A new approach for describing instantaneous line congruence

Rashad A. Abdel-Baky; Ashwaq J. Al-Bokhary

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 3, page 223-236
  • ISSN: 0044-8753

Abstract

top
Based on the E. Study’s map, a new approach describing instantaneous line congruence during the motion of the Darboux frame on a regular non-spherical and non-developable surface, whose parametric curves are lines of curvature, is proposed. Afterward, the pitch of general line congruence is developed and used for deriving necessary and sufficient condition for instantaneous line congruence to be normal. In terms of this, the derived line congruences and their differential geometric invariants were examined.

How to cite

top

Abdel-Baky, Rashad A., and Al-Bokhary, Ashwaq J.. "A new approach for describing instantaneous line congruence." Archivum Mathematicum 044.3 (2008): 223-236. <http://eudml.org/doc/250499>.

@article{Abdel2008,
abstract = {Based on the E. Study’s map, a new approach describing instantaneous line congruence during the motion of the Darboux frame on a regular non-spherical and non-developable surface, whose parametric curves are lines of curvature, is proposed. Afterward, the pitch of general line congruence is developed and used for deriving necessary and sufficient condition for instantaneous line congruence to be normal. In terms of this, the derived line congruences and their differential geometric invariants were examined.},
author = {Abdel-Baky, Rashad A., Al-Bokhary, Ashwaq J.},
journal = {Archivum Mathematicum},
keywords = {lines of curvature; line congruence; E. Study’s map; instantaneous revolution axis; lines of curvature; line congruence; E. Study's map; instantaneous revolution axis},
language = {eng},
number = {3},
pages = {223-236},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A new approach for describing instantaneous line congruence},
url = {http://eudml.org/doc/250499},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Abdel-Baky, Rashad A.
AU - Al-Bokhary, Ashwaq J.
TI - A new approach for describing instantaneous line congruence
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 3
SP - 223
EP - 236
AB - Based on the E. Study’s map, a new approach describing instantaneous line congruence during the motion of the Darboux frame on a regular non-spherical and non-developable surface, whose parametric curves are lines of curvature, is proposed. Afterward, the pitch of general line congruence is developed and used for deriving necessary and sufficient condition for instantaneous line congruence to be normal. In terms of this, the derived line congruences and their differential geometric invariants were examined.
LA - eng
KW - lines of curvature; line congruence; E. Study’s map; instantaneous revolution axis; lines of curvature; line congruence; E. Study's map; instantaneous revolution axis
UR - http://eudml.org/doc/250499
ER -

References

top
  1. Abdel-Baky, R. A., On the congruences of the tangents to a surface, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 136 (1999), 9–18. (1999) Zbl1017.53003MR1908813
  2. Abdel-Baky, R. A., On instantaneous rectilinear congruences, J. Geom. Graph. 7 (2) (2003), 129–135. (2003) Zbl1066.53036MR2071274
  3. Abdel Baky, R. A., Inflection and torsion line congruences, J. Geom. Graph. 11 (1) (2004), 1–14. (2004) MR2364050
  4. Abdel-Baky, R. A., 10.1016/S0096-3003(03)00541-1, Appl. Math. Comput. 151 (2004), 849–862. (2004) Zbl1058.53010MR2052463DOI10.1016/S0096-3003(03)00541-1
  5. Blaschke, W., Vorlesungen über Differential Geometrie, Dover Publications, New York, 1945. (1945) MR0015247
  6. Bottema, O., Roth, B., Theoretical Kinematics, North-Holland Press, New York, 1979. (1979) Zbl0405.70001MR0533960
  7. Clifford, W. K., Preliminary Sketch of bi-quaternions, Proc. London Math. Soc. 4 (64, 65) (1873), 361–395. (1873) 
  8. Eisenhart, L. P., A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969. (1969) 
  9. Gugenheimer, H. W., Differential Geometry, Graw-Hill, New York, 1956. (1956) 
  10. Gursy, O., 10.1016/0094-114X(90)90114-Y, Mech. Mach. Theory 25 (47) (1990), 131–140. (1990) DOI10.1016/0094-114X(90)90114-Y
  11. Hlavaty, V., Differential Line Geometry, Groningen, P. Noordhoff Ltd. X, 1953. (1953) Zbl0051.39101MR0057592
  12. Hoschek, J., Liniengeometrie, B.I. Hochschultaschenbuch, Mannheim, 1971. (1971) Zbl0227.53007MR0353164
  13. Karger, A., Novak, J., Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, 1985. (1985) MR0801394
  14. Koch, R., Zur Geometrie der zweiten Grundform der Geradenkongruenzen des E 3 , Verh. K. Acad. Wet. Lett. Schone Kunsten Belg., Kl. Wet. 43 (162) (1981). (1981) MR0629825
  15. Kose, Ö., 10.1016/S0094-114X(96)00034-1, Mech. Mach. Theory 32 (2) (1997), 261–277. (1997) DOI10.1016/S0094-114X(96)00034-1
  16. Mc-Carthy, J. M., 10.1115/1.3258772, ASME, J. Mech. Transmiss. Automation in Design 109 (1987), 101–106. (1987) DOI10.1115/1.3258772
  17. Muller, H. R., Kinematik Dersleri, Ankara University Press, 1963. (1963) MR0157519
  18. Schaaf, J. A., Curvature theory of line trajectories in spatial kinematics, Doctoral dissertation, University of California, Davis (1988). (1988) MR2636385
  19. Schaaf, J. A., 10.1016/S0167-8396(97)00032-0, Comput. Aided Geom. Design 15 (1998), 289–310. (1998) Zbl0903.68192MR1614079DOI10.1016/S0167-8396(97)00032-0
  20. Stachel, H., Instantaneous spatial kinematics and the invariants of the axodes, Tech. report, Institute für Geometrie, TU Wien 34, 1996. (1996) 
  21. Veldkamp, G. R., 10.1016/0094-114X(76)90006-9, Mech. Mach. Theory 11 (1976), 141–156. (1976) DOI10.1016/0094-114X(76)90006-9
  22. Weatherburn, M. A., Differential Geometry of Three Dimensions, Cambridge University Press, 1, 1969. (1969) 
  23. Yang, A. T., Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms, Doctoral dissertation, Columbia (1967). (1967) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.