On rectifying curves in Euclidean 3-space

Abstract

First, we study rectifying curves via the dilation of unit speed curves on the unit sphere S 2 in the Euclidean space E 3. Then we obtain a necessary and sufficient condition for which the centrode d(s) of a unit speed curve α(s) in E 3 is a rectifying curve to improve a main result of [4]. Finally, we prove that if a unit speed curve α(s) in E 3 is neither a planar curve nor a helix, then its dilated centrode β(s) = ρ(s)d(s) , with dilation factor ρ , is always a rectifying curve, where ρ is the radius of curvature of α .

Some characterization of curves in $\widetilde{\mathbf{SL}_{2}\mathbb{R}}$

“…Rectifying curves in the three-dimensional sphere, Euclidean space and Minkowski space are studied in [2,14,3,6,13]. If the position vector ϕ lies on its rectifying plane then ϕ(s) is called rectifying curve.…”

Space curves defined by curvature-torsion relations and associated helices

“…A rectifying curve α(s) satisfies the condition α(s), n(s) ≡ 0, where n(s) is the principal normal, i.e., at each point the position vector lies in the rectifying plane, spanned by tangent and binormal. Rectifying curves are of interest in analyzing joint kinematics, due to their close relationship with the centrode of a curve [4,5,7,13,27].…”

Conjugate mates for non-null Frenet curves

“…These notions gave us a certain method constructing the general helices and the slant helices (see [4,5]). In [3], Deshmukh, Chen and Alghanemi studied some new type associated curve called as the natural mate and the conjugate mate of a Frenet curve in Euclidean 3space, closely related with the principal (binormal)directional curve defined in [4,5] and also the adjoint curve in [14] . In [4,5,14], authors characterized these curves and also gave new results for them.…”