Interference of the kings

Montse asked me a few days ago if there wasn't a simple method to map out the interference by two kings. Since then I played thru about a hundred postions.
From that I distilled two simple empyric rules.
I can't give a 100% guarantee that everything is covered with these two rules but at least it comes pretty close.
I will put them to the test with real positions in the near future.

The interference starts at the moment the kings are close to each other, at one square distance.
I haven't taken a look yet what happens when the kings are approaching each other.
There are 12 positions with the kings close two each other (12 hours of the clock).
I have analysed two positions: 3 o'clock and 4 o'clock.
All other positions can be derived by mirroring.

The two kings have each their own target where they head for.
On the paths to their targets the kings can shoulder away each other.

diagram 1





















White to move.
red = the target of the black king
blue = a blue square can be conquered by the white king in time. That is to say, at the moment that black conquers the red square.
yellow = the white king comes one move to late due to being shouldered away. Would be blue if there wasn't a black king
none = out of reach of the white king, no matter if there is a black king in the way or not.
position = 3 o'clock

The distance from the black king to its target (red square) = 3
Diameter of the blue area = 3 + 1 (premove) = 4
Distance of the yellow file from the white king = 4

Empyric rule #1 (position 3 o'clock):
The yellow area with targets that white cannot reach in time has the shape of a file at distance x+1 from the white king.

Where x = distance black king to target.
No matter where the target of the black king is.
That is to say, as long as it is on the files a, b, c or d

Another example:

diagram 2





















White to move.
x = black king - red square = 4
yellow file = 4 + 1 = 5 moves away from the white king
ofcourse the red square is surrounded by yellow squares, since the white king can't reach them due to the black king.
So that is all very simple.

Now let's have a look at the 4 o'clock position.

diagram 3





















red = target black king
blue = targets white, white is in time
yellow = white comes 1 move to late due to shouldering
none = out of reach anyway

rule #2 (4 o'clock):
The yellow area can be calculated exactly the same way as in rule #1, with one difference:
The southern part (1 rank below the red target) can be reached in time, hence that part of the file is blue.

And again, the red target can be anywhere on the a, b, c or d file!
That's all folks!