Slant helix of order n and sequence of Darboux developables of principal‐directional curves

In this paper, we consider the sequence of the principal‐directional curves of a curve γ and define the slant helix of order n ( n‐SLH ) of the curve in Euclidean 3‐space. The notion is an extension of the notion of slant helix. We present an important formula that determines if the n th principal‐directional curve of γ can be the slant helix of order n ( n ≥ 1). As an application of singularity theory, we study the singularities classifications of the Darboux developable of n th principal‐directional curve of γ . It is demonstrated that the formula plays a key role in characterizing the singularities of the Darboux developables of the n th principal‐directional curve of a curve γ .