# Why isn’t there any curvature of the horizon in a horizontally oriented picture?

Because, if you look out horizontally, ie perpendicular to the radius and parallel to the surface of the Earth (or any other sphere), then no matter which horizontal direction you look, the angle of depression down to the horizon will be the same. So the horizon should appear perfectly straight in any undistorted truly horizontal view. It will appear curved up at the edges if your line of sight is pointing upwards and down at the edges only if you are looking downwards (which is why all those pictures taken from airplanes claiming to refute “flat earthers” are so silly).

What does show the curvature of the Earth (albeit in a way that is very hard to notice even at the highest commercial airplane altitudes) is the fact that, when you look out horizontally, the horizon is below the midpoint of the picture. This is impossible to demonstrate convincingly with a possibly cropped photograph, but if you are far enough from the sphere that the only way any part of it can be in your field of view is by looking downwards (as in the various pictures of the whole Earth taken from space), then you will indeed see the curvature.

In particular, even at an altitude of 54 km (which is well above most commercial airline cruising heights) we have the following:

An altitude of 54km is just a bit less than 1% of the radius of the Earth, so the angle of depression of the horizon is about arccos(0.99) (*see note below if you need an explanation), which is approximately 8 degrees. Since that is about 10% of the 80 degree viewing angle of a 24mm camera lens, a horizontally oriented camera with a wide angle lens at an altitude of 54 km should produce an image of a perfectly straight horizon line about 10% below the middle of the frame.

But if the camera is pointed down so that the middle of the horizon is at the centre of the frame then there should be a (just barely) noticeable curvature. (Since the half-view width of 40 degrees is less than a half of the 90 degree sweep to the side of the camera, the edges of the part of the horizon in the picture will be less than 3% of the image height below the middle.) And with a telephoto lens of viewing angle just 10–20 degrees, the horizon should only be completely straight if it is right near the bottom of the picture.

(*)-Note: The picture below shows that the angle of depression, $\delta$, is the same as the angle subtended at the Earth’s centre between the observer and the point seen on the horizon, whose cosine is just the ratio $\frac{R}{R+a}$ where $R$ is the radius and $a$ is the altitude of the observer.