Normal Developable Surfaces of A Surface Along A Direction Curve

Rashad Abdel-Baky Abdel-Sattar, Yasin Unluturk


We construct a developable surface normal to a surface along a curve on the surface. As differs from the work Hananoi, we choose the curve as the normal direction curve on which the new surface is formed in Euclidean space. We obtain some results about the uniqueness and the singularities of such developable surfaces. We also give two invariants of curves on a surface which characterize singularities.


Normal developable surfaces, Curves on surfaces, Ruled surfaces, Direction curve, Singularities

Full Text:



E. Abbena, S. Salamon, Alfred Gray, Modern differential geometry of curves and surfaces with Mathematica, Chapman and Hall/CRC; 3 edition (June 21, 2006).

R. Cipolla, P. J. Giblin, Visual Motion of Curves and Surfaces, Cambridge Univ. Press, 2000.

F. Hathout, M. Bekar, and Y. Yaylı, Ruled surfaces and tangent bundle of unit 2-sphere, International Journal of Geometric Methods in Modern Physics, 14(10) (2017),1750145.

S. Izumiya and N. Takeuchi, Geometry of Ruled Surfaces, Applicable Mathematics in the Golden Age (ed. J. C. Misra), 305-308, Narosa Publishing House, New Delhi, 2003.

S. Izumiya, N. Takeuchi, Special curves and ruled surfaces, Beitr. Algebra Geom. 44(1) (2003), 203-212.

S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk J Math, 28(2) (2004), 153-163.

S. Izumiya and S. Otani, Flat approximations of surfaces along curves, Demonstr. Math. 48(2) (2015) 1-7.

S. Hananoi, S. Izumiya, Normal developable surfaces of surfaces along curves. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 147(1) (2017), 177-203.

I. Markina, M. Raffaelli, Flat approximations of hypersurfaces along curves, manuscripts math. (2018).

R. S. Milman, and G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey. (1977).

I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces (second edition). Cambridge University Press, Cambridge, (2001).

I. Vaisman, A First Course in Differential Geometry, Pure and Applied Mathematics, A Series of Monograph and Textbooks, Marcel Dekker, (1984)



  • There are currently no refbacks.

Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243

p-ISSN: 2086-8952 | e-ISSN: 2460-0245

Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats