Abstract
In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve. We obtain the Frenet apparatus of these spherical indicatrices and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrices and their associated curves.
Highlights
We study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve
The theory of curves is a subbranch of geometry which deals with curves in Euclidean space or other spaces by using differential and integral calculus
A general helix in E3 is defined as: its tangent vector field makes a constant angle with a fixed direction
Summary
The theory of curves is a subbranch of geometry which deals with curves in Euclidean space or other spaces by using differential and integral calculus. If the principal normal vector field makes a constant angle with a fixed direction, it is called slant helix. They gave the relation of curvature and torsion between the principal-direction curve and its mate curve They defined a new curve called PD-rectifying curve and gave a new characterization of a Bertrand curve by means of the PD-rectifying curve. They made an application of associated curves and studied a general helix and slant helix as principal-donor and second principal-donor curve of a plane, respectively. Kula et al (2010) gave some characterizations for a unit speed curve in R3 to being a slant helix by using its tangent, principal normal and binormal indicatrix. We give the characterizations of being general helix and slant helix in terms of these image curves
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