This paper derives the expressions for spacelike ruled surfaces with stationary Disteli-axis by means of the E. Study map. This provides the ability to compute a set of Lorentzian curvature functions which define the local shape of spacelike ruled surfaces. Consequently, some well-known formulae of surface theory at Lorentzian line space and their geometrical explanations are obtained and examined. Lastly, a characterization for a spacelike line trajectory to be a constant Disteli-axis is derived and investigated in detail.

This paper derives the expressions for kinematic-geometry of lines with special trajectories in spatial kinematics by means of the E. Study map. A particular assurance goes to the 2nd order movement characteristics for extracting a new proof of the Disteli formulae of the axodes. Meanwhile, a new height dual function is defined and utilized to investigate the geometrical properties of the Disteli-axis. Consequently, as an implementation, the spatial equivalent of the cubic of stationary curvature is established and researched. Finally, Disteli formulae of a line-trajectory are extracted and inspected in detail.

In this paper using the Blaschke approach we generalized the Bertrand curves to spacelike ruled and developable surfaces. It is proved that, every spacelike ruled surface have a Bertrand offset if and only if an equation should be fulfilled among their dual integral invariants. Consequently, some new relationships and theorems for the developability of the Bertrand offsets of spacelike ruled surfaces are outlined.

In this treatise, several relationships are improved for the axodes of one-parameter spatial movements. Results are devised in some theorems which characterize many kinematical and geometrical properties of the movements employing the geometrical data of the stationary and movable axodes. An example illustrates the application of the formulae derived. Our findings contribute to a greater understanding of the similarities between spatial movements and axodes, with possible applications in fields such as mechanical engineering.