The Generalization of Helices

Document Type: Research articles


Meitoku High School, Nagaoka Niigata, Japan



We defined new special curves in Euclidean 3-space which refer to clad helices and found the geometric invariants of clad helices [17]. These notions are generalizations of the notion of cylindrical helices and slant helices. Using the geometric invariants of clad helices in this article, we proposed approaches to construct examples of clad helices in E3 and on S2. Moreover, we obtained the classi cation of special developable surfaces under the condition of the existence of clad helices as a geodesic and existence of slant helices as a line of curvature.


Article Title [Persian]


Abstract [Persian]


[1] A.T.Ali, Position vectors of slant helices in Euclidean space E3. Journal of the Egyptian Mathematical Society, 20 (2012), 1-6.
[2] B.Altunkaya, F.Kahraman, L.Kula and C.Aytekin, On rectifying slant helices in Euclidean 3-space. Konuralp J. Math. 4 (2016), 17-24.
[3] F.Do˘gan and Y.Yayli, On isophote curves and their characterizations. Turk J. Math, 39 (2015), 650-664.
[4] J.Monterde, Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion. Comput Aided Geomet Design, 26 (2009), 271-278.
[5] L.Kula and Y.Yayli, On slant helix and its spherical indicatrix. Applied Mathematics and Computation, 169 (2005), 600-607.
[6] M.Altinok and L.Kula, Slant helices generated by plane curves in Euclidean 3-space. Palestine Journal of Mathematics, 5(2) (2016), 164-174.
[7] M.I.Munteanu and A.I.Nistor, A New Approach on Constant Angle Surfaces in E3. Turk J Math, 33 (2009), 169-178.
[8] Paul D. Scofield, Curves of Constant Precession. The American Mathematical Monthly, 102 (1995), 531-537.
[9] P.Lucas and J.A.Ortega-Yagues, Slant helices in the Euclidean 3-space revisited. Bull. Belg. Math.Soc.Simon Stevin, 23 (2016), 133-150.
[10] S.Izumiya, H.Katsumi and T.Yamasaki, The rectifying developable and the spherical Darboux image of a space curve, Geometry and topology of caustics-Caustics ’98-Banach Center Publications 50 (1999), 137-149.
[11] S.Izumiya, K.Saji and N.Takeuchi, Flat surfaces along cuspidal edges, Journal of Singularities, 16 (2017), 73-100.
[12] S.Izumiya and N.Takeuchi, Generic properties of Helices and Bertrand curves. Journal of Geometry, 74 (2002), 97-109
[13] S.Izumiya and N.Takeuchi, New special curves and developable surfaces. Turk J Math, 28 (2004), 153-163.
[14] S.Izumiya and N. Takeuchi, Special Curves And Ruled Surface. Beitr¨age zur Algebra und Geometrie Contributions to Algebra and Geometry, 44 (2003), 203-212.
[15] S.Kaya, O.Ate¸s, I.Gok and Y.Yayli, Timelike clad helices and developable surfaces in Minkowski 3 space, Rend. Circ. Mat. Palermo, II. Ser (2018).
[16] S.Kaya and Y.Yayli, Generalized helices and singular points, Casp. J. Math. Sci., 6 (2) (2017), 121-132.
[17] T.Takahashi, N.Takeuchi. Clad helices and developable surfaces. Bulletin of Tokyo Gakugei University Division of Natural Sciences, 66 (2014), 1-9.