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Forget about the center of this or that your just over thinking it.
Usually it’s a bit forward of midships. Exactly where varies with the shape of the vessel particularly the underwater shape.
Generally, the finer the lines, the faster the vessel the further forward. Length and Beam play a part as well.
Plaining vessels it may be aft of midships while planing and forward when displacing more at low speed.
For a typical fin keel sail boat just a bit forward of midships.

KISS
The Pivot Point is the point about which a vessel or boat turns
The Pivot Point moves.
When power is applied. The Pivot Point moves in the direction the power is applied.
Apply Ahead thrust or power the Pivot Point moves forward.
Apply Astern thrust or Power the Pivot Point moves Aft.

When the acceleration (or deceleration) is over. The Pivot Point returns back to its original position just forward of amidships.

When a vessel is not moving the Pivot Point doesn’t move it will be close to midship.

Now I will complicate things a bit.
Think of a boat with its bow against a dock unable to move forward.
In this case the Pivot Point is right at the bow.
You can create the same situation with a spring line.
(A midship spring its mishap)

Same boat remove the dock and or spring boat stopped in water clear of any obstruction.
Apply a short burst of power. For a very brief moment until the vessel starts to move the Pivot Point will be right at the bow.
As the momentum increases the Pivot Point quickly moves back to roughly midships
This sudden movement of the Pivot Point happens when the vessel is moving. If an increase in power is applied.
You can use this phenomena to help you turn.

You can tighten a turn by applying a brief burst of power temporarily moving the Pivot Point forward.
This effect is most pronounced when the vessel is stopped or nearly stopped.

Often referred to as turning short round.
Vessel stopped or nearly stopped. Put Rudder hard Over, Apply brief burst of ahead trust.
Just as the vessel starts to move ahead. Stop motor and apply a brief burst of Astern
Most of the force will turn the vessel
Usually works better to Starboard as prop walk helps the turn when going astern.
 
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