Relationships between the characteristics of mutually orthogonal families of curves and surfaces

In the Euclidean space E3, we consider the family {Lτ} of the curves Lτ with the tangent unit vector τ = τ (x, y, z) and the family {Sτ} of the surfaces Sτ with the unit normal τ which are orthogonal to the curves Lτ , i.e., to the field τ. Each of these families continuously fills in a domain D in E3. We have obtained formulas which express the classical characteristics of the surfaces Sτ : the principal directions, the principal curvatures, the mean curvature, and the Gaussian curvature in terms of the classical characteristics of the curves Lτ , i.e., their Frenet basis, the first curvature, and the second curvature. A new proof for the equality of the non-holonomicity values of the fields of principal directions has been obtained. The proofs are based on the fact that the principal curvatures are stationary values of the normal curvature at each point of the surface Sτ.