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*.

Abstract

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43-50