On the Line Congruences

The purpose of this paper is to find the quantities and surfaces of a line congruence via

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  • [1] Abdel-Baky, R.A., Al-Bokhary, A.J., A new approach for describing instantaneous line congruence, Archivum Mathematicum, (2008), pp. 223-236.
  • [2] Ball, R.S., A Treatise on the Theory of Screws, Cambridge University Press, 1900 (reprinted 1999).
  • [3] Blaschke, W., Vorlesungen über differential geometrie I, Springer-Verlag, 1945, Berlin.
  • [4] Bottema, O., Roth, B., Theoretical Kinematics, Dover Publications, 1990.
  • [5] Clifford, W.K., Preliminary sketch of bi-quaternions, Proceedings of the Royal Society of London, (1873), pp. 381-395.
  • [6] Dimentberg, F.M., The Screw Calculus and Its Applications in Mechanics, Foreign Technology Division translation, FTD-HT-1632-67, 1965.
  • [7] Eisenhart, L.P., A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969.
  • [8] Guggenheimer, H.W., Differential geometry, Dover Publications, Inc. New York, 1977.
  • [9] Gürsoy, O. J Geom (1990) 39: 80. https://doi.org/10.1007/BF01222141.
  • [10] Gürsoy, O., Some results on closed ruled surfaces and closed space curves, Mech. Mach. Theory, (1992), pp.323-330.
  • [11] Hlavaty, V., Differential line geometry, Groningen, P.Noordhoff Ltd. X, 1953.
  • [12] Hoschek, J., Liniengeometrie, B.I. Hochschultaschenbuch, Mannheim, 1971.
  • [13] Hunt, K.H., Kinematic Geometry of Mechanisms, Oxford Press, 1978.
  • [14] Jüttler, B., Rittenschober, K., Using line congruences for parametrizing special algebraic surfaces, The Mathematics of Surfaces X, Lecture Notes in Computer Science, vol.2768, Springer-Berlin, 2003, pp. 223-243.
  • [15] Köse, Ö., A method of the determination of a developable ruled surface, Mech. Mach. Theory, (1999), pp. 1187-1193.
  • [16] Larochelle, P.M., Vance, J.M., Kihonge, J.N., Interactive visualization of line congruences for spatial mechanism design, Journal of Computing and Information Science in Engineering, (2002), pp. 208-215.
  • [17] Odehnal, B., On rational isotropic congruences of lines, Journal of Geometry, (2004), pp. 126-138.
  • [18] Plücker, J., On a new geometry of space, Proceedings of the Royal Society of London, (1865), pp. 53-58.
  • [19] Pottmann, H., Wallner, J., Computational line geometry, Mathematics and Visualization, Springer-Verlag Berlin Heidelberg, 2001.
  • [20] Selig, J.M., Geometrical Methods in Robotics, Springer, New York, 1994.
  • [21] Shepherd, M.D., Line congruences as surfaces in the space of lines, Differential Geometry and its Applications, (1999), pp. 1-26.
  • [22] Study, E., Geometry der dynamen, Leipzip, 1903.
  • [23] Veldkamp, G.R., On the use of dual numbers vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, (1976), pp. 141-156.