The Imaginary Center of the Imaginary
The useful fiction that displaced water creates an upwards force inside its volume becomes even more valuable when we try to figure where that upwards force is acting. The pressure of the water is always acting perpendicular to the surface of the hull. If you visualize the curved surfaces of a typical boat or ship, and then remember that the pressure is different at every depth, you'll see that there are an almost infinite number of pressure quantities and directions that would need to be all summed up in order to directly calculate the amount of the upwards force. It is indeed a fortunate thing that volume equivalency works so well. Now add the need to figure out the point that corresponds to the imaginary point where gravity seems to act and the job of calculating all those pressure force vectors becomes even more unmanageable.
We need to know the location of the Center of Buoyancy, CB, for the same reason we need to know the Center of Gravity. It is just too complex to deal with each unit of displaced volume individually, even after adopting the simplifying fiction that it is the volume which is generating the upwards force. The buoyancy will act as if it is pushing up through a single point exactly as gravity acts as if it is pulling down. One magic string up and one magic string down.
It is actually quite a bit easier to calculate the CB than the CG at the design stage. Instead of vast tabulations of every item in the vessel and their locations, you only need to make some calculations of the shape of the underwater hull. The calculations are conceptually more complex but there are far, far, fewer of them. Instead of a phone book sized document, you need only a single page worksheet if doing it by hand.
In order to be able to understand stability at the gut level, you need to be able to form a clear mental visualization of the hole that the boat makes in the water. A good way to do this is to imagine the water being frozen solid and a crane lifting out the boat. This leaves a hole. Now fill the hole level with the surface with something of uniform density like wax or concrete and lift it up in the air. You will have a solid object the shape of the hole the boat makes in the water. THE CENTER OF BUOYANCY WILL BE AT THE CENTER OF GRAVITY OF THIS SHAPE. This is the key point, the big enchilada of stability. The CG of this shape, the Center of Buoyancy, will have both a fore and aft and a vertical location. If the boat wasn't built carefully and the two sides are different, it may also be slightly off the centerline but we'll assume proper boatbuilding for the remainder of this discussion.
Why is the CB important? Well, one thing we can be sure of when we see a boat sitting quietly in still water is that the center of gravity is directly in line, vertically, with the center of buoyancy. If you have two forces acting in opposite directions and not in line, something has to give. The offset produces rotation and the boat will therefore heel or pitch until the two centers are again aligned, if they can be. If they can't, there will be bottom paint drying in the sunlight. If you want a boat you are designing to end up floating so that the boot top is parallel with the water, you have to predict where both the CG and CB will be as well as making sure that the structural and other weights will correspond to the weight of the displaced water.
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