Canal Surface Whose Center Curve is a Hyperbolic Curve with Hyperbolic Frame

Year 2021, Volume: 14 Issue: 1, 106 - 120, 15.04.2021

Ali Uçum

https://doi.org/10.36890/iejg.829766

Cited By: 1

Abstract

In this paper, we obtain the parametrization of the canal surfaces whose center curves are the hyperbolic curves on the hyperbolic space $H^{2}$ in $%
\mathbb{E}_{1}^{3}$. The parametrization of the canal surface is expressed according to the hyperbolic frame given in [10]. Then, the parallel
surface of this surface is studied. Also, we define the notion of the associated canal surface. Lastly, we give the geometric properties of these
surfaces such that Weingarten surface, $\left( X,Y\right) $-Weingarten surface and linear Weingarten surface.

Keywords

Canal surfaces, tubular surfaces, hyperbolic curve, Weingarten surface, (X,Y )-Weingarten surface

References

• [1] Chen B.-Y.: Geometry of Submanifolds. Dekker, New York (1973).
• [2] Gök İ.: Quaternionic approach of canal surfaces constructed by some new ideas. Adv. Appl. Clifford Algebras. 27 (2), 1175-1190 (2017).
• [3] İlarslan K. and Boyacıoğlu Ö.: Position vectors of a spacelike W-curve in Minkowski space E₁³, Bulletin of the Korean Mathematical Society. 44 (3), 429-438 (2007).
• [4] İlarslan K. and Boyacıoğlu Ö.: Position vectors of a timelike and a null helix in Minkowski 3-space. Chaos, Solitons & Fractals, 38 (5), 1383–1389 (2008).
• [5] Karacan M. K., Es H. and Yaylı, Y.: Singular points of the Tubular Surfaces in Minkowski 3-space. Sarajevo Journal of Mathematics. 14, 73–82 (2006).
• [6] Karacan M. K., Yoon D.W. and Tuncer Y.: Tubular Surfaces of Weingarten Types in Minkowski 3-space. Gen. Math. Notes. 22, 44–56 (2014).
• [7] Karacan M. K. and Bukcu B.: An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space. Rendiconti del Circolo Matematico di Palermo.57, 193–201 (2008).
• [8] Karacan M. K. and Tuncer Y.: Tubular Surfaces of Weingarten types in Galilean and Pseudo-Galilean. Bulletin of Mathematical Analysis and Applications. 5, 87–100 (2013).
• [9] Kocakuşaklı E., Tuncer O. O., Gök İ. and YaylıY.: A new representation of canal surfaces with split quaternions in Minkowski 3-space. Adv. Appl. Clifford Algebras. 27 (2), 1387–1409 (2017).
• [10] Liu H.: Curves in three dimensional Riemannian space forms. Results. Math. 66, 469–480 (2014).
• [11] Petrović-Torgašev M. and Šućurović E.: W-curves in Minkowski space-time. Novi Sad J. Math. 32 (2), 55-65 (2002).
• [12] Ro J. S. and Yoon D. W.: Tubes of Weingarten types in a Euclidean 3-space. Journal of the Chungcheong mathematical society. 22 (3), 359-366 (2009).
• [13] Uçum A., İlarslan, K.: New types of canal surfaces in Minkowski 3-space. Adv. Appl. Clifford Algebras. 26, 449–468 (2016).
• [14] Yoon D. W.: On non-developable ruled surfaces in Euclidean 3-spaces. Indian Journal of Pure and Applied Mathematics. 38, 281–290 (2007).