Some thoughts

 The following diagram is intended to show that the amount of LoAs is finite. I left out the knights to keep the amount of arrows in check.

A line of attack ends near the opposing king.

An assault on the enemy king must make use of a LoA. This way, you can prune the things you have to think about.

The pawn landscape cuts down the amount of LoA's to consider even more.

As you can see, the pawns are limiting both your own attackers AND can prevent the defenders from reaching the LoAs. We must grab any opportunity to get rid of thing to think about. Hence, focus on the LoAs and the typical scenarios that belong to the battle of the LoAs!

White to move

r4rk1/pbqn1ppp/1p1bpn2/2ppN3/3P1P2/1P1BP3/PBPN2PP/R2Q1RK1 w - - 0 1

Here the line of attack b2 g7 looks very blockaded. But that is an optical illusion. With moves like Nxd7 and dxc5, the long diagonal is cleared all of a sudden. While Bxf6 removes a defender of the focal point h7.

Look at the same time at the black defenders that are blockaded by th black pawns, and have trouble to reach the LoAs.

Positionally, a move like dxc5 normally is forbidden. But for opening up the long diagonal with tempo, we make an exception.

This position stems from a Colle - Zukertort, by the way.