From one space dimension to two space dimensions in special relativity
  Authors derive the Lorentz-Einstein transformation for the space-time coordinates starting with a one-space dimension approach. They add to the results the invariance of the space coordinates measured perpendicular to the direction of relative motion. Students often ask if the transition does not affect the transformation equation for the time coordinate. Our paper shows that the transformation of the time coordinate depends only on the x space coordinate. 
Bernhard Rothenstein