The main goal of this paper, some characteristic properties of a ruled surface in E 3 are given related to the geodesic curvature and the second fundamental form of its.

Abstract:The relaxed elastic line of second kind on an oriented surface in the Minkowski space was defined and for the relaxed elastic line of second kind which is lying on an oriented surface the Euler-Lagrange equations were derived. Furthermore, whether these curves lie on a curvature line or not was investigated and some applications were given.

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The purpose of this paper, first, is to give a definition of the inverse surface of a given regular surface with respect to a unit sphere in E3. Second, some characteristic properties of the inverse surface are to express depending on the algebraic invariants of the original surface. In the last part of the study, we gave examples supporting our claims and plotted their graphics with the help of Maple software program.