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Slant helix of order n and sequence of Darboux developables of principal‐directional curves
Author(s) -
Li Yanlin,
Wang Zhigang,
Zhao Tiehong
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6663
Subject(s) - mathematics , gravitational singularity , sequence (biology) , helix (gastropod) , singularity , order (exchange) , principal (computer security) , euclidean geometry , euclidean space , geometry , family of curves , mathematical analysis , pure mathematics , computer science , ecology , genetics , finance , snail , economics , biology , operating system
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 .