How distance is calculated: range of defense buildings. When will TC's units spawn?

When you're studying an opponent in War many times you are not sure if you can rally to a certain point because you don't know if mortars or cannons will attack you there. Or you may not be sure if TC's troops will come out or when will they come out.

If you zoom in a tower, you will see it is a 3x3 matrix, TC is 6x6, ballistas are 2x2, and so on. The whole map is a matrix and thus you can always calculate who's gonna attack what.

Distances are calculated different for units and buildings:

1. Units:  To know how far a unit can hit u can just count. Think of the unit's position as distance 0 and count the spaces towards the building. If the distance is < than range you can hit it, if its > or equal, then u can't. Let's see a couple examples:
1. To say that a Fusilier has a 1.5 range means that he can attack a building  (next to the wall) from behind a wall because walls are 1x1 buildings. Since the building is 1 unit, you can hit it. In earlier ages soldiers have a range of 1, and since 1 is just the same as the distance you have to cover (walls are 1x1) they can't hit buildings that are right behind walls
2. Archers: normal archers have a range of 4. Castles are 4x4. If you want to atack a building that is behind a castle, you won't be able to unless you surround the castle from the sides (range 4 = distance 4 --> cant attack it). But if you're British you will have range 5 and thus will be able to attack that building.
2. Buildings: Building's range is calculated from the center of the building, instead of the side of it. Examples:
1. Tower: range of 7, but since it is a 3x3 building, the center is 1.5 units away from the side of the tower, so 7-1.5 = 5.5 range. A tower will attack anything that is <5.5 units away from the center of any of its sides. Its important to notice that all defensive buildings have less range from their corners than from the sides. When you click on them you see the circle around them, well, the maximum distance from the circle to the buildings is the line from the circle to the center of any of its sides.
2. Catapult: range of 10, but its 3x3, so it attacks 8.5 units away from its sides.
3. Town Center: IMPORTANT!! This is an exception, the TC has a range of 5 units, but thats counting from the center of its sides, not from the center of the building. So if you stay further than or just 5 units away from its sides units won't spawn, the second you move to 4.9 units away, they will. 
1. IMPORTANT: normal defenders from the TC spawn in a few seconds, but troops donated from allies spawn almost immediately, in about 1-2 secs.