eng
Shahid Bahonar University of Kerman
Journal of Mahani Mathematical Research Center
2251-7952
2251-7952
2017-05-01
6
2
57
72
10.22103/jmmrc.2017.10079.1038
1866
Construction of a surface pencil with a common special surface curve
Onur Kaya
onur.kaya@cbu.edu.tr
1
Mehmet Önder
mehmetonder197999@gmail.com
2
Manisa Celal Bayar University
Independent Researcher
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 $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. 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.
http://jmmrc.uk.ac.ir/article_1866_f228056f4057aee18d10d5b8fc41f15a.pdf
Surface pencil
$D$-type curve
Parametric representation
Marching-scale function
Surface curve