Contributions to the theory of integral invariants of a closed ruled surface

On the Geometry of Closed Timelike Ruled Surfaces in Dual Lorentzian Space

“…is timelike. In addition, unit dual timelike vector − → V with respect to dual orthonormal frame D * by equations (9) and (10) also is written as follows:…”

“…In recent years, by using the real integral invariants of closed ruled surfaces, many investigations are done, [3,9,23]. In addition, by using Lorentzian metric, ruled surfaces in Lorentz space have been studied [1,11,12,15,16,20].…”

On the Determination of Ruled and Developable Surfaces in Euclidean 3-space E³

“…Previously, mathematical techniques that relay on the so-called E. Study's map has been used to deal with ruled and developable surfaces in computer-aided geometric design (CAGD) [10][11][12][13][14][15]. The E. Study's map concludes: All oriented lines set in Euclidean 3-space E 3 is in one-to-one correspondence with all points set of the dual unit sphere in the dual 3-space D 3 .…”

Lamarle Formula in 3-Dimensional Lorentz Space

“…Projective geometry, [13], Computeraided design, [11], etc.) Also, it is well known that the geometry of ruled surface is very important of kinematics or spatial mechanisms in R 3 , [4], [7]. A ruled surface is one which can be generated by sweeping a line through space.…”