To add to this (I don't have my rulebook handy, but I can look it up if you need references), first off, the ball needs to break the vertical plane of the goal line. Note that on a properly marked field, the GL is approximately 4" wide. The actual, technical goal line, is the theoretical, vertical plane on the side of the line facing the field of play. The remaining 3.9999999999 inches of chalk are technically in the end zone.
Next, you might have heard (or eventually hear) the term "extended goal line." Basically, the goal line plane per se does not end at the sideline, but extends indefinitely. As an example, the GL pylon, when properly placed, is located out of bounds at the intersection of the GL and the sideline, however it is "in" the GL extended.
Why does this matter?
Because certain rules apply based on whether the runner is airborne or not when the ball crosses the GL or the extended.
First, if the runner is NOT airborne, then the extended GL is applicable. Basically imagine a runner approaching the end zone near the pylon. If the runner crosses the GL inbounds, however the ball crosses the extended GL outside the sideline (assume the runner is holding the ball out over the sideline in his outside hand so that a defender cannot reach it and try to bat it, etc.), then a touchdown is scored. The key is that if the runner is airborne, then the ball must break the GL plane inside the sidelines.
Second, if a runner is airborne, then the forward progress of the runner (the ball) is determined by the forward point of the ball when it crosses the sideline. In other words, a player can't launch himself diagonally forward and across the sideline at the 2 yard line and 'cross' the extended GL outside the pylon and expect to score a touchdown. In this case, the covering official must decide where the ball crossed the sideline when the runner was in the air. This can be difficult as theoretically the best angle would be a birds-eye view, however with practice and experience, this can be done by being in a position that is close to parallel yardline-wise with the runner, either in front of or in back of the runner (be careful if you are in front, as in being a deep official (FJ/SJ) that is at the pylon with the play coming toward him. conversely, you cannot make this call if you are a wing official (HL/LJ or FJ/SJ that got beat) if you are trailing by several yards and are hence "straightlined" i.e., looking parallel to forward progress as you can see the ball cross the sideline laterally, but you cannot see which yardline is being crossed as the ball is crossing the sideline.
Hopefully this helps...
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