Legendrian dualities between spherical indicatrixes of curves and surfaces according to Bishop frame

Legendrian dualities between spherical indicatrixes of curves in Euclidean 3-space are investigated by using the theory of Legendrian duality. Moreover, the singularities of the ruled surfaces according to Bishop frame which are deeply related to space curves are classified from the viewpoints of wave fronts. We also give some more detail descriptions on the conditions of those singularities