

We construct a surface family possessing an involute of a given curve as an asymptotic curve. We express necessary and sufficient conditions for that curve with the above property. We also present natural results for such ruled surfaces. Finally, we illustrate the method with some examples, e.g. circles and helices as given curves.

scholarly journals NC -Smarandache Ruled Surface and NW -Smarandache Ruled Surface according to Alternative Moving Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Soukaina Ouarab
Keyword(s):  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Moving Frame ◽  
Necessary And Sufficient

In this study, original definitions of NC − Smarandache ruled surface and NW − Smarandache ruled surface according to the alternative moving frame are introduced in E 3 . The main results of the study are presented in theorems that give necessary and sufficient conditions for those special surfaces to be developable and minimal. Finally, an example with illustrations is presented.

THE GEOMETRICAL PROPERTIES OF BERRY’S PHASE

1989 ◽  
Vol 04 (06) ◽  
pp. 1453-1465 ◽  
Author(s):  
S. GILER ◽  
P. KOSIŃSKI ◽  
L. SZYMANOWSKI
Keyword(s):  
Parameter Space ◽  
Spectral Decomposition ◽  
Parallel Transport ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Geometrical Properties ◽  
Riemannian Connection ◽  
Necessary And Sufficient ◽  
Relationship Of ◽  
The Relationship

Using the spectral decomposition of the Hamiltonian, we clarify the relationship of Berry’s effect with the parallel transport in the parameter space of this Hamiltonian. We derive the necessary and sufficient conditions under which this parallel transport is described by a natural, Riemannian connection. The considerations are illustrated by several examples.

scholarly journals Darboux Iso-Geodesic Special Curve in Euclidean Space

2019 ◽  
Vol 13 (9) ◽  
pp. 98
Author(s):  
M. M. Wageeda ◽  
E. M. Solouma ◽  
M. Bary
Keyword(s):  
Euclidean Space ◽  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Family ◽  
Darboux Frame ◽  
Necessary And Sufficient ◽  
Special Curve

In this paper, by using Darboux frame we scrutinize the issues of reconstructing surfaces with given some unusual Smarandache curves in Euclidean 3-space, we make manifest the family of surfaces as a linear combination of the components of this frame and derive the necessary and sufficient conditions for coefficients to satisfy both the iso-geodesic and iso-parametric requirements.

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