The horizon paradox!

Above is a photo of the sun disappearing over the horizon. You may also have seen photos of sailing ships disappearing over the horizon when they get a few miles away from the beach. The horizon is part of our daily experience, but exactly what is the horizon? Where is the horizon? And if the Earth is curved enough to block our view of boats more than a few miles away, then why does the horizon look so flat side to side?

This is the horizon paradox. Let's solve it.

Attack of the tall woman!

Tall woman at the North Pole, and how much of the Earth she can see

An absurdly tall woman stands at the North Pole of a perfectly spherical Earth (left picture in the figure above). She's 625 miles in height, big enough to threaten cities and attract film crews from low-budget Japanese movie studios. Suppose she looks south. This is easy to do at the North Pole because any direction is south, so she randomly picks a meridian, say longitude 60E. She can see only to the point P where her line of sight is tangent to the meridian. At her eye height, that point is on the 60N parallel (latitude circle). As she looks in any direction, anything south of the 60N parallel is blocked from her view.  So her horizon is that circle. Also, she has to dip her line of sight downward by 30 degrees to focus on this horizon.

Fortunately, a scientist has invented a ray gun that shrinks her. After the first shot, she shrinks to 256 miles in height  (right picture in the figure above). Now she can see only to the 70N parallel, and her dip angle has decreased to 20 degrees.

To save humankind, we now shoot her repeatedly with the ray gun until she's the size of a normal human being. Then she can be taken into custody when she's done helping us with this thought experiment. As she shrinks, so does her dip angle and the radius of her horizon circle. Suppose she ends up measuring 5 feet from toes to eyes. Her horizon is now the circle of latitude 89.96N, which is 2.8 miles in radius, 10 feet below her eye level, and at a dip angle of 0.04 degrees.  (If you are interested, see the math below.)

This is a very small dip angle. Compare it with the sizes of the moon and of an index finger held at arm's length, as they would project on a camera's image sensor or the retina.

Comparative sizes of objects as they would be projected onto a camera's image sensor or the retina

A 0.04-degree dip angle is too small to be distinguished from "eye level" by the naked eye, so she perceives the horizon circle as if she were viewing it from its center. She will see the horizon as a straight line at eye level across her field of view.

If we try to add a third drawing to the tall woman figure, we get this:

Woman and horizon point appear to be on the same horizontal line

Not very helpful. The picture is 6 miles wide, but the woman's height and the downward curve of the Earth are each less than 0.001 mile. So let's stretch the vertical scale 300x to see what's happening:

5 foot woman at the North Pole, but with vertical scale expanded

This picture greatly exaggerates the dip angle and the vertical distances, but now we can mark the measurements. The Earth's bulge will start to block her view of boats that sail beyond the horizon. But at 2.8 miles away, the 10-foot drop from her light of sight to the horizon subtends this undetectable 0.04-degree dip angle.

Now we can resolve the horizon paradox. The distance at which a sailboat starts to disappear corresponds to the radius of the horizon circle. But we don't detect curvature side to side because our eyes are viewing the circle from too close to its center. It is at eye level, curving around us. So there's no need after all to go organize the Cylindrical Earth Society.

Eye at the center of a hoop

By the way, the woman at the North Pole wisely turned herself in to the Royal Canadian Mounted Police rather than wandering into Russia. Good call.

What happens when we get high?

No, I don't mean that kind of high. I mean high altitudes. The 256-mile-tall woman above is seeing things from the same height as the International Space Station. In my post on perspective, I've explained how to predict a camera image, given a scene and some camera specifications. The equations are based on comparing similar triangles and are simple to implement on your favorite computer tool. Let's take a camera with a 6.3-mm focal length and a square 5.744-mm x 5.744-mm image sensor and mathematically take it up into space. At each point in our ascent, we can calculate the horizon curve when photographed with the camera held at eye level. Here's what I get using Wolfram's Mathematica.

Shape of horizon as photographed from height ranging from 5 feet up to the International Space Station

So, the horizon's curvature should be visible to the naked eye, if you go high enough and have a wide enough field of view. And it is visible. In 2009, Justin Hamel and Chris Thompson used a Canon Powershot A470, to take this photograph of the horizon from a balloon at 100,000 feet. This camera had the same focal length and image sensor width as my mathematical camera. (Camera lenses can introduce distortion that makes a straight line look curved. I used Canon's test data for the A470 to correct for this and verify that the curvature is authentic.)

If a satellite or spacecraft is far enough away to photograph the "entire" Earth, its view is still limited by its horizon. Here is a picture to show this. A satellite flying 1,000,000 miles away (right) would see almost an entire hemisphere. But a satellite 1000 miles above the Earth's surface (left) would see substantially less.

The views of the Earth from satellites flying at 1000, 5000, and 1,000,000 miles above the ground. Top: Green area shows what can be seen from the satellite. Bottom: Wolfram Mathematica maps showing what the Earth would look like from each distance

To summarize, the horizon is always a circle, but depending on how high the eye is above ground, it can be anything from an equator-sized circle viewed from far away to a tiny hoop viewed from its center. Here's a video recap showing the journey of an eyeball from the moon to 5 feet above the Earth.

How the horizon changes as one approaches the Earth

If you fancy math...

Here are the equations to get the horizon distance and dip angle.