On the striction curves of involute and Bertrandian Frenet ruled surfaces in E^3

On the singularity and distribution parameters of involutive and Bertrandian Frenet ruled surfaces in $\mathbb{E}^{3}$

“…are the parametrization of the ruled surface which are called involutive tangent ruled surface, involutive normal ruled surface, involutive binormal ruled surface and involutive Darboux ruled surface, respectively, [5]. Normal vector fields η * 1 , η * 2 , η * 3 , and η * 4 of ruled surfaces ϕ * 1 , ϕ * 2 , ϕ * 2 , and ϕ * 4 respectively, along the curve involute α * , can be expressed by the following matrix, [6] Proof.…”

“…are the parametrization of the ruled surface which are called Bertrandian tangent ruled surface, Bertrandian normal ruled surface, Bertrandian binormal ruled surface and Bertrandian Darboux ruled surface, respectively, [5]. …”