Designing Surface Pencils from Bertrand Pairs as Iso-Curvature Lines via the Modified Orthogonal Frenet Frame

In this paper, we investigate the problem of designing surface pencil pairs through Bertrand curves as iso-curvature lines in three-dimensional Euclidean space. By using the modified orthogonal frame and suitable marching-scale functions, we construct a pair of parametric surface pencils along a prescribed Bertrand pair. Necessary and sufficient conditions are derived for the Bertrand curves to be iso-curvature lines (isoparametric curves and curvature lines) on their corresponding surface pencils. The special case in which the curves also satisfy the isogeodesic condition is also discussed. In addition, several cases related to the torsion functions of the prescribed curves are analyzed. The construction is also extended to ruled surface pencil pairs, where the ruling directions are expressed with respect to the modified orthogonal frame. Finally, examples involving helical and planar Bertrand pairs are provided to show the effect of the controlling functions on the generated surface pencils.