The method of hanging or balancing the sail shape produces a geometric CE, which is only an approximation for the aerodynamic CE.
The aerodynamic CE is quite moveable. The obvious case is changing the sheeting angle – to go downwind, let the sails out, and the aerodynamic CE moves forward and the boat should change direction. Conversely, sheet in, and the aerodynamic CE moves aft and the boat changes to weather. See diagrams below that show the sail profiles for centred, 45 degrees and 90 degrees. Changing mast rake or position obviously will have a similar effect of moving the geometric CE.
What is not so obvious and is mostly neglected, are changes to the aerodymic CE due to sail shape or trim. If we change the twist of the mainsail of a boat going upwind by increasing tension on the mainsheet, mast ram or in most cases by tensioning the vang, we would barely change the sail profile as shown on the left, hence barely move the geometric CE, but we would move the aerodynamic CE aft and increase weather helm. Similarly, if we made a fuller mainsail, we would again barely change the sail profile shown on the left, hence barely move the geometric CE, but we would move the aerodynamic CE aft and increase weather helm.
See http://en.wikipedia.org/wiki/Forces_on_sails
Similarly, the method of hanging or balancing the underwater shape of the hull will give an approximation for the hydrodynamic CLR.
If we have 2 identical underwater profiles of a hull, then obviously the “geometric” CLR’s are the same. However, if the hull shapes are different (one has hard flat sections aft, soft and rounded forward – the other soft and rounded aft, hard V forward) then I would think the swimming pool method of pushing/pulling the hull sideways (which again is only an approximation) would show a difference. See diagrams below.
If our boat is sailing forwards, then it is heeling. As our hull heels, the underwater shape usually changes, and the hydrodynamic CLR usually changes position also.
Therefore yacht designers who use both these “geometric” methods are just using approximations of the actual centres – hence “on water tuning” is often required.