Research Article
Year 2020,
Volume: 69 Issue: 1, 450 - 460, 30.06.2020
Abstract
References
- Abdel-Baky, R. A. On the Congruences of the Tangents to a Surface, Osterreich. Akad. Wiss. Math-Natur Kl. Sitzungsber, Anzeiger Abt., 11 (136) (1999) 9-18.
- Abdel-Baky, R. A. On Instantaneous Rectilinear Congruences, J. for Geometry and Graphics, V. 7, No. 2, (2003), 129-135.
- Abdel-Baky, R. A. Inflection and Torsion line Congruences, J. for Geometry and Graphics, V. 11, No. 1, (2007), 1-14.
- Abdel-Baky, R. A. and Bochary, A. J. A new approach for describing Instantaneous line congruences, Arch. Math., Tom. 44, (2008), 237-250.
- Chern, S. S., Terng,C. L. An analogue of Bäcklund theorem in affine geometry, Rocky Mountain Journal of Mathematics, Vol.10,No 1, (1980).
- Eisenhart, L. P. A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969.
- Hilbert, D. and Cohn-Vossen, S. Geometry and Imagination, Chelsea, New York, NY, U.S.A., 1952.
- Koch, R. Zur Geometrie der zweiten Grunform der Geradenkongruenzen des E³, Acad. V. Wetenshch. V. Belgie, Brussel, (1981).
- Nomizu,K.,Sasaki,T. Affine Differential Geometry, Cambridge University Press, 1994.
- Odehnal, B. Geometric Optimization Methods for Line Congruences, Ph. D. Thesis, Vienna University of Technology, 2002.
- Odehnal, B. and Pottmann, H. Computing with discrete models of ruled surfaces and line congruences, Proc. of the 2^{nd} workshop on computational kinematics, Seoul 2001.
- Pottman, H. and Wallner, J. Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, 2001.
- Schaaf, J. A. Curvature Theory of Line Trajectories in Spatial Kinematics, Doctoral dissertation, University of California, Davis, CA, 1988.
- Weatherburn, M. A. Differential Geometry of three dimensions, Vol. 1, Cambridge University Press, 1969.
Year 2020,
Volume: 69 Issue: 1, 450 - 460, 30.06.2020
Abstract
By utilizing the Darboux frames, along with a regular surface whose parametric curves are lines of curvature, we analyzed the normal line congruence which preserves the asymptotic curves between its focal surfaces. This allows deriving systems of partial differential equations through which the problem of determining the director surface and the corresponding normal line congruence could be solved. Moreover, a necessary and sufficient condition that the focal surfaces of the normal line congruence are degenerates into curves is derived. As a result, the middle focal surface of the normal line congruence is presented as a new surface interrogation tool.
References
- Abdel-Baky, R. A. On the Congruences of the Tangents to a Surface, Osterreich. Akad. Wiss. Math-Natur Kl. Sitzungsber, Anzeiger Abt., 11 (136) (1999) 9-18.
- Abdel-Baky, R. A. On Instantaneous Rectilinear Congruences, J. for Geometry and Graphics, V. 7, No. 2, (2003), 129-135.
- Abdel-Baky, R. A. Inflection and Torsion line Congruences, J. for Geometry and Graphics, V. 11, No. 1, (2007), 1-14.
- Abdel-Baky, R. A. and Bochary, A. J. A new approach for describing Instantaneous line congruences, Arch. Math., Tom. 44, (2008), 237-250.
- Chern, S. S., Terng,C. L. An analogue of Bäcklund theorem in affine geometry, Rocky Mountain Journal of Mathematics, Vol.10,No 1, (1980).
- Eisenhart, L. P. A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969.
- Hilbert, D. and Cohn-Vossen, S. Geometry and Imagination, Chelsea, New York, NY, U.S.A., 1952.
- Koch, R. Zur Geometrie der zweiten Grunform der Geradenkongruenzen des E³, Acad. V. Wetenshch. V. Belgie, Brussel, (1981).
- Nomizu,K.,Sasaki,T. Affine Differential Geometry, Cambridge University Press, 1994.
- Odehnal, B. Geometric Optimization Methods for Line Congruences, Ph. D. Thesis, Vienna University of Technology, 2002.
- Odehnal, B. and Pottmann, H. Computing with discrete models of ruled surfaces and line congruences, Proc. of the 2^{nd} workshop on computational kinematics, Seoul 2001.
- Pottman, H. and Wallner, J. Computational Line Geometry, Springer-Verlag, Berlin, Heidelberg, 2001.
- Schaaf, J. A. Curvature Theory of Line Trajectories in Spatial Kinematics, Doctoral dissertation, University of California, Davis, CA, 1988.
- Weatherburn, M. A. Differential Geometry of three dimensions, Vol. 1, Cambridge University Press, 1969.
There are 14 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 30, 2020 |
| Submission Date | April 7, 2019 |
| Acceptance Date | November 19, 2019 |
| Published in Issue | Year 2020 Volume: 69 Issue: 1 |
Cite
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