Quadric Surfaces in Terms of Coordinate Finite II-type

b19c854f-d6eb-4006-934f-1c42aad2408320250226073311677wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpQuadric Surfaces in Terms of Coordinate Finite II-typeHamzaAlzaareerDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANHassanAlzoubiDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANWaseemAlmashalehDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JORDANMutazAlsabbaghDepartment of Basic Sciences and Humanities, Imam Abdulrahman bin Faisal University, SAUDI ARABIAQuadric surfaces of finite type are a class of three-dimensional surfaces in geometry that are defined by second-degree equations in three variables, which are an essential part of the study of conic sections, and they exhibit a wide range of interesting geometric properties and real-world applications. This paper explores the intriguing domain of quadric surfaces, particularly emphasizing those of finite type. This will start by defining the ideas of the second Laplace-Beltrami operators, involving a surface's second fundamental form (II) in the Euclidean space E3. Then, we characterize the coordinate finite type quadrics involving the second fundamental form.2262025226202569749https://wseas.com/journals/mathematics/2025/a185106-005(2025).pdf10.37394/23206.2025.24.9https://wseas.com/journals/mathematics/2025/a185106-005(2025).pdfB.-Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math, Vol. 17, 1991. H. Alzaareer, H. Al-Zoubi, F. Abed Al-Fattah, Quadrics with finite Chen-type Gauss map, Journal of Prime Research in Mathematics, Vol.18 No. 1, 2022, 96-107. D.-S. Kim, On the Gauss map of quadric hypersurfaces, J. Korean Math. Soc., Vol. 31 (1994), 429–437. B.-Y. Chen, F. Dillen, Quadrics, of finite type, J. Geom. Vol. 38, 1990, 16-22. 10.3836/tjm/1270128488Ch. Baikoussis, B.-Y. Chen, L. Verstraelen, Ruled Surfaces and tubes with finite type Gauss map, Tokyo J. Math., Vol. 16, (1993) 341-349. 10.1007/s00022-012-0136-0M. Bekkar, B. Senoussi, Translation surfaces in the 3-dimensional space satisfying Δ IIIri=ωiri , J. Geom., 103, (2012), 367-374. 10.21099/tkbjm/1496162870S. M. Choi, On the Gauss map of Ruled Surfaces in a 3-dimensional Minkowski space, Tsukuba J. Math., Vol. 19, 285-304 (1995) 10.1017/s0004972700028616B.-Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Ruled surfaces of finite type, Bull. Austral. Math. Soc., 42, 447-553 (1990). F. Dillen, J. Pass, L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica, Vol. 18, 1990, 239- 246. 10.2996/kmj/1138038815O. Garay, On a certain class of finite type surfaces of revolution, Kodai Math. J. Vol. 11, 1988, 25-31. 10.21099/tkbjm/1496162145B.-Y. Chen, S. Ishikawa, On classification of some surfaces of revolution of finite type, Tsukuba J. Math., Vol. 17, 287-298 (1993). 10.1007/bf03322254Ch. Baikoussis, L. Verstraelen, The Chentype of the spiral surfaces, Results. Math., Vol. 28, 1995, 214-223. 10.1017/s001708950003055xF. Denever, R. Deszcz, L. Verstraelen, The Chen type of the noncompact cyclides of Dupin, Glasg. Math. J., 36, 1994, 71-75. 10.1007/bf01230997F. Denever, R. Deszcz, L. Verstraelen, The compact cyclides of Dupin and a conjecture by B.-Y Chen, J. Geom. Vol. 46, 1993, 33-38. Ch. Baikoussis, L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Semi. Mat. Messina Ser II Vol. 16, 1993, 31-42. B. Senoussi and M. Bekkar, Helicoidal surfaces with △ J r = Ar in 3-dimensional Euclidean space, Stud. Univ. Babes-Bolyai Math. 60 2015, pp 437–448. 10.1007/bf03322734S. Stamatakis, H. Al-Zoubi, On surfaces of finite Chen type. Result. Math. Vol. 43, 2003, 181-190. B.-Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, Singapore, 1984. M. Mhailan, M. Abu Hammad, M. Al Horani, R. Khalil, On fractional vector analysis, J. Math. Comput. Sci. \Vol 10 (2020), 2320- 2326. http://dx.doi.org/10.28919/jmcs/4863. I. Batiha, J. Oudetallah, A. Ouannas, A. AlNana, I. Jebril. Tuning the fractional-order PID-controller for blood Glucose level of diabetic patients, Int. J. Adv. Soft Comp. App., 13 No. 2 1-10 (2021). 10.15849/ijasca.220720.07I. Batiha, S. Njadat R. Batyha. A. Zraiqat, A. Dabbabneh, Sh. Momani. Design fractionalorder PID controller for single-joint robot arm model. Int. J. Adv. Soft Comp. App., 14 No.2 96-114 (2022).