The observation that different effectors can execute the same movement suggests functional equivalences driven by limb independent representation of action in the central nervous system. A common invariant motor behavior is the speed and curvature coupling (the 1/3 power law), a low dimensional (abstract) descriptor of movement which is resilient to different sensorimotor contexts. Our purpose is to verify the consistency of such motor equivalence during a drawing task, by testing the effect of manual dominance and movement speed on motor performance. We hypothesize that abstract kinematic variables are not the most resistant to speed or limb effector changes. The results show specific effects of speed and hand side on the drawing task. Movement duration, speed-curvature covariation, and maximum velocity were not significantly affected by hand side, while geometrical features were strongly speed and limb dependent. However, intra-trial analysis performed over the successive drawing movements reveals a significant hand side effect on the variability of movement vigor and velocity-curvature relationship (the 1/3 PL). The identified effects of speed and hand dominance on the kinematic parameters suggest different neural strategies, in a pattern that does not go from the most abstract to the least abstract component, as proposed by the traditional hierarchical organization of the motor plan.