scholarly journals Simultaneous Developability of Partner Ruled Surfaces according to Darboux Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Soukaina Ouarab
Keyword(s):  
Ruled Surfaces ◽  
Regular Surface ◽  
Darboux Frame

In this paper, we introduce original definitions of Partner ruled surfaces according to the Darboux frame of a curve lying on an arbitrary regular surface in E 3 . It concerns T g Partner ruled surfaces, T n Partner ruled surfaces, and g n Partner ruled surfaces. We aim to study the simultaneous developability conditions of each couple of two Partner ruled surfaces. Finally, we give an illustrative example for our study.

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 Some Special Ruled Surfaces Generated by a Direction Curve according to the Darboux Frame and their Characterizations

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Nidal Echabbi ◽  
Amina Ouazzani Chahdi
Keyword(s):  
Vector Field ◽  
Ruled Surfaces ◽  
Geodesic Curve ◽  
Regular Surface ◽  
Principal Line ◽  
Asymptotic Line ◽  
Darboux Vector ◽  
Base Curve ◽  
Darboux Frame ◽  
Made In

In this work, we consider the Darboux frame T , V , U of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a V -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.

Bertrand Offsets of Ruled Surfaces with Darboux Frame

2016 ◽  
Vol 72 (3) ◽  
pp. 1151-1159 ◽  
Author(s):  
Gülsüm Yeliz Şentürk ◽  
Salim Yüce
Keyword(s):  
Ruled Surfaces ◽  
Darboux Frame

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.

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