THE RELATIVELY OSCULATING DEVELOPABLE SURFACES OF A SURFACE ALONG A DIRECTION CURVE

We construct a developable surface tangent to a surface along a curve on the surface. We call this surface as relatively osculating developable surface. We choose the curve as the tangent normal direction curve on which the new surface is formed in the Euclidean 3-space. We obtain some results about the existence and uniqueness, and the singularities of relatively osculating developable surfaces. We also give two invariants of curves on a surface which determine these singularities. We present two results for special curves such as asymptotic line and line of curvature which are rulings of the relatively osculating surface