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

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 nth principal‐dire...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 43; no. 17; pp. 9888 - 9903
Main Authors: Li, Yanlin, Wang, Zhigang, Zhao, Tiehong
Format: Journal Article
Language:English
Published: Freiburg Wiley Subscription Services, Inc 30-11-2020
Subjects:
Darboux developable
Euclidean geometry
n‐SLH
principal‐directional curve
Singularity (mathematics)
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Summary: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 nth 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 nth 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 nth principal‐directional curve of a curve γ.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.6663