Ruled surfaces generated by elliptic cylindrical curves in the isotropic space

References

[1] P. Alegre, K. Arslan, A. Carriazo, C. Murathan and G. Öztürk, Some special types of developable ruled surface, Hacet. J. Math. Stat. 39 (2010), no. 3, 319–325. Suche in Google Scholar

[2] A. T. Ali, H. S. Abdel Aziz and A. H. Sorour, Ruled surfaces generated by some special curves in Euclidean 3-space, J. Egyptian Math. Soc. 21 (2013), no. 3, 285–294. 10.1016/j.joems.2013.02.004Suche in Google Scholar

[3] L. J. Alías and N. Gürbüz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113–127. 10.1007/s10711-006-9093-9Suche in Google Scholar

[4] M. E. Aydin and M. Ergut, Composite functions with Allen determinants and their applications to production models in economics, Tamkang J. Math. 45 (2014), no. 4, 427–435. 10.5556/j.tkjm.45.2014.1569Suche in Google Scholar

[5] M. E. Aydin and A. Mihai, Classification of quasi-sum production functions with Allen determinants, Filomat 29 (2015), no. 6, 1351–1359. 10.2298/FIL1506351ASuche in Google Scholar

[6] B.-Y. Chen, On submanifolds of finite type, Soochow J. Math. 9 (1983), 17–33. Suche in Google Scholar

[7] B.-Y. Chen, Total mean Curvature and Submanifolds of Finite Type, Ser. Pure Math. 1, World Scientific Publishing, Singapore, 1984. 10.1142/0065Suche in Google Scholar

[8] B.-Y. Chen, On some geometric properties of quasi-sum production models, J. Math. Anal. Appl. 392 (2012), no. 2, 192–199. 10.1016/j.jmaa.2012.03.011Suche in Google Scholar

[9] B.-Y. Chen, Solutions to homogeneous Monge–Ampère equations of homothetic functions and their applications to production models in economics, J. Math. Anal. Appl. 411 (2014), no. 1, 223–229. 10.1016/j.jmaa.2013.09.029Suche in Google Scholar

[10] B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type: Recent development, Tamkang J. Math. 45 (2014), no. 1, 87–108. 10.5556/j.tkjm.45.2014.1564Suche in Google Scholar

[11] B.-Y. Chen, S. Decu and L. Verstraelen, Notes on isotropic geometry of production models, Kragujevac J. Math. 38 (2014), no. 1, 23–33. 10.5937/KgJMath1401023CSuche in Google Scholar

[12] M. Choi and Y. H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38 (2001), no. 4, 753–761. Suche in Google Scholar

[13] M. Crâşmăreanu, Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Appl. 7 (2002), no. 1, 37–42. Suche in Google Scholar

[14] S. Decu and L. Verstraelen, A note on the isotropical geometry of production surfaces, Kragujevac J. Math. 37 (2013), no. 2, 217–220. Suche in Google Scholar

[15] F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), no. 3, 307–320. 10.1007/s002290050142Suche in Google Scholar

[16] Z. Erjavec, B. Divjak and D. Horvat, The general solution of Frenet’s system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space, Int. Math. Forum 6 (2011), no. 17–20, 837–856. Suche in Google Scholar

[17] O. J. Garay, An extension of Takahashi’s theorem, Geom. Dedicata 34 (1990), no. 2, 105–112. 10.1007/BF00147319Suche in Google Scholar

[18] T. Hasanis and T. Vlachos, Hypersurfaces of En+1 satisfying Δ⁢x=A⁢x+B, J. Aust. Math. Soc. Ser. A 53 (1992), no. 3, 377–384. 10.1017/S1446788700036545Suche in Google Scholar

[19] S. Izumiya, H. Katsumi and T. Yamasaki, The rectifying developable and the spherical Darboux image of a space curve, Geometry and Topology of Caustics—CAUSTICS ’98 (Warsaw 1998), Banach Center Publ. 50, Polish Academy of Sciences, Warsaw (1999), 137–149. 10.4064/-50-1-137-149Suche in Google Scholar

[20] S. Izumiya and N. Takeuchi, Special curves and ruled surfaces, Beitr. Algebra Geom. 44 (2003), no. 1, 203–212. Suche in Google Scholar

[21] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turkish J. Math. 28 (2004), no. 2, 153–163. Suche in Google Scholar

[22] Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), no. 3–4, 191–205. 10.1016/S0393-0440(99)00063-7Suche in Google Scholar

[23] J. J. Koenderink and A. van Doorn, Image processing done right, European Conference on Computer Vision, Lecture Notes in Comput. Sci. 2350, Springer, Berlin (2002), 158–172. 10.1007/3-540-47969-4_11Suche in Google Scholar

[24] B. Ons and P. Verstraelen, A geometrical description of visual sensation II, Kragujevac J. Math. 34 (2010), 5–24. Suche in Google Scholar

[25] H. Pottmann, P. Grohs and N. J. Mitra, Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math. 31 (2009), no. 4, 391–419. 10.1007/s10444-008-9076-5Suche in Google Scholar

[26] H. Pottmann and Y. Liu, Discrete surfaces of isotropic geometry with applications in architecture, Mathematics of Surfaces XII, Lecture Notes in Comput. Sci. 4647, Springer, Berlin, Heidelberg (2007), 341–363. 10.1007/978-3-540-73843-5_21Suche in Google Scholar

[27] H. Pottmann and K. Opitz, Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces, Comput. Aided Geom. Design 11 (1994), no. 6, 655–674. 10.1016/0167-8396(94)90057-4Suche in Google Scholar

[28] H. Pottmann and M. Peternell, Applications of Laguerre geometry in CAGD, Comput. Aided Geom. Design 15 (1998), no. 2, 165–186. 10.1016/S0167-8396(97)00023-XSuche in Google Scholar

[29] H. Sachs, Isotrope Geometrie des Raumes, Friedr. Vieweg & Sohn, Braunschweig, 1990. 10.1007/978-3-322-83785-1Suche in Google Scholar

[30] Ž. M. Šipuš, Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hungar. 68 (2014), no. 2, 160–175. 10.1007/s10998-014-0027-2Suche in Google Scholar

[31] Ž. M. Šipuš and B. Divjak, Mappings of ruled surfaces in simply isotropic space I31 that preserve the generators, Monatsh. Math. 139 (2003), no. 3, 235–245. 10.1007/s00605-002-0526-8Suche in Google Scholar

[32] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. 10.2969/jmsj/01840380Suche in Google Scholar

[33] G. E. Vîlcu, A geometric perspective on the generalized Cobb–Douglas production functions, Appl. Math. Lett. 24 (2011), no. 5, 777–783. 10.1016/j.aml.2010.12.038Suche in Google Scholar

[34] G. Vrănceanu, Sur les courbes cylindriques fermées, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 5, 601–607. Suche in Google Scholar

[35] X. Wang and Y. Fu, Some characterizations of the Cobb–Douglas and CES production functions in microeconomics, Abstr. Appl. Anal. 2013 (2013), Article ID 761832. 10.1155/2013/761832Suche in Google Scholar