Why parallel lines appear to converge

Sunrays, train tracks, and shadows of columns

In photographs (and in live viewing) we see parallel lines that appear to converge. Above are three examples: sunrays through broken clouds, train tracks, and shadows cast by columns.  I've encountered a lot of misconceptions about this subject. I hope to convince you that the same thing is happening in these three examples. It's an effect that is intrinsic to the geometry of image formation in cameras and in our eyes. It does not matter what is making the parallel lines. It could be light rays, tracks, trees, shadows, rows of seats in a theater, lanes on a highway, or even lines of marching penguins occupying your town.

Let's consider the tracks. First, a simple and succinct explanation for the impatient: If an object of a given size is moved farther away from a viewer, it will appear smaller because its angular size will become smaller. The same thing happens with the tracks; the farther away you look, the smaller the apparent spacing between the rails. Now, a more detailed explanation based on the functioning of a camera.

Looking at train tracks

Here are pictures of train tracks viewed from different angles. When looking straight down (A), the tracks appear parallel. As the line of sight tilts increasingly off vertical (B and C) the apparent convergence of the tracks gets more and more pronounced.

Train tracks viewed from straight above (A) and from increasingly oblique views (B and C)

Here's an animated version, showing how the appearance of a set of tracks would change as we continuously tilt our line of sight from A to C:

Parallel lines as seen from different view angles

The key thing that's changing is the angle between the line of sight and the plane containing the parallel lines. Let's look at the reason for this.

The camera boiled down to its essence

A camera is conceptually an extremely simple device, with only two parts: the lens and the image sensor (or film if you're old). The lens takes in light from the scene and refocuses it onto the image sensor to form an image:

Photographing a heart

To locate the image of a point on the object, such as the tip of the heart, just draw a line that starts at the object point, passes through the center of the lens, and then continues to the image sensor. The lens flips the image upside down, but the display on a camera presents it to us right side up. The image is also smaller than the object by the ratio of their distances from the lens center. For example, if the image sensor is 1 foot from the lens, and the object is in a plane 100 feet away, then the image will be 100 times smaller than the object. There's nothing deep about this; this is just the ratio of the sizes of the similar right triangles we see in the figure above.

Back to the track

Let's now look at photographing a section of train track, first from 100 feet straight above and then from an oblique view. In the straight view, the entire track is in a plane 100 feet from the lens. For a 1-foot image sensor distance, the image is 100 times smaller.

Train track in a plane 100 feet from the plane of the lens

Let's now tilt the line of sight so that the nearest part of the track (N) is in a plane 100 feet away from the lens and the farthest part of the track (F) is in a plane 200 feet away:

Train track in a plane oblique from the plane of the lens

The image of N is 100 times smaller but the image of F is 200 times smaller. When the image is turned right side up, the rails will appear to be converging. Note again that the important thing here is the ratios. F is twice as far away as N, so it will be smaller by a factor of two in the image.

If the track were to keep going off to infinity, the image would become infinitely smaller. The rails in the image would converge to a point, known as the vanishing point. Here's a picture of parallel shadows showing that in the image, they all point to the same vanishing point. Artists and architects use this to give their drawings a sense of depth.

Shadows of columns appear to converge toward a vanishing point

Sunrays are no different!

The distance to the sun is over 11,000 times the diameter of the Earth. Sunrays are therefore extremely close to being parallel. Imagine an observer looking at two sunrays that are 1 yard apart. It's late enough in the afternoon that the sunrays come through the clouds at an oblique angle, giving her an oblique view of the plane containing the sunrays. In the figure below, I've placed yardsticks between the rays. Yardstick C is twice as far away from her eyes as yardstick A, so that it will be smaller by a factor of two in the image projected on her retina.. The sunrays will appear as though they are diverging from a closer sun.

Left: Side view of an observer looking at sunrays. Right: What she sees

Now suppose she views the sunrays at high noon. Now her line of sight is perpendicular to the plane containing the sunrays and the three yardsticks:

Left: Side view of an observer looking at sunrays. Right: What she sees

As in the case of the train tracks, she sees the sunrays as they really are - parallel. Here's an animation of how such rays would change in appearance as the sun climbs from dawn to high noon. This animation was created in Wolfram's Mathematica using the equations that describe the action of a camera lens:

Sunrays as they appear at different view angles

If you search the internet for pictures of sunrays, you'll find that the degree of convergence varies a great deal, depending on the time of day and the orientation of the gap in the clouds. There are examples, such as this one, where the sunrays are parallel or close to it:

Parallel crepuscular rays