A new characterization for inclined curves by the help of spherical representations according to bishop frame

Ghadami R., YAYLI Y.

International Journal of Pure and Applied Mathematics, cilt.74, sa.4, ss.455-463, 2012 (Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 74 Sayı: 4
  • Basım Tarihi: 2012
  • Dergi Adı: International Journal of Pure and Applied Mathematics
  • Derginin Tarandığı İndeksler: Scopus, zbMATH
  • Sayfa Sayıları: ss.455-463
  • Anahtar Kelimeler: Harmonic curvature, Inclined curve, Ordinary helix, Slant helix, Spherical helix
  • Ankara Üniversitesi Adresli: Evet

Özet

In this paper we investigate spherical images the N 1and 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 C→ = →w/||→w||, where w→ = -k 2N 1 + k 1N 2 is the Darboux vector field of a space curve α in E 3are calculated. Let us denote the spherical representation of T→, N→ 1, N 2 and C→ by (T→), (N→ 1, (N→ 2) and C→ respectively. The arc element ds c of the spherical representation (C→) expressed in terms of the harmonic curvature H = k 2/k 1= const is slant helix of bishop frame. Thus the following characterization is given. The curve α ⊂ E 3 is an inclined curve if and only if the arc length s c of the Darboux spherical representation (C→) of α is constant. © 2012 Academic Publications, Ltd.