To get a better picture of the interference between the kings in the endgame, it is useful to have a few expedients for better communication.
The first is the windrose.
A king can walk in only 8 directions. From now on I call these directions as their name in the windrose.
Most activity in endings take place with the kings circleing around each other at one square distance. It is useful to have a convenient naming system for all 12 different rotation positions.
To that end we think of it as a clock with the black king (the opponent) in the middle.
The white king is at the end of the short hand.
Horizontal opposition is 3 o'clock or 9 o'clock.
Vertical oppositon is 6 or 12 o'clock and so on.
The movements of the kings can be described with vectors (length = # moves, direction = according the windrose) and rotation.
With these tools we can easy describe new rules.
Black to move.
Begin position = Kg8, Kg6 (= position 12 o'clock)
When the black king walks south west and the white king follows at 1 square distance, there can be no rotation. When the black king arrives at b1, the white king will still be in the 12 o'clock position.
Rule #1: when two kings walk in the same direction at close distance, they can't encircle each other.
And derived from rule #1:
Rule #2: When you want to encircle the enemy king, you can only do so at the expense of a tempo.
White to move.
Blue = move 1
Red = move 2
Green = move 3 - 5
White uses a tempo to rotate from 12 o'clock to 11 o'clock.
Black starts to walk in SE-direction.
White follows at 1 o'clock, which will be the endposition too.
Let's have a look at a real position.
White to move and draw.
It is clear white can't keep his pawn on the board.
How many extra tempo's has white?
The first spare tempo is the start tempo.
When both kings walk in the same direction, there can be no rotation.
The last 2 moves of black are in western direction, which provides white with another spare 2 tempo's. Don't bother to much about this, a little further this will be shown in a few diagrams.
Thanks to Takchess we now all know that you have to imagine the final position first.
See diagram 6.
Kings 90 degrees rotated
When the white king reaches the c2 right after the black king has eaten the white pawn, white can hold the draw.
In the start position the white king is at 12 o'clock and in the final position at 3 o'clock.
So white has to rotate with the black king in the center 3 times (3 x 30 degrees = 90 degrees).
At the expense of 3 spare tempi.
Hence the start move of white must be 1. Kh8!!
I'll give you all the important moves in the next diagrams.
1st rotation from 12 o'clock to 1 o'clock at the expense of the start tempo.
See diagram 7
2. ... Kf5
3. Kg7 Ke4
4. Kf6 Kd3
5. Ke5 Kc2
Translation over 4 moves in SW-direction.
No rotation. See diagram 8
6. ... Kb2
The translation of the black king to the west (Kc2-Kb2-Kxa2) will cost him 2 additional tempo's
The white king rotates around the black king from 1 o'clock to 2 o'clock at the cost of the first additional tempo. See diagram 9.
7. ... Kxa2
8. Kc2 =
The white king rotates further from 2 o'clock to 3 o'clock.
See diagram 10.
Can you see for yourself now why 1. Kf8? loses the game?