Ruled Surfaces Constructed by Planar Curves In Euclidean 3-Space With Density

Year 2020, Volume: 16 Issue: 1, 81 - 88, 27.03.2020

Mustafa Altın , Ahmet Kazan , H.bayram Karadağ

Abstract

In the
present study, firstly we recall the parametric expressions of planar curves
with zero



















phi-curvature in Euclidean 3-space with density
e^ax_1
and with the aid of the
Frenet frame of these planar curves, we obtain the Smarandache curves of them.
After that, we study on ruled surfaces which is constructed by curves with zero


-curvature in Euclidean 3-space with density

e^ax_1 and their Smarandache curves by giving
the striction curves, distribution parameters, mean curvature and Gaussian
curvature of these ruled surfaces. Also, we give some examples for these
surfaces by plotting their graphs. We use Mathematica
when we are plotting the graphs of examples.

Keywords

Ruled surfaces, Smarandache curves, Weighted curvature

Supporting Institution

Inonu University

Project Number

FDK-2018-1349

Thanks

We're very much thankful to Inonu University BAP (Scientific Research Projects) unit for their support to our this study.

References

• 1. Brauner H. and Kickinger W. Baugeometrie 1. Bauverlag, 1977.
• 2. Hanbay K., Alpaslan N., Talu M.F. and Hanbay D. 2016. Principal curvatures based rotation invariant algorithms for efficient texture classification. Neurocomputing; 199: 77–89.
• 3. Ravani B. and Wang J.W. 1991. Computer Aided Geometric Design of Line Constructs. ASME Journal of Mechanical Design; 113(4): 363-371.
• 4. Gromov M. 2003. Isoperimetry of Waists and Concentration of Maps. Geometric and Functional Analysis; 13: 178–215.
• 5. Corwin I., Hoffman N., Hurder S., Sesum Vojislav and Xu Y. 2006. Differential Geometry of Manifolds with Density. RoseHulman Und. Math. J.; 1–15.
• 6. Alarcón A., Ferrer L. and Martín F. 2008. Density theorems for complete minimal surfaces in 𝑅3 . Geometric and Functional Analysis; 18(1): 1–49.
• 7. Belarbi L. and Belkhelfa M. 2015. Some Results in Riemannian Manifolds with Density. Analele Universitatii Oradea Fasc. Matematica; XXII(2): 81–86.
• 8. Engelstein M., Marcuccio A, Quinn M.. and Pritchard T. 2009. Isoperimetric problems on the sphere and on surfaces with density. New York Journal of Mathematics; 15: 97–123.
• 9. Hieu D.T. and Nam T. Le. 2008. On the four vertex theorem in planes with radial density 𝑒𝜙(𝑟). Colloquium Mathematicum; 113(1): 119–149.
• 10. Hieu D.T. and Nam T. Le. 2009. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure and Applied Analysis; 243(2): 277–285.
• 11. Hieu D.T. and Hoang N.M. 2009. Ruled minimal surfaces in 𝑅3 with density 𝑒 𝑧 . Pacific Journal of Mathematics; 243(2): 277–285.
• 12. Belarbi L. and Belkhelfa M. 2012. Surfaces in 𝑅3 with density. IManager’s Journal on Mathematics; 1(1): 34–48.
• 13. Morgan F., Hutchings M. and Howards H. 2000. The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature. Transactions of the American Mathematical Society; 352(11): 4889–4909.
• 14. Morgan F. 2005. Manifolds with Density. Notices Amer Math Soc; 52(8): 853–858.
• 15. Morgan F. 2009. Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture. The American Mathematical Monthly; 116(2): 134–142.
• 16. Rosales C., Cañete A., Bayle V. and Morgan F. 2008. On the isoperimetric problem in Euclidean space with density. Calculus of Variations and Partial Differential Equations; 31(1): 27–46.
• 17. Yıldız Ö. 2018. Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density. Mathematics; 7(27): 1–10.
• 18. Yıldız Ö.G., Hızal S. and Akyiğit M. 2018. Type 𝐼+ helicoidal surfaces with prescribed weighted mean or Gaussian curvature in Minkowski space with density. Analele Stiintifice Ale Universitatii Ovidius Constanta, Seria Matematica; 26(3): 99–108.
• 19. Yoon D.W. 2017. Weighted minimal translation surfaces in the Galilean space with density. Open Mathematics; 15(1): 459–466.
• 20. Yoon D.W., Kim D.-S., Kim H. Y. and Lee, J. W. 2017. Constructions of Helicoidal Surfaces in Euclidean Space with Density. Symmetry; 9(173): 1–9.
• 21. Yoon D.W. and Yuzbasi Z.K. 2018. Weighted Minimal Translation Surfaces in Euclidean Space with Density. International Journal of Geometric Methods in Modern Physics; 15(11).
• 22. Altın M., Kazan A. and Karadağ, H.B. 2019. Ruled Surfaces in 𝐸3 with Density. Honam Mathematical Journal; 41(4): 683-695.
• 23. Altın M., Kazan A. and Karadağ, H.B. 2020. Monge Hypersurfaces in Euclidean 4-Space with Density. Journal of Polytechnic; 23(1): 207-214.
• 24. Rutter J.W. Geometry of Curves (1st), New York: Chapman and Hall/CRC, 2000.
• 25. Hacısalihoğlu H.H. Differensiyel Geometri, Ankara: Gazi Üniversitesi Basın Yayın Yüksekokulu Basımevi, 1983.
• 26. O’Neill B. Elementary Differential Geometry, Academic Press Inc, 1966.
• 27. Altın M., Kazan A. and Karadağ, H.B. 2019. Rotational Surfaces Generated by Planar Curves in 𝐸3 with Density. Int. J. of Anal. and Appl.; 17(3): 311-328.