A Parallax to Grind

Marika and I have an Apple TV, and if we leave it paused for a while, it goes into a screensaver panning over various scenes. One of these caught my eye, a view of the Isle of Skye in Scotland. If you skip near the end, you'll see an interesting effect where two pieces of land are moving by at different speeds, even though the camera is moving at a constant speed. This is due to the parallax effect, and I was curious what I could learn from their movement, similar to my earlier post on the angle of the Moon.

When light hits our eye, our brains translate the angle the light enters into a position. The angle of an object at x horizontal to our eye, and a distance d away will appear at an angle off center of

If we move horizontally, we change x, so for a constant velocity v the angle will change by

What this means is that even for the same camera position and speed, two objects will have different angular velocities based on their distance. We can visualize this with an animated plot:

On the left, we have the top-down view, where we see two objects moving past the camera at the bottom. On the right is the camera's view, where we can see that even though the blue begins ahead of the red, as we pass the objects, the closer red one races ahead. We can try to relate their speeds by using the equation for theta's change twice: the velocity and horizontal positions are the same for both, but the distances are different. Putting the two equations together gives

which says that if we take measurements at a few different horizontal positions, we'll be able to find a ratio between the distances, but not their individual values.

You may have noticed earlier I specified a single eye – Because we have two eyes, our brains can use this process without needing to move to find relative distances. This method also connects to astronomy: One way to measure the distance to a star is to observe it at different points in the Earth's revolution around the Sun – A planet-sized pair of binoculars!