On the developable time-like trajectory ruled surfaces in a Lorentz 3-space

On the trajectory ruled surfaces of framed base curves in the Euclidean space

“…So trajectory ruled surfaces are one of the most important topics of ruled surface theory to study spatial mechanism and space kinematic. For more details on ruled surfaces and its applications, see previous studies 1–6 …”

“…For more details on ruled surfaces and its applications, see previous studies. [1][2][3][4][5][6] In differential geometry, the curve theory is highly important and a classical subject. Generally, geometers consider regular curve, if they work on curve theory.…”

Singularities of Non-Developable Ruled Surface with Spacelike Ruling

“…Timelike ruled surface with timelike rulings have been studied by Abdel-All and others [7]. Küçük has obtained some results on the developable timelike ruled surfaces in the same space [8]. Furthermore, Ugurlu and Onder have introduced Frenet frames and Frenet invariants of timelike ruled surfaces in Minkowski 3-space [9].…”

Non-null slant ruled surfaces

“…The function defined by , , , dk q dq dq dq δ = is called the distribution parameter (or drall) of the ruled surface. Then, N is called developable surface if and only if 0 δ = [12,16,18]. Then at all points of same ruling, the tangent planes are identical, i.e., tangent plane contacts the surface along a ruling.…”