A new method for designing involute trajectory timelike ruled surfaces in Minkowski 3-space
Abstract
In this study, we present the new concept of involute trajectory ruled surface in Minkowski 3-space. The involute trajectory timelike ruled surface is a surface generated by the motion of a timelike oriented line X along the spacelike involute curve γ(s) of a given timelike base curve r(s). The main purpose of this article is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. These surfaces are formed depending on the angle θ between the Darboux vector D and the binormal vector b of the evolute curve r(s). Also, some new results and theorems related to the developability of the involute trajectory timelike ruled surfaces are obtained. Finally, we illustrate these surfaces by presenting one example.
Downloads
References
Bilici M, Calıskan M (2002) Some characterizations for the pair of involute-evolute curves in Euclidean space. Bull of Pure and Appl Sci 21E(2):289-294
Bilici M (2009) Ph.d. Dissertation, Ondokuz Mayıs University, Institute of Science and Technology, Samsun 3. Bayram E, M. Bilici M (2016) Surface family with a common involute asymptotic curve. Int. J. Geom. Methods Mod. Phys, 13(5):1650062 (9 pages) https://doi.org/10.1142/S0219887816500626
Ravani B, Ku TS (1991) Bertrand Offsets of ruled and developable surfaces. Comput. Aided Geom. Design 23(2):145-152 https://doi.org/10.1016/0010-4485(91)90005-H
Kasap E, Akyıldız FT (2006) Surfaces with common geodesics in Minkowski 3-space. Appl. Math. Comput. 177(1):260-270 https://doi.org/10.1016/j.amc.2005.11.005
Chen Y J, Ravani B (1987) Offset surface generation and contouring in computer aided design. J Mech Des 109:133-142. https://doi.org/10.1115/1.3258777
Farouki R T (1986) The approximation of non-degenerate offset surfaces. Comput. Aided Geom. Design 3(1):15-43. https://doi.org/10.1016/0167-8396(86)90022-1
Turgut A, Hacısalihoglu HH (1997) Timelike ruled surfaces in the Minkowski 3-Space. Far East J. Math. Sci 5(1):83-90. https://doi.org/10.1501/Commua1_0000000427
Yaylı Y, Saracoglu S (2012) On developable ruled surfaces in Minkowski space. Adv Appl Clifford Algebr 22(2):499-510 https://doi.org/10.1007/s00006-011-0305-5
Woestijne VI (1990) Minimal surfaces of the 3- dimensional Minkowski Space. World Scientific Publishing, Singapore, pp 344-369.
Kim YH, Yoon D W (2004) Classification of ruled surfaces in Minkowski 3-spaces. J Geom Phys 49:89-100. https://doi.org/10.1016/S0393-0440(03)00084-6
Gursoy O, Kucuk A (1999) On the invariants of trajectory surfaces. Mech and Mach Theory 34 (4):587-597 https://doi.org/10.1016/S0094-114X(98)00042-1
Kucuk A (2004) On the developable timelike trajectory ruled surfaces in Lorentz 3-space R3 1 . Appl Math Comput 157(2):483-489 https://doi.org/10.1016/j.amc.2003.09.001
Kucuk A, Gursoy O (2004) On the invariants of Bertrand trajectory surface offsets. Appl Math Comput 151(3):763-773 https://doi.org/10.1016/S0096-3003(03)00534-4
Orbay K, Aydemir I (2010) The ruled surface generated by Frenet vectors of a curve in R3 1 . C.B.U. J of Sci 6(2):155-160
O'Neill B (1983) Semi-Riemannian Geometry, Academic Press, New York
Ugurlu H H (1997) On The Geometry of Time-like Surfaces. Commun Fac Sci Univ Ank Ser A1 Math Stat 46:211-223 https://doi.org/10.1501/Commua1_0000000438
Ratcliffe J G (1994) Foundations of Hyperbolic Manifolds, Springer-Verlag, New York https://doi.org/10.1007/978-1-4757-4013-4
Bilici M, Calıskan M (2011) Some new notes on the involutes of the timelike curves in Minkowski 3-space. Int J Contemp Math Sci 6(41):2019-2030.
Kaya, F. E.,On involute and evolute of the curve and curve-surface pair in Euclidean 3-space, Pure and Applied Mathematics Journal, 4(1-2),6-9, 2015. https://doi.org/10.11648/j.pamj.s.2015040102.12
Senyurt S., Gur S., Spacelike surface geometry, International Journal of Geometric Methods in Modern Physics Vol. 14, No. 9 (2017) 1750118 (16 pages) https://doi.org/10.1142/S0219887817501183
M. Petrovic and E. Sucurovic , Some characterizations of the spacelike, the timelike and the null curves on the pseudohyperbolic space H20 in E31, Kragujevac J. Math. 22 (2000), 71-82, 2000
Kılıcoglu S¸, Senyurt S and Calıskan A, On the Tangent Vector Fields of Striction Curves Along the Involute and Bertrandian Frenet Ruled Surfaces, International J.Math. Combin. Vol.2(2018), 33-43
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
