Paper's Title:

On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space

Author(s):

M. Khalifa Saad and R. A. Abdel-Baky

Department of Mathematics, Faculty of Science,
Islamic University of Madinah,
KSA.
Department of Mathematics, Faculty of Science,
Sohag University, Sohag,
EGYPT.
E-mail: mohamed_khalifa77@science.sohag.edu.eg, mohammed.khalifa@iu.edu.sa

Department of Mathematics, Faculty of Science,
Assiut University, Assiut,
EGYPT.
E-mail: rbaky@live.com

Abstract:

This paper aims to study the skew ruled surfaces by using the quasi-frame of Smarandache curves in the Euclidean 3-space. Also, we reveal the relationship between Serret-Frenet and quasi-frames and give a parametric representation of a directional ruled surface using the quasi-frame. Besides, some comparative examples are given and plotted which support our method and main results.

Paper's Title:

Sweeping Surfaces with Darboux Frame in Euclidean 3-space E3

Author(s):

F. Mofarreh, R. Abdel-Baky and N. Alluhaibi

Mathematical Science Department, Faculty of Science,
Princess Nourah bint Abdulrahman University
Riyadh  11546,
Saudi Arabia.
E-mail: fyalmofarrah@pnu.edu.sa

 
Department of Mathematics, Faculty of Science,
University of Assiut,
Assiut 71516,
Egypt.
E-mail: rbaky@live.com

Department of Mathematics Science and Arts, College Rabigh Campus,
 King Abdulaziz University
Jeddah,
Saudi Arabia.
E-mail: nallehaibi@kau.edu.sa

Abstract:

The curve on a regular surface has a moving frame and it is called Darboux frame. We introduce sweeping surfaces along the curve relating to the this frame and investigate their geometrical properties. Moreover, we obtain the necessary and sufficient conditions for these surfaces to be developable ruled surfaces. Finally, an example to illustrate the application of the results is introduced.

Paper's Title:

On the Equiform Geometry of the Involute-evolute Curve Couple in Hyperbolic and de Sitter Spaces

Author(s):

M. Khalifa Saad, H. S. Abdel-Aziz and A. A. Abdel-Salam

Department of Mathematics,
Faculty of Science,
Islamic University of Madinah,
KSA.
E-mail: mohammed.khalifa@iu.edu.sa

Department of Mathematics,
Faculty of Science,
Sohag University, Sohag,
EGYPT.
E-mail: habdelaziz2005@yahoo.com

Department of Mathematics,
Faculty of Science,
Sohag University, Sohag,
EGYPT.
E-mail: asem2e@yahoo.com

Abstract:

In this paper, we aim to investigate the equiform differential geometric properties of the involute-evolute curve couple with constant equiform curvatures in three-dimensional hyperbolic and de Sitter spaces. Also, we obtain some relations between the curvature functions of these curves and investigate some special curves with respect to their equiform curvatures. Finally, we defray two computational examples to support our main findings.

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