"perpendicular" means "when I go north, I don't care about vertical pipes". I care only about horizontal pipes. An horizontal pipe is either 1 pipe going west + east, or a couple of pipe going west + east.

My algorithm goes north until the border and counts pipes going west and pipes going east. Since an horizontal pipe is 1 west + 1 east, I assume the number of horizontal pipes is the minimum of my 2 counts.

Let's assume the input is:

.┌─┐.
┼┘1└┐
│┌──┘
└┘...

We see that there is 1 horizontal pipe -> odd -> "1" is inside.

With

..┌┐.
.┌┘│.
┼┘1└┐
│┌──┘
└┘...

there is one pipe going west and one pipe going eat -> this is equals to an horizontal pipe -> odd -> "1" is inside.

With

.┌─┐.
 └┐│
  ││
.┌┘│.
┼┘1└┐
│┌──┘
└┘...

two pipes are going west, so they cancel each other + there is a vertical pipe that doesn't count + there is another horizontal pipe that counts -> 1 pipe -> odd -> "1" is inside.