Natural mates of Frenet curves in Euclidean 3-space

Authors : SHARIEF DESHMUKH, BANG-YEN CHEN, AZEB ALGHANEMI

Pages : 2826-2840

View : 19 | Download : 4

Publication Date : 0000-00-00

Article Type : Research Paper

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.
Keywords : Rectifying curves, natural mate, spherical curves, conjugate mate, helix, slant helix