In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.
Citation: Rashad A. Abdel-Baky, Monia F. Naghi. A study on a line congruence as surface in the space of lines[J]. AIMS Mathematics, 2021, 6(10): 11109-11123. doi: 10.3934/math.2021645
In this work, we introduce a line congruence as surface in the space of lines in terms of the E. Study map. This provides the ability to derive some formulae of surfaces theory into line spaces. In addition, the well known equation of the Plucker's conoid has been obtained and its kinematic-geometry are examined in details. At last, an example of application is investigated and explained in detail.
[1] | L. P. Eisenhart, A treatise in differential geometry of curves and surfaces, London and Boston: Ginn, 1909. |
[2] | B. Jüttler, K. Rittenschober, Using line congruences for parametrizing special algebraic surfaces, In: The mathematics of surfaces, Lecture Notes in Computer Science, Berlin: Springer, 2768 (2003), 223-243. |
[3] | J. A. Schaaf, B. Ravani, Geometric continuity of ruled surfaces, Comput. Aided Geom. Des., 15 (1998), 289-310. doi: 10.1016/S0167-8396(97)00032-0 |
[4] | M. D. Shepherd, Line congruences as surfaces in the space of lines, Differ. Geom. Appl., 10 (1999), 1-26. doi: 10.1016/S0926-2245(98)00025-4 |
[5] | B. Odehnal, H. Pottmann, Computing with discrete models of ruled surfaces and line congruences, Proceedings of the 2nd workshop on computational kinematics, Seoul, 2001. |
[6] | H. Pottmann, J. Wallner, Computational line geometry, Springer Science & Business Media, 2009. |
[7] | B. Odehnal, Geometric optimization methods for line congruences, Ph. D. Thesis, Vienna University of Technology, 2002. |
[8] | O. Bottema, B. Roth, Theoretical kinematic, North-Holland Series in Applied Mathematics and Mechanics, North-Holland Publishing Company, 1979. |
[9] | A. Karger, J. Novak, Space kinematics and Lie groups, New York: Routledge, 1985. |
[10] | R. A. Abdel-Baky, The relation among Darboux vectors of ruled surfaces in a line congruence, Riv. Mat. Univ. Parma, 5 (1997), 201-211. |
[11] | W. Blaschke, Vorlesungen über differential geometrie I, Springer-Verlag Berlin Heidelberg, 1945. |
[12] | Y. L. Li, S. Y. Liu, Z. G. Wang, Tangent developables and Darboux developables of framed curves, Topol. Appl., (2020), 107526. |
[13] | Y. L. Li, Z. G. Wang, T. H. Zhao, Geometric algebra of singular ruled surfaces, Adv. Appl. Clifford Algebras, 31 (2021), 19. doi: 10.1007/s00006-020-01097-1 |
[14] | O. Gursoy, The dual angle of pitch of a closed ruled surface, Mech. Mach. Theory, 25 (1990), 131-140. doi: 10.1016/0094-114X(90)90114-Y |
[15] | R. A. Abdel-Baky, On a line congruence which has the parameter ruled surfaces as principal ruled surfaces, Appl. Math. Comput., 151 (2004), 849-862. |
[16] | R. A. Abdel-Baky, A. J. Al-Bokhary, A new approach for describing instantaneous line congruence, Arch. Math., 44 (2008), 223-236. |