Normal Developable Surfaces of A Surface Along A Direction Curve

Rashad Abdel-Baky Abdel-Sattar; Yasin Unluturk

doi:10.22342/jims.26.3.872.319-333

Rashad Abdel-Baky Abdel-Sattar ⁽¹⁾ , Yasin Unluturk ⁽²⁾

(1) ,

(2)

https://doi.org/10.22342/jims.26.3.872.319-333

Issue
VOLUME 26 NUMBER 3 (NOVEMBER 2020)

Submitted
September 13, 2019

Accepted
June 16, 2020

Published
November 5, 2020

Keywords:

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

PDF

Abstract

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.

Full text article

Generated from XML file

References

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). https://doi.org/10.1007/s00229-018-1072-6.

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)

Authors

Rashad Abdel-Baky Abdel-Sattar

Yasin Unluturk

yasin_unluturk@yahoo.com (Primary Contact)

Abdel-Sattar, R. A.-B., & Unluturk, Y. (2020). Normal Developable Surfaces of A Surface Along A Direction Curve. Journal of the Indonesian Mathematical Society, 26(3), 319–333. https://doi.org/10.22342/jims.26.3.872.319-333