Geometric analysis of translation surfaces based on special curves in the modified orthogonal frame

The geometry of translation surfaces generated by curves related by the modified orthogonal frame in Euclidean 3-space was investigated in this paper. We described the geometric features leading to minimality and developability by taking into account particular pairs as Bertrand, Mannheim, and involute-evolute curves. Furthermore, we revealed new connections between surface behavior and curvature requirements by incorporating adjoint curves into the building process. Understanding translation surfaces in the wider context of differential geometry has advanced considerably due to the examples and theorems that have been presented.