IJPAM: Volume 74, No. 4 (2012)

A NEW CHARACTERIZATION FOR INCLINED CURVES BY
THE HELP OF SPHERICAL REPRESENTATIONS
ACCORDING TO BISHOP FRAME

Raheleh Ghadami$^1$, Yusuf Yayli$^2$
$^1$Department of Mathematics
Islamic Azad University Urmia Branch
Urmia, IRAN
$^2$Department of Mathematics
Faculty of Science
University of Ankara
Tandoğan, Ankara, TURKEY


Abstract. In this paper we investigate spherical images the $N_{1}$and $N_{2}$ indicatrix of a slant helix. We obtain that the spherical images are spherical helices. Moreover, arc lengths of spherical representations of tangent vector field $T,$ vector field $N_{1},$ vector field $N_{2}$ and the vector field $\overrightarrow{C}=\frac{\overrightarrow{w}}{\Vert \overrightarrow{w}\Vert }$ , where $\overrightarrow{w}% =-k_{2}N_{1}+k_{1}N_{2} $ is the Darboux vector field of a space curve $% \alpha $ in $E^{3}$are calculated. Let us denote the spherical representation of $\overrightarrow{T},\overrightarrow{N_{1}},\overrightarrow{% N_{2}}$ and $\overrightarrow{C}$ by $\left( \overrightarrow{T}\right) ,\left( \overrightarrow{N_{1}}\right) ,\left( \overrightarrow{N_{2}}\right) $ and $\overrightarrow{C}$ respectively.The arc element $ds_{c}$ of the spherical representation $\left( \overrightarrow{C}\right) $ expressed in terms of the harmonic curvature $H=\frac{k_{2}}{k_{1}}=const$ is slant helix of bishop frame. Thus the following characterization is given. The curve $% \alpha \subset E^{3}$ is an inclined curve if and only if the arc length $% s_{c}$ of the Darboux spherical representation $\left( \overrightarrow{C}% \right) $ of $\alpha $ is constant.

Received: October 31, 2011

AMS Subject Classification: 53A04, 53A99

Key Words and Phrases: inclined curve, harmonic curvature, ordinary helix, slant helix, spherical helix

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 74
Issue: 4