Inextensible Flows of Space Curves According to a New Orthogonal Frame with Curvature in $\mathbb{E}_{1}^{3}$

Using a new orthogonal frame with curvature in $E_1^3$ and we put forth a new general formulation for inextensible flows of space curves in this work. We demonstrate presufficient conditions and prove the necessary conditions for inextensible curve flow which is a partial differential equations (PDE) incorporating the curvatures and torsion.

Keywords:

Inextensible flow, Frenet-Serret frame modified orthogonal frame, Minkowski 3-Space,

PDF

___

[1] Abdel-All, N., Mohamed, S., Al-Dossary, M.: Evolution of generalized space curve as a function of its local geometry. Applied Mathematics. 5 (15),(2014) 2381-2392.
[2] Bukcu, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space. Mathematical Sciences and Applications E-Notes. 4 (1), (2016) 184-188.
[3] Desbrun, M., Cani-Gascuel, M. P.: Active implicit surface for animation. Procedings of Graphics Interface. 18 (1), (1998) 143-150.
[4] Hartmand, P., Winter, A.: On the fundamental equations of differential geometry. American Journal Mathematics. 72 (4), (1950) 757-774.
[5] Gurbuz, N.: Inextensible flows of spacelike, timelike and null curves. International Journal of Contemporary Mathematical Sciences. 4 (32), (2009) 1599-1604.
[6] Inoguchi, J., and Lee, S., Null curves in Minkowski 3-space, International Electronic Journal of Geometry, 2 (2008) 40-83.
[7] Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision. 1 (4), (1988) 321-331.
[8] Kwon, D. Y., Park, F. C., Chi, D. P.: Inextensible flows of curves and developable surfaces. Applied Mathematics Letter. 18 (10), (2005) 1156-1162.
[9] Kwon, D. Y., Park, F. C.: Evolution of inelastic plane curves. Applied Mathematics Letter. 12 (6), (1999) 115-119.
[10] Latifi, D., Razavi, A.: Inextensible flows of curves in Minkowskian space. Advanced Studies in Theoretical Physics. 2 (16), (2008) 761-768.
[11] Lopez, R.: Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 7 (1) (2014) 44-107.
[12] Sasai, T.: The fundamental theorem of analytic apace curves and apparent singularities of Fuchsian differential equations, Tohoku Mathematical Journal, 36 (1), (1984) 17-24.
[13] Yıldız, Ö. G., Okuyucu, O. Z., Inextensible Flows of Curves in Lie Groups. Caspian Journal of Mathematical Sciences, 2 (1),(2013) 23-32.
[14] Yıldız, Ö. G., Tosun, M.:A note on evolution of curves in the Minkowski spaces. Advances in Applied Clifford Algebras. 27 (3),(2017) 2873-2884.
[15] Yüzbaşı, Z. K., Yoon D. W.:Inextensible Flows of Curves on Lightlike Surfaces. Mathematics 6 (11),(2018) 224
[16] Walrave, J.: Curves and surfaces in Minkowski space. Ph.D. thesis, Leuven University (1995).