A new approach for describing instantaneous line congruence

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

Department of Mathematics Faculty of Science, University of Assiut Assiut 71516, Egypt
Department of Mathematics Girls College of Education/Science Department Um alqura University, Makkah, Saudi Arabia

E-mail: rbaky@hotmail.com

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.

AMSclassification: primary 53A04; secondary 53A05, 53A17.

Keywords: lines of curvature, line congruence, E. Study’s map, instantaneous revolution axis.