In this paper, the relationships among the instantaneous invariants of a one-parameter Lorentzian spatial movement and the
local invariants of the axodes are studied. New proofs for Euler–Savary, and Disteli formulae are given which demonstrate
the elegance and efficiency of the E. Study map in Lorentzian spatial kinematics. Consequently, two spacelike line congruences
are introduced and their spatial equivalent are examined in detail.