# IJPAM: Volume 74, No. 4 (2012)

**A NEW CHARACTERIZATION FOR INCLINED CURVES BY**

THE HELP OF SPHERICAL REPRESENTATIONS

ACCORDING TO BISHOP FRAME

THE HELP OF SPHERICAL REPRESENTATIONS

ACCORDING TO BISHOP FRAME

Department of Mathematics

Islamic Azad University Urmia Branch

Urmia, IRAN

Department of Mathematics

Faculty of Science

University of Ankara

Tandoğan, Ankara, TURKEY

**Abstract. **In this paper we investigate spherical images the and
indicatrix of a slant helix. We obtain that the spherical images are
spherical helices. Moreover, arc lengths of spherical representations of
tangent vector field vector field vector field and the
vector field
, where
is the Darboux vector field of a space curve in are calculated. Let us denote the spherical
representation of
and
by
and
respectively.The arc element of the
spherical representation
expressed in
terms of the harmonic curvature
is slant helix
of bishop frame. Thus the following characterization is given. The curve
is an inclined curve if and only if the arc length of the Darboux spherical representation
of 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