CHARACTERIZATION OF THE SLANT HELIX AS SUCCESSOR CURVE OF THE GENERAL HELIX

Year 2014, , 84 - 91, 30.10.2014

Anton Mennınger

https://doi.org/10.36890/iejg.593986

Cited By: 16

Abstract

Keywords

Frenet curves, Frenet frame, Bishop frame, Natural equations, Successor curves, Salkowski curves, Closed curves

References

• [1] Ali, A. T. Position vectors of slant helices in Euclidean 3-space. Journal of the Egyptian Mathematical Society, 20:1–6, 2012, DOI: 10.1016/j.joems.2011.12.005.
• [2] Bilinski, S. Eine Verallgemeinerung der Formeln von Frenet und eine Isomorphie gewisser Teile der Differentialgeometrie der Raumkurven. Glasnik Math.-Fiz. i Astr., 10:175–180. 1955
• [3] Bilinski, S. Über eine Erweiterungsmöglichkeit der Kurventheorie. Monatshefte fr Mathe- matik, 67:289–302, 1963, eudml.org/doc/177219.
• [4] Bishop, R. L. There is more than one way to frame a curve. Amer. Math. Monthly, 82:246– 251, 1975, www.jstor.org/stable/2319846.
• [5] Camci, C., Kula, L., and Altinok, M. On spherical slant helices in euclidean 3-space. arXiv:1308.5532 [math.DG], 2013.
• [6] Hoppe, R. U¨ ber die Darstellung der Curven durch Krmmung und Torsion. Journal fr die reine und angewandte Mathematik, 60:182–187, 1862, http://eudml.org/doc/147848.
• [7] Hoschek, J. Eine Verallgemeinerung der B¨oschungsfl¨achen. Mathematische Annalen, 179:275– 284, 1969, http://eudml.org/doc/161782.
• [8] Izumiya, S. and Takeuchi, N. New Special Curves and Developable Surfaces. Turk J Math, 28:153–163, 2004, journals.tubitak.gov.tr/math/issues/mat-04-28-2/mat-28-2-6-0301-4.pdf.
• [9] Kühnel, W. Differential Geometry: Curves - Surfaces - Manifolds, Second Edition. American Mathematical Society, 2006.
• [10] Kula, L., Ekmekci, N., Yaylı, Y., and İlarslan, K. Characterizations of slant helices in Eu- clidean 3-space. Turk. J. Math., 34(2):261–274, 2010, DOI: 10.3906/mat-0809-17.
• [11] Kula, L. and Yayli, Y. On slant helix and its spherical indicatrix. Appl. Math. Comput., 169(1):600–607, 2010, DOI: 10.1016/j.amc.2004.09.078.
• [12] Menninger, A. Frenet Curves and Successor Curves. arXiv:1302.3175 [math.DG], 2013.
• [13] Monterde, J. Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion. Computer Aided Geometric Design, 26:271278, 2009, DOI: 10.1016/j.cagd.2008.10.002
• [14] Nomizu, K. On Frenet equations for curves of class C∞. Tˆohoku Math. J., 11:106–112, 1959, projecteuclid.org/euclid.tmj/1178244631.
• [15] Salkowski, E. Zur Transformation von Raumkurven. Mathematische Annalen 66: 517–557, 1909, eudml.org/doc/158392.
• [16] Scofield, P. D. Curves of Constant Precession. Amer. Math. Monthly, 102:531–537, 1995, www.jstor.org/stable/2974768.
• [17] Wintner, A. On Frenet’s Equations. Amer. J. Math., 78:349–355, 1956, www.jstor.org/stable/2372520.
• [18] Wong, Y.-C. and Lai, H.-F. A Critical Examination of the Theory of Curves in Three Dimensional Differential Geometry. Tˆohoku Math. J., 19:1–31, 1967, projecteu- clid.org/euclid.tmj/1178243344.