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Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$

Year 2023, Volume: 6 Issue: 3, 114 - 121, 30.09.2023

Betül Bulca Sokur Tuğçe Dirim

https://doi.org/10.32323/ujma.1330866

Abstract

In this paper, we study the conchodial surfaces in 3-dimensional Euclidean space with the condition $\Delta x_{i}=\lambda _{i}x_{i}$ where $\Delta $ denotes the Laplace operator with respect to the first fundamental form. We obtain the classification theorem for these surfaces satisfying under this condition. Furthermore, we have given some special cases for the classification theorem by giving the radius function $r(u,v)$ with respect to the parameters $u$ and $v$.

Keywords

Conchoid Gaussian curvature Laplace operator Mean curvature

References

  • [1] E.H. Lockwood, A Book of Curves, Cambridge University Press, 1961.
  • [2] A. Albano, M. Roggero, Conchoidal transform of two plane curves, AAECC, 21(2010), 309-328.
  • [3] J.R. Sendra, J. Sendra, An algebraic analysis of conchoids to algebraic curves, AAECC, 19(2008), 413-428.
  • [4] A. Sultan, The Limacon of Pascal: Mechanical generating fluid processing, J. Mech. Eng. Sci., 219(8)(2005), 813-822.
  • [5] R.M.A. Azzam, Limacon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light, J. Opt. Soc. Am. Opt. Imagen Sci. Vis., 9(1992), 957-963.
  • [6] D. Gruber, M. Peternell, Conchoid surfaces of quadrics, J. Symbolic Computation, 59(2013), 36-53.
  • [7] B. Odehnal, Generalized conchoids, KoG, 21(2017), 35-46.
  • [8] B. Odehnal, M. Hahmann, Conchoidal ruled surfaces, 15. International Conference on Geometry and Graphics, 1-5 August 2012, Montreal, Canada.
  • [9] M. Peternell, D. Gruber, J. Sendra, Conchoid surfaces of rational ruled surfaces, Comput. Aided Geom. Design, 28(2011), 427-435.
  • [10] M. Peternell, D. Gruber, J. Sendra, Conchoid surfaces of spheres, Comput. Aided Geom. Design, 30(2013), 35-44.
  • [11] M. Peternell, L. Gotthart, J. Sendra, J. R. Sendra, Offsets, conchoids and pedal surfaces, J. Geo., 106(2015), 321-339.
  • [12] B. Bulca, S.N. Oruç, K. Arslan, Conchoid curves and surfaces in Euclidean 3-Space, J. BAUN Inst. Sci. Technol., 20(2) (2018), 467-481.
  • [13] M. Dede, Spacelike Conchoid curves in the Minkowski plane, Balkan J. Math., 1(2013), 28–34.
  • [14] M.Ç . Aslan, G.A. S¸ekerci, An examination of the condition under which a conchoidal surfaces is a Bonnet surface in the Euclidean 3-Space, Facta Univ. Ser. Math. Inform., 36(2021), 627–641.
  • [15] S. C¸ elik, H.B. Karada˘g, H.K. Samanci, The conchoidal twisted surfaces constructed by anti-symmetric rotation matrix in Euclidean 3-Space, Symmetry, 15(6)(2023), 1191.
  • [16] O.J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata, 34(1990), 105-112.
  • [17] R. Lopez, Minimal translation surfaces in hyperbolic space, Beitr. Algebra Geom., 52(1) (2011), 105-112. [18] M. Bekkar, H. Zoubir, Surfaces of revolution in the 3-Dimensional Lorentz-Minkowski space satisfying Dri liri, Int. J. Contemp. Math. Sciences, 3(24) (2008), 1173 - 1185.
  • [19] M. Bekkar, B. Senoussi, Factorable surfaces in three-dimensional Euclidean and Lorentzian spaces satisying Dri = liri, Int. J. Geom., 103(2012), 17-29.
  • [20] S.A. Difi, H. Ali, H. Zoubir, Translation-Factorable surfaces in the 3-dimensional Euclidean and Lorentzian spaces satisfying Dri = liri, EJMAA, 6(2) (2018), 227-236.
  • [21] H. Al-Zoubi, A.K. Akbay, T. Hamadneh, M. Al-Sabbah, Classification of surfaces of coordinate finite type in the Lorentz–Minkowski 3-Space, Axioms, 11(7) (2022), 326.
  • [22] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CCR Press, 1997.
  • [23] B. O’Neill, Elementary Differential Geometry, Academic Press, USA, 1997.
  • [24] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1983.
  • [25] B.Y. Chen, Finite Type Submanifolds and Generalizations, Universita degli Studi di Roma La Sapienza, Istituto Matematico Guido Castelnuovo, Rome, 1985.
  • [26] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18(1966), 380-385.

Year 2023, Volume: 6 Issue: 3, 114 - 121, 30.09.2023

Betül Bulca Sokur Tuğçe Dirim

https://doi.org/10.32323/ujma.1330866

Abstract

References

  • [1] E.H. Lockwood, A Book of Curves, Cambridge University Press, 1961.
  • [2] A. Albano, M. Roggero, Conchoidal transform of two plane curves, AAECC, 21(2010), 309-328.
  • [3] J.R. Sendra, J. Sendra, An algebraic analysis of conchoids to algebraic curves, AAECC, 19(2008), 413-428.
  • [4] A. Sultan, The Limacon of Pascal: Mechanical generating fluid processing, J. Mech. Eng. Sci., 219(8)(2005), 813-822.
  • [5] R.M.A. Azzam, Limacon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light, J. Opt. Soc. Am. Opt. Imagen Sci. Vis., 9(1992), 957-963.
  • [6] D. Gruber, M. Peternell, Conchoid surfaces of quadrics, J. Symbolic Computation, 59(2013), 36-53.
  • [7] B. Odehnal, Generalized conchoids, KoG, 21(2017), 35-46.
  • [8] B. Odehnal, M. Hahmann, Conchoidal ruled surfaces, 15. International Conference on Geometry and Graphics, 1-5 August 2012, Montreal, Canada.
  • [9] M. Peternell, D. Gruber, J. Sendra, Conchoid surfaces of rational ruled surfaces, Comput. Aided Geom. Design, 28(2011), 427-435.
  • [10] M. Peternell, D. Gruber, J. Sendra, Conchoid surfaces of spheres, Comput. Aided Geom. Design, 30(2013), 35-44.
  • [11] M. Peternell, L. Gotthart, J. Sendra, J. R. Sendra, Offsets, conchoids and pedal surfaces, J. Geo., 106(2015), 321-339.
  • [12] B. Bulca, S.N. Oruç, K. Arslan, Conchoid curves and surfaces in Euclidean 3-Space, J. BAUN Inst. Sci. Technol., 20(2) (2018), 467-481.
  • [13] M. Dede, Spacelike Conchoid curves in the Minkowski plane, Balkan J. Math., 1(2013), 28–34.
  • [14] M.Ç . Aslan, G.A. S¸ekerci, An examination of the condition under which a conchoidal surfaces is a Bonnet surface in the Euclidean 3-Space, Facta Univ. Ser. Math. Inform., 36(2021), 627–641.
  • [15] S. C¸ elik, H.B. Karada˘g, H.K. Samanci, The conchoidal twisted surfaces constructed by anti-symmetric rotation matrix in Euclidean 3-Space, Symmetry, 15(6)(2023), 1191.
  • [16] O.J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata, 34(1990), 105-112.
  • [17] R. Lopez, Minimal translation surfaces in hyperbolic space, Beitr. Algebra Geom., 52(1) (2011), 105-112. [18] M. Bekkar, H. Zoubir, Surfaces of revolution in the 3-Dimensional Lorentz-Minkowski space satisfying Dri liri, Int. J. Contemp. Math. Sciences, 3(24) (2008), 1173 - 1185.
  • [19] M. Bekkar, B. Senoussi, Factorable surfaces in three-dimensional Euclidean and Lorentzian spaces satisying Dri = liri, Int. J. Geom., 103(2012), 17-29.
  • [20] S.A. Difi, H. Ali, H. Zoubir, Translation-Factorable surfaces in the 3-dimensional Euclidean and Lorentzian spaces satisfying Dri = liri, EJMAA, 6(2) (2018), 227-236.
  • [21] H. Al-Zoubi, A.K. Akbay, T. Hamadneh, M. Al-Sabbah, Classification of surfaces of coordinate finite type in the Lorentz–Minkowski 3-Space, Axioms, 11(7) (2022), 326.
  • [22] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CCR Press, 1997.
  • [23] B. O’Neill, Elementary Differential Geometry, Academic Press, USA, 1997.
  • [24] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1983.
  • [25] B.Y. Chen, Finite Type Submanifolds and Generalizations, Universita degli Studi di Roma La Sapienza, Istituto Matematico Guido Castelnuovo, Rome, 1985.
  • [26] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18(1966), 380-385.
There are 25 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Articles
Authors

Betül Bulca Sokur 0000-0001-5861-0184

Tuğçe Dirim 0000-0001-5893-0401

Early Pub Date September 21, 2023
Publication Date September 30, 2023
Submission Date July 21, 2023
Acceptance Date September 18, 2023
Published in Issue Year 2023 Volume: 6 Issue: 3

Cite

APA Bulca Sokur, B., & Dirim, T. (2023). Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$. Universal Journal of Mathematics and Applications, 6(3), 114-121. https://doi.org/10.32323/ujma.1330866
AMA Bulca Sokur B, Dirim T. Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$. Univ. J. Math. Appl. September 2023;6(3):114-121. doi:10.32323/ujma.1330866
Chicago Bulca Sokur, Betül, and Tuğçe Dirim. “Conchoidal Surfaces in Euclidean 3-Space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$”. Universal Journal of Mathematics and Applications 6, no. 3 (September 2023): 114-21. https://doi.org/10.32323/ujma.1330866.
EndNote Bulca Sokur B, Dirim T (September 1, 2023) Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$. Universal Journal of Mathematics and Applications 6 3 114–121.
IEEE B. Bulca Sokur and T. Dirim, “Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$”, Univ. J. Math. Appl., vol. 6, no. 3, pp. 114–121, 2023, doi: 10.32323/ujma.1330866.
ISNAD Bulca Sokur, Betül - Dirim, Tuğçe. “Conchoidal Surfaces in Euclidean 3-Space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$”. Universal Journal of Mathematics and Applications 6/3 (September 2023), 114-121. https://doi.org/10.32323/ujma.1330866.
JAMA Bulca Sokur B, Dirim T. Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$. Univ. J. Math. Appl. 2023;6:114–121.
MLA Bulca Sokur, Betül and Tuğçe Dirim. “Conchoidal Surfaces in Euclidean 3-Space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$”. Universal Journal of Mathematics and Applications, vol. 6, no. 3, 2023, pp. 114-21, doi:10.32323/ujma.1330866.
Vancouver Bulca Sokur B, Dirim T. Conchoidal Surfaces in Euclidean 3-space Satisfying $\Delta x_{i}=\lambda _{i}x_{i}$. Univ. J. Math. Appl. 2023;6(3):114-21.
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