The above diagram shows a grid of lines at 35,000 feet above the ground, viewed from ground level. Each line spacing is five miles apart. I've previously shown how to use the elevation of the horizontal lines to get a ballpark estimate of distance. But what about the sloped lines, what can they tell us?
We've all seen a plane flying through the sky at what seems like a sharp angle, but it's actually just perspective. A plane flying directly towards you at a constant altitude will appear to be rising straight up in the air, then as it passes you and flies away from you will seem to be nose-diving towards the horizon.
Look at the grid above, the vertical line on the right represents that plane flying directly overhead of you, the angle between the track of that plane and the horizon is 90°.
The next line over to the left is just five miles away, it's parallel to the overhead track, but because of perspective it's tilted away from vertical. This time the angle between the track and the horizon is 55°. The next line over at 10 miles is at 35°.
Note the way the lines curve as they go over the horizon. Note how they initially seem to be converging at a vanishing point, but then appear to curve away from it. This is due to the curvature of the earth. This makes the "angle" of the line a bit tricky to measure when very close to the horizon, however it's quite rare to be able to see much of a contrail beyond 100 miles, and up to that distance the curve of the line does not diverge much from a simple straight line.
So in a simplified flat earth model (accurate at <100 miles), we can calculate the angle between the track and the horizon as ATAN(Altitude/Distance). Note that the "Distance" we are using here is the distance from a parallel track that passes directly overhead. It's the closest distance the plane will get to you (assuming it keeps going on the same track).
So more usefully, if we can measure the angle, then we can get a very rough estimate of the closest distance the plane will get to us. This can be useful in tracking down the plane in a flight tracker, and also gives a good sense of when a distant plane is going to get very close to you (handy for contrail photography).
Here's a chart of distances and angles.
Really the key thing to remember is that when the angle wth the horizon is 45°, then the plane will pass you at horizontal distance equal to its altitude. So within 6 or 7 miles. Any angle steeper than 45° is going to be a very close plane, and a good one to watch for photography. Conversely, at the angle gets more and more shallow the distance increases quite rapidly. At just 30 degrees the distance has nearly doubled to 12 miles, and at 15 degrees it has quadrupled to 24 miles. Beyond that the angle cannot really be usefully estimated.
Really though the only practically useful rule of thumb is
"Angle of an approaching plane above 45 degrees = good photos"