Research Article
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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.

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

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.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Yasin Ünlütürk 0000-0002-6395-5272

Rashad Abdel-baky 0000-0001-7016-9280

Publication Date June 30, 2020
Submission Date February 14, 2019
Acceptance Date December 13, 2019
Published in Issue Year 2020 Volume: 69 Issue: 1

Cite

APA Ünlütürk, Y., & Abdel-baky, R. (2020). The relatively osculating developable surfaces of a surface along a direction curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 511-527. https://doi.org/10.31801/cfsuasmas.527231
AMA Ünlütürk Y, Abdel-baky R. The relatively osculating developable surfaces of a surface along a direction curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2020;69(1):511-527. doi:10.31801/cfsuasmas.527231
Chicago Ünlütürk, Yasin, and Rashad Abdel-baky. “The Relatively Osculating Developable Surfaces of a Surface Along a Direction Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 1 (June 2020): 511-27. https://doi.org/10.31801/cfsuasmas.527231.
EndNote Ünlütürk Y, Abdel-baky R (June 1, 2020) The relatively osculating developable surfaces of a surface along a direction curve. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 1 511–527.
IEEE Y. Ünlütürk and R. Abdel-baky, “The relatively osculating developable surfaces of a surface along a direction curve”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 1, pp. 511–527, 2020, doi: 10.31801/cfsuasmas.527231.
ISNAD Ünlütürk, Yasin - Abdel-baky, Rashad. “The Relatively Osculating Developable Surfaces of a Surface Along a Direction Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/1 (June 2020), 511-527. https://doi.org/10.31801/cfsuasmas.527231.
JAMA Ünlütürk Y, Abdel-baky R. The relatively osculating developable surfaces of a surface along a direction curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:511–527.
MLA Ünlütürk, Yasin and Rashad Abdel-baky. “The Relatively Osculating Developable Surfaces of a Surface Along a Direction Curve”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 1, 2020, pp. 511-27, doi:10.31801/cfsuasmas.527231.
Vancouver Ünlütürk Y, Abdel-baky R. The relatively osculating developable surfaces of a surface along a direction curve. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(1):511-27.

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