A GENERALIZATION OF THE BERTRAND CURVES AS GENERAL INCLINED CURVES IN En

Smarandache curves for spherical indicatrix of the Bertrand curves pair

“…Bertrand curves have been studied by some authors [5,6,7]. Bertrand curves detected by J. Bertrand in 1850 are one of the important and interesting matters of classical special curve theory.…”

“…Let α : I → E 3 unit speed and α * : I → E 3 be the C 2 -class differentiable two curves and the amounts of {T (s), N (s), B(s)} and {T * (s * ), N * (s * ), B * (s * )} are entirely Frenet-Serret frames of the curves α Bertrand and the Bertrand partner α * , respectively, then [5] T * = cos θT − sin θB, N * = N, B * = sin θT + cos θB (1.6)…”

“…From the equation (1.10), we have the following spherical Frenet formulae of (T * ), (N * ) and (B * ) are, respectively, 5) and…”

Bertrand and Mannheim partner -curves on parallel surfaces

“…The well-known examples of such curves are Bertrand curves, involute-evolute curves and Mannheim curves which are studied by many mathematicians in different spaces [2,3,4,5,10,12,14,16]. Moreover, some new definitions of special curve pairs have been given by Kazaz and et al They have considered the notions of Bertrand curve and Mannheim curve for the curves lying fully on regular surfaces and called these curve pairs as Bertrand partner D-curves and Mannheim partner D-curves.…”

Some characterizations of Mannheim partner curves in the Minkowski 3-spaceE13

“…Bertrand partner curves are studied in refs [1][2][3][4]13,15]. Especially the partner curves, i.e., the curves which are related to each other at the corresponding points, have attracted the attention of many mathematicians.…”