Conoid Surfaces with Given Mean or Gaussian Curvature in Euclidean 3-Space
Abstract
In this paper, we construct a right conoid surface in the Euclidean $3$-space $\mathbb{R}^{3}$ with a given mean curvature or a given Gaussian curvature satisfying certain conditions. We also provide examples of right conoid surfaces corresponding to given mean and Gaussian curvatures satisfying these conditions.
Keywords
References
- M. Berger, B. Gostiaux, Geometrie Differentielle: Varietes, Courbes et Surfaces, Presses Univ. France, France, 1987.
- M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, USA, 1976.
- A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, Boca Raton, FL, USA, 1997.
- A. Kreyszig, Differential Geometry, Dover Publications, New York, USA, 1991.
- E. Güler, O. Kişi, Conoid surfaces family in three-dimensional Euclidean space, Proceedings of the 5th International Conference on Applied Engineering and Natural Sciences, 1(1) (2023), 139-146. https://doi.org/10.59287/icaens.982
- E. Güler, O. Kişi, Conoid surfaces family in Minkowski 3-Space, Proceedings of the 5th International Conference on Applied Engineering and Natural Sciences, 1(1) (2023), 147-154. https://doi.org/10.59287/icaens.983
- M. K. Choi, Y. H. Kim, H. Liu, et al., Helicoidal surfaces and their Gauss map in Minkowski 3-space, Bull. Korean Math. Soc., 47(4) (2010), 859–881. https://doi.org/10.4134/BKMS.2010.47.4.859
- Ch. Delaunay, Sur la surface de r´evolution dont la courbure moyenne est constante, J. Math. Pures Appl., 6 (1841), 309–315.
- M. P. do Carmo, M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J., 34(3) (1982), 425–435. https://doi.org/10.2748/tmj/1178229204
- R. Lopez, E. Demir, Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, Open Math., 12(9) (2014), 1349–1361. https://doi.org/10.2478/s11533-014-0415-0