On the characterizations of special adjoint curves and surface family for the modified orthogonal frame with torsion
Gülnur Şaffak Atalay
Received May 15, 2025. Published online November 19, 2025.
Abstract: In this context, the main interest of this paper is to construct a family of surfaces in $E^3$ space which is the adjoint pair of a given curve. Then, using the modified orthogonal frame with torsion, we express the necessary conditions for the surface family that recognize the adjoint curve as a geodesic curve, an asymptotic curve and a line of curvature. Also, the singular points of the surface given by the parametric equation are expressed. It is concluded that the adjoint curve of the alpha curve whose torsion is not constant can not be geodesic on the surface under any circumstances. In addition, the case where the adjoint curve of the unit velocity alpha curve with nonconstant torsion with respect to the modified orthogonal frame with torsion is the Salkowski curve, is also exemplified. By the way, this work is illustrated by several examples.
Keywords: the modified orthogonal frame with torsion; adjoint curve; Salkowski curve; geodesic curve; asymptotic curve; line of curvature; surface family
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