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.

scholarly journals The Frenet Vector Fields and the Curvatures of the Natural Lift Curve

2012 ◽  
Vol 2 ◽  
pp. 38-43 ◽  
Author(s):  
Evren Ergün ◽  
Mustafa Bilici ◽  
Mustafa Çalişkan
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Similar Calculation ◽  
Darboux Vector ◽  
Lift Curve ◽  
Curvature And Torsion ◽  
The Given ◽  
Made In

In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in 3/1 . Also, a similar calculation is made in 3/1 considering timelike or spacelike Darboux vector field.

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 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 A new aspect of rectifying curves and ruled surfaces in Galilean 3-space

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2953-2962
Author(s):  
Çetin Demir ◽  
İsmail Gök ◽  
Yusuf Yayli
Keyword(s):  
Vector Fields ◽  
The Other ◽  
Ruled Surfaces ◽  
Position Vector ◽  
Darboux Vector ◽  
Minkowski Spaces ◽  
Base Curve ◽  
Rectifying Curve

A curve is named as rectifying curve if its position vector always lies in its rectifying plane. There are lots of papers about rectifying curves in Euclidean and Minkowski spaces. In this paper, we give some relations between extended rectifying curves and their modified Darboux vector fields in Galilean 3-Space. The other aim of the paper is to introduce the ruled surfaces whose base curve is rectifying curve. Further, we prove that the parameter curve of the surface is a geodesic.

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space

2019 ◽  
Vol 16 (05) ◽  
pp. 1950076 ◽  
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
Null Curve ◽  
Base Curve

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

scholarly journals Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 514
Author(s):  
Miekyung Choi ◽  
Young Kim
Keyword(s):  
Circular Cylinder ◽  
Eigenvalue Problems ◽  
Gauss Map ◽  
Natural Generalization ◽  
Ruled Surfaces ◽  
Usual Sense ◽  
Infinite Type ◽  
Base Curve ◽  
Smooth Map ◽  
Nice Tool

A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from the eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with a generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of G-type.

scholarly journals A Note upon some Properties of the Curve of Striction

1925 ◽  
Vol 44 ◽  
pp. 90-97
Author(s):  
H. W. Richmond
Keyword(s):  
Ruled Surfaces ◽  
Mathematical Work ◽  
Fundamental Quantity ◽  
Made In

Certain properties of Ruled Surfaces relating to their Curves of Striction are suggested and immediately proved by the use of a particular kind of coordinates, known as Dual Coordinates, for the Generating Lines. These coordinates, were introduced by Prof. E. Study; the theory of them is fully explained and many beautiful applications are made in his treatise, Geometrie der Dynamen, Teubner, 1903, and in his article, Complexe Grössen in the Encyclopädie der math. Wissenschaften, Bd. I., pp. 147–183, and specially p. 166. In applying the method here I have ventured to regard the fundamental quantity c in a light which, if less rigorous, has the advantage of being familiar to most of us in other mathematical work.

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