Rashad A. Abdel-Baky

On the one-parameter dual spherical motions

Based on E. Studys dual line coordinates, explicit expressions were developed for the differential properties of one-parameter dual spherical motions that are coordinate systems independent. This eliminates the need of laborious coordinate transformations necessary in the determination of the canonical systems. With the proposed technique, the Disteli formulae of the axodes were derived and the connections between kinematic geometry of a line trajectory were researched. Meanwhile, a ruled analogy of the curvature circle of a curve in planar kinematics was extended into general spatial kinematics. Then a lot of efforts have been directed to establish dual versions of the Euler-Savary equations of a line trajectory resulting in two different expressions depending on the axodes. As a result, theoretical expressions for the direction of a line congruence were revealed. Finally, an example illustrating the application of the obtained formulae was introduced.

The Backlund's theorem in Minkowski 3-space R31

In this paper we present the Minkowski versions of the Backlunds theorem and its application by using the method of moving frames.

Holditch's theorem for one-parameter closed motions

Abstract In this paper, we consider some dual closed spherical and closed spatial motions as in H. H. Hacisalihoǧlu, Mech. Mach. Theory , 1972, 7 , 291–305 and O. Gursoy, Mech. Mach. Theory , 1990, 25 , 131–140. Then we give some Holditch-type results and relations between the integral invariants of corresponding ruled surfaces and spherical surface areas.

On a line congruence which has the parameter ruled surfaces as principal ruled surfaces

In this paper, by using dual vector calculus, a line congruence which has the parameter ruled surfaces as principal ruled surfaces, is introduced. Then some well-known formulas of surfaces theory are proved. Moreover, geometric explanations of these formulas are given.

On the Null Scrolls in Minkowski 3-space

We study ruled surfaces with lightlike ruling in Minkowski 3-space which are said to be null-scrolls. Even that the result is a consequence of some well-known results involving the Gauss map, we give another approach to classify all null-scrolls under the condition where is the Laplace operator with respect to the first fundamental form and the set of 3 real matrices.