Characterizations of Adjoint Curves in Euclidean 3-Space

Conjugate mates for non-null Frenet curves

“…Proof. The proof of this corollary is proved exactly the same way as in the Euclidean case such as, for instance, the elemantry proof given in [14]. Note that the proof of the followings can be made in exactly the same way as the previous theorem.…”

“…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. In this paper, we will recall the concept of conjugate mate for non-null Frenet curves in Minkowski 3-space by moving from the notion of conjugate mate defined in [3].…”

New version of Bäcklund transformations in Euclidean 3‐space

“…Assume that is a regular curve in 3‐dimensional space and is the Frenet frame of the curve α . Then the adjoint curve ϵ of the any space curve α is defined by …”

Naturel mates of equiaffine space curves in affine 3-space

“…On the other hand, there are a lot of studies under the concepts of binormal-direction curve, W-direction curve, conjugate mate curve, principal normal adjoint curve and binormal adjoint curve, etc. in different spaces denoted with metric, by using a curve on a region consisting of that curve but Deshmukh, Chen, and Alghanemi defined the natural mate of a space curve in Euclidean 3-space as tangent to the principal normal vector field of given space curve, along the curve not for a curve on a region [7][8][9][10][11]. We use this definition and extend affine tangent, affine normal and affine binormal mates of space curves in affine 3-space, by using equiaffine frame and giving equiaffine curvatures.…”