Pivot Point
In this text we will try to keep the theory of pivot point very simple and will not discuss about ship turning circle diagram as our aim is to reach to practical application of pivot point.
Below is important info about pivot point:

It’s an imaginary point at which a ship/vessel is pivoted while turning.

Its position is not stationary and will change with vessel’s speed over water.
Position of pivot point(V/l at even keel)

While v/l is stationary,
The pivot point will be at ships center of gravity, approximately amidship.
2. V/l Just started to move ahead
As soon as a force is applied (like engine kick), the bow section faces a resistance from stationary water wall hence momentarily pivot point will reach/shoot to the bow position(shown as P1).
3. V/l moving ahead
As vessel picks up ahead speed the pivot point will settle roughly 1/4thof ship length from bow(shown as P).
4. V/l falling back /sternway
If vessel is having sternway, the pivot point will settle roughly 1/4thof ship length from stern.
There is more to the Pivot point concept i.e. shift of pivot point with application of force (tugs or bow thruster use), Normally the pivot point will move away from the point of application of force towards the center of gravity of vessel. We will discuss same in last chapter.
However for now the above concept will suffice for you to understand and go ahead with practical application of pivot point concept in next chapter where we have discussed the most important application of pivot point theory, i.e. Turning lever and turning moment.