Research article

A study on a line congruence as surface in the space of lines

  • Received: 07 February 2021 Accepted: 21 July 2021 Published: 03 August 2021
  • MSC : 53A04, 53A05, 53A17

  • 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

    Related Papers:

  • 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.



    加载中


    [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.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1524) PDF downloads(78) Cited by(2)

Article outline

Figures and Tables

Figures(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog