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

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

Department of Mathematics
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.

AMS Subject Classification: 53A04, 53A99

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