Turkish Journal of Mathematics
Abstract
For each Frenet curve $\alpha $ in the Euclidean 3-space $\mathbb{E}^{3}$, there exists a unique unit speed curve $\beta$ tangent to the principal normal vector field of $\alpha $. We simply call this curve $\beta $ the natural mate of $\alpha$. The main purpose of this paper is to prove some relationships between a Frenet curve and its natural mate. In particular, we obtain some necessary and sufficient conditions for the natural mate of a Frenet curve to be a helix, a spherical curve, or a curve of constant curvature. Several applications of our main results are also presented.
DOI
10.3906/mat-1712-34
Keywords
Rectifying curves, natural mate, spherical curves, conjugate mate, helix, slant helix
First Page
2826
Last Page
2840
Recommended Citation
DESHMUKH, SHARIEF; CHEN, BANG-YEN; and ALGHANEMI, AZEB
(2018)
"Natural mates of Frenet curves in Euclidean 3-space,"
Turkish Journal of Mathematics: Vol. 42:
No.
5, Article 58.
https://doi.org/10.3906/mat-1712-34
Available at:
https://journals.tubitak.gov.tr/math/vol42/iss5/58
