Some Characterizations of Spherical Indicatrix Curves Generated by Sannia Frame

Year 2021, Volume: 9 Issue: 2, 222 - 232, 15.10.2021

Süleyman Şenyurt , Kebire Hilal Ayvacı , Davut Canlı

Abstract

In this study, we have first provided the relations between the Frenet frame and Sannia frame on the striction points of four ruled surfaces of
each formed by taking the basis as the tangent, normal, binormal and Darboux vector. Second, we have defined the relations between the
Sannia vectors and their derivatives. For each Sannia frame, we have calculated the Darboux frame and expressed those in terms of Frenet
frame. Last, we have obtained the arc lengths and the geodesic curvatures according to both Euclidean space E3 and unit sphere S2 of Sannia
vectors for each four of ruled surfaces.

Keywords

Ruled surface, Sannia frame, Striction curve, Spherical indicatrix, Geodesic curvature.

References

• [1] Bilici, M. The Curvatures and the natural lifts of the spherical indicator curves of the involute-evolute curve. Master Thesis, Ondokuz Mayıs University, The Institute of Science, Samsun, 1999.
• [2] Bishop, R.L. There is more than one way to Frame a curve. American Mathematical Monthly, 82(3), (1975), :246-251.
• [3] C¸ apın, R. Spherical Indicator Curves In Minkowski Space. Master Thesis, Gaziantep University, The Institute of Science, Gaziantep, 2016.
• [4] Fenchel, W. On The Differential Geometry of Closed Space Curves, Bulletin of American Mathematical Society, 57, (1951), (44-54).
• [5] G¨okyes¸il, D.Characterizations Of Some Curves According To Dual Bishop Frame In Dual Space. Master Thesis, Manisa Celal Bayar University, The Institute of Science, Manisa, 2018.
• [6] Hacısalihog˘lu, H. H., Diferensiyel geometri, Cilt I-II, AnkaraU¨ niversitesi, Fen Faku¨ltesi Yayınları, 2000.
• [7] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
• [8] Pottmann, H., and Wallner, J. Computational line geometry. Springer Science & Business Media, 2009.
• [9] S¸enyurt, S. Natural lifts and the geodesic sprays for the spherical indicatrices of the Mannheim partner curves in E3. International Journal of Physical Sciences, 7(23), (2012), 2980-2993.
• [10] S¸ enyurt S. and O¨ zgu¨ner Z. The Natural Lift Curves And Geodesic Curvatures Of The Spherical Indicatrices Of The Bertrand Curve Couple. Ordu Univ. J. Sci. Tech., 3(2), (2013), 58-81.
• [11] Yılmaz, S., O¨ zyılmaz, E. and Turgut, M. New Spherical Indicatrices and Their Characterizations. An. S¸ t. Univ. Ovidius Constanta, 18(2), (2010), 337-354.