[1] Abdel-Baky, R.A.; The relation among Darboux vectors of ruled surfaces in a line congruence, Riv. Mat., Univ. Parama, (1997), (5)6, pp.201-
211.
[2] Abdel-Baky, R.A. On the congruences of the tangents to a surface, Anz. Osterreich. Akad. Wiss. Math.-Natur. Kl., (1999), 136: 9–18.
[3] Abdel-Baky, R.A. . On instantaneous rectilinear congruences, J. Geom. Graph., (2003), 7(2): 129–135.
[4] Abdel-Baky, R. A. On a line congruence which has the parameter ruled surfaces as principal ruled surfaces, Applied mathematics and
computation, 151.3 (2004): 849-862.
[5] Blaschke, W.; Vorlesungen uber Differential Geometrie, Bd 1, Dover Publications, New York, pp.260-277, 1945.
[6] Bottema, O. and Roth, B. Theoretical Kinematics, North-Holland Press, New York, 1979.
[7] Odehnal, B. and Pottmann, H. Computing with discrete models of ruled surfaces and line congruences, Proceedings of the 2nd workshop
on computational kinematics, Seoul 2001.
[8] Odehnal, B. Geometric Optimization Methods for Line Congruences, Ph. D. Thesis, Vienna University of Technology, 2002.
[9] O’Neil, B. Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London, 1983.
[10] Papadopoulou, D. and Koltsaki, P. Triply orthogonal line congruences with common middle surface, J. Geom. Graph., (2010), 14(1): 45–58.
[11] Pottman, H. and Wallner, J. Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, 2001.
[12] F. Ta¸s, 2016, "On the Computational Line Geometry," Ph.D. thesis, Institute of Graduate Studies in Science and Engineering, Istanbul
University, Istanbul.
[13] F. Taş, O. Gürsoy, 2018, On the Line Congruences, International Electronic Journal of Geometry, Vol. 11(2), pp. 47-53.
[14] Tsagas, Gr. "On the rectilinear congruences of Lorentz manifold establishing an area preserving representation." Tensor, NS 47 (1988):
127-139.
[15] Tunahan, T., Nurai, Y. and Nihat, A. On pseudohyperbolic space motions, Turk J Math., (2015), 39: 750-762, TUBITAK doi:10.3906/mat-
1503-37.
[16] Uğurlu, H.H.; Çalışkan, A. The Study Mapping for Directed Space-Like and Time-Like in Minkowski 3-Space R13. Math. Comput. Appl.,
(1996), 1, 142-148.
[17] Veldkamp, G.R.; On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics, Mech. and Mach. Theory, (1976),
V.11, pp.141-156.
[18] Yaglom, I.M. A Simple Non-Euclidean Geometry and Its Physical Basis. Springer-Verlag, New York, 1979.
[19] Yayli,Y, Caliskan A. and Ugurlu, H.H. The E study maps of circles on dual hyperbolic and Lorentzian unit spheresH20
and S^2_1, Mathematical Proceedings of the Royal Irish Academy, (2002), 102A: 37-47.
[20] Zhong, H. and Hual, Li, L. A kind of rectilinear congruences in the Minkowski 3-Space, J. of Mathematical Research & Exposition, Nov.,
2008, Vol. 28, No. 4 pp. 911–918.
Year 2019,
Volume: 12 Issue: 1, 135 - 143, 27.03.2019
[1] Abdel-Baky, R.A.; The relation among Darboux vectors of ruled surfaces in a line congruence, Riv. Mat., Univ. Parama, (1997), (5)6, pp.201-
211.
[2] Abdel-Baky, R.A. On the congruences of the tangents to a surface, Anz. Osterreich. Akad. Wiss. Math.-Natur. Kl., (1999), 136: 9–18.
[3] Abdel-Baky, R.A. . On instantaneous rectilinear congruences, J. Geom. Graph., (2003), 7(2): 129–135.
[4] Abdel-Baky, R. A. On a line congruence which has the parameter ruled surfaces as principal ruled surfaces, Applied mathematics and
computation, 151.3 (2004): 849-862.
[5] Blaschke, W.; Vorlesungen uber Differential Geometrie, Bd 1, Dover Publications, New York, pp.260-277, 1945.
[6] Bottema, O. and Roth, B. Theoretical Kinematics, North-Holland Press, New York, 1979.
[7] Odehnal, B. and Pottmann, H. Computing with discrete models of ruled surfaces and line congruences, Proceedings of the 2nd workshop
on computational kinematics, Seoul 2001.
[8] Odehnal, B. Geometric Optimization Methods for Line Congruences, Ph. D. Thesis, Vienna University of Technology, 2002.
[9] O’Neil, B. Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London, 1983.
[10] Papadopoulou, D. and Koltsaki, P. Triply orthogonal line congruences with common middle surface, J. Geom. Graph., (2010), 14(1): 45–58.
[11] Pottman, H. and Wallner, J. Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, 2001.
[12] F. Ta¸s, 2016, "On the Computational Line Geometry," Ph.D. thesis, Institute of Graduate Studies in Science and Engineering, Istanbul
University, Istanbul.
[13] F. Taş, O. Gürsoy, 2018, On the Line Congruences, International Electronic Journal of Geometry, Vol. 11(2), pp. 47-53.
[14] Tsagas, Gr. "On the rectilinear congruences of Lorentz manifold establishing an area preserving representation." Tensor, NS 47 (1988):
127-139.
[15] Tunahan, T., Nurai, Y. and Nihat, A. On pseudohyperbolic space motions, Turk J Math., (2015), 39: 750-762, TUBITAK doi:10.3906/mat-
1503-37.
[16] Uğurlu, H.H.; Çalışkan, A. The Study Mapping for Directed Space-Like and Time-Like in Minkowski 3-Space R13. Math. Comput. Appl.,
(1996), 1, 142-148.
[17] Veldkamp, G.R.; On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics, Mech. and Mach. Theory, (1976),
V.11, pp.141-156.
[18] Yaglom, I.M. A Simple Non-Euclidean Geometry and Its Physical Basis. Springer-Verlag, New York, 1979.
[19] Yayli,Y, Caliskan A. and Ugurlu, H.H. The E study maps of circles on dual hyperbolic and Lorentzian unit spheresH20
and S^2_1, Mathematical Proceedings of the Royal Irish Academy, (2002), 102A: 37-47.
[20] Zhong, H. and Hual, Li, L. A kind of rectilinear congruences in the Minkowski 3-Space, J. of Mathematical Research & Exposition, Nov.,
2008, Vol. 28, No. 4 pp. 911–918.
Taş, F., & Abdel-baky, R. A. (2019). On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces. International Electronic Journal of Geometry, 12(1), 135-143. https://doi.org/10.36890/iejg.545870
AMA
Taş F, Abdel-baky RA. On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces. Int. Electron. J. Geom. March 2019;12(1):135-143. doi:10.36890/iejg.545870
Chicago
Taş, Ferhat, and Rashad A. Abdel-baky. “On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces As Principal Ruled Surfaces”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 135-43. https://doi.org/10.36890/iejg.545870.
EndNote
Taş F, Abdel-baky RA (March 1, 2019) On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces. International Electronic Journal of Geometry 12 1 135–143.
IEEE
F. Taş and R. A. Abdel-baky, “On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 135–143, 2019, doi: 10.36890/iejg.545870.
ISNAD
Taş, Ferhat - Abdel-baky, Rashad A. “On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces As Principal Ruled Surfaces”. International Electronic Journal of Geometry 12/1 (March 2019), 135-143. https://doi.org/10.36890/iejg.545870.
JAMA
Taş F, Abdel-baky RA. On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces. Int. Electron. J. Geom. 2019;12:135–143.
MLA
Taş, Ferhat and Rashad A. Abdel-baky. “On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces As Principal Ruled Surfaces”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 135-43, doi:10.36890/iejg.545870.
Vancouver
Taş F, Abdel-baky RA. On a Spacelike Line Congruence Which Has the Parameter Ruled Surfaces as Principal Ruled Surfaces. Int. Electron. J. Geom. 2019;12(1):135-43.