Evolutions of the Ruled Surfaces via the Evolution of Their Directrix Using Quasi Frame along a Space Curve

Abstract

In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.

Timelike Surfaces of Evolution in Minkowski 3-Space

“…Surfaces constructed by the evolution of the spherical indicatrices of a space curve are studied by Soliman in [1]. In that study, they used the Frenet frame of the curves in Euclidean 3-space.…”

“…The following definitions can be given according to quasi frame for evolving curves in considering the references [1,4,7]. From now on we study timelike surfaces in .…”

“…Especially, the curves obtained with the help of a given space curve and special curve couples are studied by many researchers. Bertrand curve couples, involute-evolute curve couples, Mannheim curve couples and spherical indicatrices of a space curve can be given as examples of these curves, [1].…”

“…They derived the time evolution equations for the Frenet frame of binormal spherical indicatrix and gave some geometric properties of these surfaces. In [1], Soliman studied surfaces constructed by the evolution of the spherical indicatrices of a space curve. In that study, they used the Frenet frame of the curves in Euclidean 3-space.…”

“…In that study, they used the Frenet frame of the curves in Euclidean 3-space. In [4], Soliman studied evolution of ruled surfaces with the help of their directrix using the quasi frame along a space curve in Euclidean 3-space. In that study, he gave the time evolution equations of a space curve given with the quasi frame.…”