In this paper, based on the E. Study map, clear terms are oered for the dierential equations
of one-parameter Lorentzian spatial kinematics that are coordinate system-independent. This cancels
the request of demanding coordinate transformations that are typically required in the determination of
the canonical systems. With the suggested technique, new proofs of the Disteli formulae of a spacelike
line trajectory are instantly gained and their spatial equivalents are studied in detail. As a consequence,
we address the kinematic geometry of a point trajectory for the one-parameter Lorentzian spherical
and planar movements.