In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $\vec{W}_{0} $ of a surface curve $\vec{r}(s)$ and unit surface normal $\vec{n} $ along the curve $\vec{r}(s)$ satisfy the condition $\left\langle \vec{n} ,\vec{W}_{0} \right\rangle =\text{constant}$. 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.
Kaya,O and Onder,M . (2017). Construction of a surface pencil with a common special surface curve. Journal of Mahani Mathematical Research, 6(2), 57-72. doi: 10.22103/jmmrc.2017.10079.1038
MLA
Kaya,O , and Onder,M . "Construction of a surface pencil with a common special surface curve", Journal of Mahani Mathematical Research, 6, 2, 2017, 57-72. doi: 10.22103/jmmrc.2017.10079.1038
HARVARD
Kaya O, Onder M. (2017). 'Construction of a surface pencil with a common special surface curve', Journal of Mahani Mathematical Research, 6(2), pp. 57-72. doi: 10.22103/jmmrc.2017.10079.1038
CHICAGO
O Kaya and M Onder, "Construction of a surface pencil with a common special surface curve," Journal of Mahani Mathematical Research, 6 2 (2017): 57-72, doi: 10.22103/jmmrc.2017.10079.1038
VANCOUVER
Kaya O, Onder M. Construction of a surface pencil with a common special surface curve. J. Mahani Math. Res.. 2017;6(2):57-72. doi: 10.22103/jmmrc.2017.10079.1038
