IIRC the "3/4" circle for passive sonar/hydrophone is (should be) more like 330 degrees...all-around except for the baffles, 15 degrees either side of dead astern....in game or real world.
You "hit the curvature of the earth" right at the point where you're standing; the distance to the horizon depends on your height of eye. If you're bobbing around in a life jacket or on a raft, the distance to the horizon is a mile or less.
An observer's height of eye of 36ft provides for distance to the horizon of 7nm; that's about correct for a lookout in the shears of a fully surfaced USN "Fleet Boat" (or on the sail of a surfaced "boomer").
To get distance to the horizon of 14nm requires a height of eye of almost 145ft. But, you should also account for the height of the object you are looking for...consider that to be the height of eye looking back at you. So, a lookout with height of eye of 36ft should be able to sight an object that is also 36ft above the surface, at a distance of 14nm (provided it's large enough to be seen at that distance to begin with).
A good lookout, well trained and experienced, should be able to pick out the masthead of a ship well over the horizon. Of course, smoke can be a dead giveaway from an even much farther distance over the horizon.