Geodesic
A new approach for describing instantaneous line congruence
Archivum mathematicum, Tome 44 (2008) no. 3, pp. 223-236 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 53A04, 53A05, 53A17
Keywords: lines of curvature; line congruence; E. Study’s map; instantaneous revolution axis 
@article{ARM_2008_44_3_a6,
     author = {Abdel-Baky, Rashad A. and Al-Bokhary, Ashwaq J.},
     title = {A new approach for describing instantaneous line congruence},
     journal = {Archivum mathematicum},
     pages = {223--236},
     year = {2008},
     volume = {44},
     number = {3},
     mrnumber = {2462978},
     zbl = {1212.53001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008_44_3_a6/}
}
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Abdel-Baky, Rashad A.; Al-Bokhary, Ashwaq J. A new approach for describing instantaneous line congruence. Archivum mathematicum, Tome 44 (2008) no. 3, pp. 223-236. http://geodesic.mathdoc.fr/item/ARM_2008_44_3_a6/