J. Korean Math. Soc. 43 (2006), No. 6, pp. 1339–1355 ON THE SCALAR AND DUAL FORMULATIONS OF THE CURVATURE THEORY OF LINE TRAJECTORIES IN THE LORENTZIAN SPACE N˙ ihat Ayyıldız and Ahmet Y¨ ucesan Abstract. This paper develops in detail the differential geometry of ruled surfaces from two perspectives, and presents the under- lying relations which unite them. Both scalar and dual curvature functions which define the shape of a ruled surface are derived. Explicit formulas are presented for the computation of these func- tions in both formulations of the differential geometry of ruled sur- faces. Also presented is a detailed analysis of the ruled surface which characterizes the shape of a general ruled surface in the same way that osculating circle characterizes locally the shape of a non-null Lorentzian curve. 1. Introduction Dual numbers were introduced by W. K. Clifford (1849 − 79) as a tool for his geometrical investigations. After him E. Study used dual numbers and dual vectors in his research on the geometry of lines and kinematics. He devoted special attention to the representation of directed lines by dual unit vectors and defined the mapping that is known by his name. There exist one-to-one correspondence between the vectors of dual unit sphere S 2 and the directed lines of space of lines R 3 [3]. If we take the Minkowski 3-space R 3 1 instead of R 3 the E. Study mapping can be stated as follows: The dual timelike and spacelike unit vectors of dual hyperbolic and Lorentzian unit spheres H 2 0 and S 2 1 at the dual Lorentzian space D 3 1 are in one-to-one correspondence with the directed timelike and spacelike lines of the space of Lorentzian lines R 3 1 , Received September 27, 2005. 2000 Mathematics Subject Classification: 53B30, 53A17. Key words and phrases: Disteli axis, ruled surface, asymptotic normal, the central normal surface, dual Lorentzian space, Frenet frame. The authors would like to thank Professor Dr. McCarthy for invaluable comments and suggestions relating to this work.