Vaderios, I believe the room you are referring to in which Vader and Veers had their discussion, is a different room altogether. In fact, it's probably a hallway instead.
Check out my screens.
First Vader is seen on the left, after which he indeed turns 180°, but then walks away from the camera, and takes a right turn behind a corner, which is the exact same corner Piett passes when walking up to Vader.
Not to mention the next scene, after the space exterior shot, then showing him immediately exiting his chamber.

EDIT: euroherbal, I still support your theory. I too believe Vader walked into a completely separate room, close to the hallway where he spoke to Piett, which is shaped like a figure 8. That room would both hold the hologram platform, and his meditation chamber.