Construction of a surface pencil with a common special surface curve

In this study, we introduce a new type of surface curves called D- type curve. This curve is de ned by the property that the unit Darboux vector W 0 of a surface curve r(s) and unit surface normal n along the curve r(s) satisfy the condition D n,W 0 E = constant. We point out that a D-type curve is a geodesic curve or an asymptotic curve in some special cases. Then, by using the Frenet vectors and parametric representation of a surface pencil as a linear combination of the Frenet vectors, we investigate necessary and sufficient condition for a curve to be a D-type curve on a surface pencil. Moreover, we introduce some corollaries by considering the D-type curve as a helix, a Salkowski curve or a planar curve. Finally, we give some examples for the obtained results.