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.
@article{119762,
author = {Rashad A. Abdel-Baky and Ashwaq J. Al-Bokhary},
title = {A new approach for describing instantaneous line congruence},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {223-236},
zbl = {1212.53001},
mrnumber = {2462978},
language = {en},
url = {http://dml.mathdoc.fr/item/119762}
}
Abdel-Baky, Rashad A.; Al-Bokhary, Ashwaq J. A new approach for describing instantaneous line congruence. Archivum Mathematicum, Tome 044 (2008) pp. 223-236. http://gdmltest.u-ga.fr/item/119762/
On the congruences of the tangents to a surface, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 136 (1999), 9–18. (1999) | MR 1908813 | Zbl 1017.53003
On instantaneous rectilinear congruences, J. Geom. Graph. 7 (2) (2003), 129–135. (2003) | MR 2071274 | Zbl 1066.53036
Inflection and torsion line congruences, J. Geom. Graph. 11 (1) (2004), 1–14. (2004) | MR 2364050
On a line congruence which has the parameter ruled surfaces as principal ruled surfaces, Appl. Math. Comput. 151 (2004), 849–862. (2004) | Article | MR 2052463 | Zbl 1058.53010
Vorlesungen über Differential Geometrie, Dover Publications, New York, 1945. (1945) | MR 0015247
Theoretical Kinematics, North-Holland Press, New York, 1979. (1979) | MR 0533960 | Zbl 0405.70001
Preliminary Sketch of bi-quaternions, Proc. London Math. Soc. 4 (64, 65) (1873), 361–395. (1873)
A Treatise in Differential Geometry of Curves and Surfaces, New York, Ginn Camp., 1969. (1969)
Differential Geometry, Graw-Hill, New York, 1956. (1956)
The dual angle of pitch of a closed ruled surface, Mech. Mach. Theory 25 (47) (1990), 131–140. (1990) | Article
Differential Line Geometry, Groningen, P. Noordhoff Ltd. X, 1953. (1953) | MR 0057592 | Zbl 0051.39101
Liniengeometrie, B.I. Hochschultaschenbuch, Mannheim, 1971. (1971) | MR 0353164 | Zbl 0227.53007
Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, 1985. (1985) | MR 0801394
Zur Geometrie der zweiten Grundform der Geradenkongruenzen des $E^3$, Verh. K. Acad. Wet. Lett. Schone Kunsten Belg., Kl. Wet. 43 (162) (1981). (1981) | MR 0629825
Contributions to the theory of integral invariants of a closed ruled surface, Mech. Mach. Theory 32 (2) (1997), 261–277. (1997) | Article
On the scalar and dual formulations of curvature theory of line trajectories, ASME, J. Mech. Transmiss. Automation in Design 109 (1987), 101–106. (1987) | Article
Kinematik Dersleri, Ankara University Press, 1963. (1963) | MR 0157519
Curvature theory of line trajectories in spatial kinematics, Doctoral dissertation, University of California, Davis (1988). (1988) | MR 2636385
Geometric continuity of ruled surfaces, Comput. Aided Geom. Design 15 (1998), 289–310. (1998) | Article | MR 1614079 | Zbl 0903.68192
Instantaneous spatial kinematics and the invariants of the axodes, Tech. report, Institute für Geometrie, TU Wien 34, 1996. (1996)
On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics, Mech. Mach. Theory 11 (1976), 141–156. (1976) | Article
Differential Geometry of Three Dimensions, Cambridge University Press, 1, 1969. (1969)
Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms, Doctoral dissertation, Columbia (1967). (1967)