The relatively osculating developable surfaces of a surface along a direction curve

Year 2020, Volume: 69 Issue: 1, 511 - 527, 30.06.2020

Yasin Ünlütürk , Rashad Abdel-baky

https://doi.org/10.31801/cfsuasmas.527231

Cited By: 1

Abstract

We construct a developable surface tangent to a surface along a curve on the surface. We call this surface as relatively osculating developable surface. We choose the curve as the tangent normal direction curve on which the new surface is formed in the Euclidean 3-space. We obtain some results about the existence and uniqueness, and the singularities of such developable surfaces. We also give two invariants of curves on a surface which characterize these singularities. We present two results for special curves such as asymptotic line and line of curvature which are rulings of the relatively osculating surface.

Keywords

Relatively osculating developable surfaces, Curves on surfaces, Ruled surfaces, Direction curve, Singularities

References

• Abbena, E., Salamon, S. and Gray, A. Modern differential geometry of curves and surfaces with Mathematica, Chapman and Hall/CRC; 3 edition, June 21, 2006.
• Cipolla, R. and Giblin, P. J., Visual Motion of Curves and Surfaces, Cambridge Univ. Press, 2000.
• Hathout, F., Bekar, M. and Yaylı, Y., Ruled surfaces and tangent bundle of unit 2-sphere, International Journal of Geometric Methods in Modern Physics, 14(10) (2017),1750145.
• Izumiya, S. and Takeuchi, N., Singularities of ruled surfaces in R³. Math. Proceedings of Cambridge Philosophical Soc.,vol 130 (2001),1--11
• Izumiya, S. and Takeuchi, N., Geometry of Ruled Surfaces, Applicable Mathematics in the Goldon Age (ed. J. C. Misra), 305--308, Narosa Pulishing House, New Delhi, 2003.
• Izumiya, S. and Takeuchi, N., Special curves and ruled surfaces, Beitr. Algebra Geom. 44(1) (2003), 203-212.
• Izumiya, S. and Takeuchi, N., New special curves and developable surfaces, Turk J Math, 28(2) (2004), 153-163.
• Izumiya S. and Otani, S., Flat approximations of surfaces along curves, Demonstr. Math. 48(2) (2015) 1--7.
• Hananoi, S. and Izumiya, S., Normal developable surfaces of surfaces along curves. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 147(1) (2017), 177-203.
• Hoschek, J. and Pottman, H., Interpolation and approximation with developable B-spline surfaces , in Mathematical Methods for curves and surfaces, ed. by M. Dæhlen, T. Lyche and L.L. Schumacker, Vanderbilt Univ. Press (1995), 255--264
• Lawrence, S., Developable Surfaces: Their History and Application, Nexus Netw. J. (2011) 13: 701-714.
• Markina, I.and Raffaelli, M., Flat approximations of hypersurfaces along curves, manuscripta math. (2018). https://doi.org/10.1007/s00229-018-1072-6.
• Milman, R. S. and Parker, G. D., Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
• Porteous, I. R., Geometric Differentiation for the Intelligence of Curves and Surfaces (second edition). Cambridge University Press, Cambridge, 2001.
• Sasaki, T., Projective Differential Geometry and Linear Homogeneous Differential Equations, Rokko Lectures in Mathematics, Kobe University, vol 5, 1999
• Schneider, M., Interpolation with Developable Strip-Surfaces Consisting of Cylinders and Cones, in Mathematical Methods for curves and surfaces II, ed. by M. Dæhlen, T. Lyche and L.L. Schumacker, Vanderbilt Univ. Press, (1998), 437--444.
• Ushakov, V., Developable surfaces in Euclidean space, J. Austral. Math. Soc. Ser. A 66 (1999), 388-402.
• Vaisman, I., A First Course in Differential Geometry, Pure and Applied Mathematics, A Series of Monograph and Textbooks, Marcel Dekker, 1984.