On the translation surfaces generated by spherical indicatrices of regular curves in Euclidean 3-space and their characterizations

In this paper, we investigate the translation surfaces generated by the tangent and normal indicatrices of two regular curves in three-dimensional Euclidean space. We establish the necessary and sufficient conditions for the generating curves of these translation surfaces to be geodesic lines, asymptotic lines, and lines of curvature. Furthermore, we identify the essential conditions for these translation surfaces to be developable or minimal. We conclude this work by generalizing the results obtained for spherical k-indicatrices.