Dual curves associated with the Bonnet ruled surfaces

An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.