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., & 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
Onur Kaya; Mehmet 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
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
VANCOUVER
Kaya, O., Onder, M. Construction of a surface pencil with a common special surface curve. Journal of Mahani Mathematical Research, 2017; 6(2): 57-72. doi: 10.22103/jmmrc.2017.10079.1038
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