# relationship between focal length, perspective projection and camera distance

This page here describes that the focal length does not result in perspective distortion, while this page here describes if you increase the focal length and camera distance to infinity then it results in orthographic projection(affecting the perspective). So, what is the mathematical relationship between focal length, camera distance and the perspective projection.

P.S: If changing the focal length doesn't affect the perspective of the image, what affects the perspective distortion of the image?

One thing and one thing only determines perspective: Subject distance. Period.

I think your basic confusion expressed in the question is due to the different ways the different sources you're looking at use the word perspective. They're not all meaning the same thing when they use the same word.

The term perspective projection isn't describing the same thing as "the perspective of the image." The term geometric projection, in my opinion, is a more accurate way of saying what is meant by perspective projection. That is, it describes how the lens projects an image of the 3D world onto a 2D medium such as film or a sensor.

The difference is covered in more detail at this question here at Photography SE:
What is the difference between perspective distortion and barrel or pincushion distortion?

Perspective describes the distance relationships between the camera and various items in the lens' field of view. As long as the camera is in the same spot and everything in front of the camera is in the same spot, then the perspective will be the same regardless of the focal length lens that is used. The focal length will determine the angle of view (as long as the film or sensor size does not change), but it won't affect the perspective at all. If you take a shot with very wide angle lens and cropped the middle of the resulting image, it would look the same as a shot taken from the very same spot with a longer focal length lens giving the narrower field of view of your crop. You'd lose resolution due to throwing away most of the camera's pixels, but the perspective would be the same.

From the description for the [Perspective] tag here at Photography SE:

Perspective is the spatial relationship between a camera and the things the camera is photographing. If two objects of the same size are in the scene and one is much closer to the camera than the other the closer object will appear to be much larger than the further object. If the two objects remain the same distance from each other but the camera distance is increased from both then the apparent size difference of the two objects will decrease.

If changing the focal length doesn't affect the perspective of the image, what affects the perspective distortion of the image?

The only thing that affects perspective is shooting distance, and the relative distances of various objects in the scene to the camera. In order to change the perspective one must change the relative distances between the camera and various things in the scene. Here's an example from an answer to this question:

Let's assume you are 10 feet away from your friend Joe and take his picture in portrait orientation with a 50mm lens. Say there is a building 100 feet behind Joe. The building is 10X the distance from the camera as Joe is, so if Joe is 6 feet tall and the building is 60 feet tall they will appear to be the same height in your photo, because both would occupy about 33º of the 40º angle of view of a 50mm lens along the longer dimension.

Now back up 30 feet and use a 200mm lens. Your total distance from Joe is now 40 feet. Since you are using a focal length that is 4X the original 50mm (50mm X 4 = 200mm), he will appear the same height in the second photo as he did in the first. The building, on the other hand, is now 130 feet from the camera. That is only 1.3X as far as it was in the first shot (100ft X 1.3 = 130ft), but you have increased the focal length by 4X. Now the 60 foot tall building will appear to be roughly 3X the height of Joe in the picture (100ft / 130ft = 0.77; 0.77 X 4 = 3.08). At least it would if all of it could fit in the picture, but it can't.

Another way to look at it is that in the first photo with the 50mm lens, the building was 10X further away than Joe was (100ft / 10ft = 10). In the second photo with the 200mm lens, the building was only 3.25X further away than Joe was (130ft / 40ft = 3.25), even though the distance between Joe and the building was the same. What changed was the ratio of the distance from the camera to Joe and the distance of the camera to the building. That is what defines perspective: The ratio of the distances between the camera and various elements of a scene.

On the other hand, even two lenses with the same focal length can have different geometric projections. This will affect the shapes of items in the image, but it won't change what the camera can and can't see from the same shooting position. If Box "A" is in front of box "B" and hiding half of box "B" from the camera, changing from a rectilinear lens to a fisheye lens will not change how much of box "B" can be seen by the camera. That's what perspective is. The different geometric projection may cause straight lines to look curved, or items at the edge of the field of view to be "stretched", but it doesn't change perspective.

In the real world, there is no such thing as a lens with a focal length of infinity. That's a figment of the imagination that can be employed in CGI. To get orthographic projection with an actual camera, one needs a telecentric lens.

To get a view where objects on the far end of a three dimensional subject appear the same size as objects on the near side of the subject, we must use a telecentric type lens that will give us an orthographic view of our subject. One of the basic requirements of a telecentric lens is that the lens must be at least as large in diameter as the subject. That tends to make them very expensive.

• @juztcode; when the subject is kept the same size, a change in FL causes a change in subject distance, which causes a change in perspective... Feb 17, 2020 at 14:31
• @juztcode Yes, if the subject is changing distance from you and you change focal length to keep it the same size in the frame, the perspective will change. Feb 17, 2020 at 18:37
• This will also happen if you don't change focal length, but will be less apparent (unless you enlarge the image, in which case it will be the same as increasing focal length) Feb 17, 2020 at 18:38
• @juztcode There are a lot of folks who think and say that focal length affects perspective. There were also a lot of folks who said the Earth is flat and the center of the universe. At one time, that is what universities taught in opposition to heretics such as Copernicus and Galileo. Focal length only indirectly affects perspective if it causes one to change the camera or subject positions, which is what directly affects perspective. Look at this link and see that focal length makes no difference. Feb 17, 2020 at 20:14
• @juztcode, no. It is the subject moving closer that causes it. I've added an answer with example pictures that should help. Feb 18, 2020 at 0:20

Just to provide visuals to the answer that only distance controls perspective. It's not "perspective distortion" nor "focal length compression" as commonly described... it is simply perspective (point of view).

Here are three images taken with three different FL lenses (82mm, 50mm, 28mm) from the same distance (copied from a lesson I created).

And this is the wider images cropped to the same final image... the perspective is the same in all of them.

These images are taken from different distances but keep the subject approximately the same size (different FL lenses were used, but they could be crops as well).

It is the change in relative distances that causes the change in perspective. I.e. if you are very close to the subject then the background is relatively much father away from you. But if you move far enough away, the subject and background are much closer to each other than they are to you; enough so that they seem to be nearly on the same plane.

• Your models are amazing for their ability to stand still for so long! Feb 18, 2020 at 2:08

Most images do well even if their perspective is incorrect, some images fare poorly. To view a photograph with the correct perspective:

View the camera’s image made the same size as the film / imaging chip, from a distance equal to the focal length setting. Such a viewpoint is likely not possible because modern cameras are petite thus the focal length used will be very short. Keep in mind that the closest the average person can read or view is 250mm (10 inches).

Because the modern camera makes tiny images we view them enlarged on our computer screen, TV, or as a print on paper. For these viewing methods we need to take into account the magnification applied to the final image.

Suppose an FX (full frame) 35mm size camera images the human face (portrait) taken with a 100mm lens. We enlarge this image and make an 8x10 inch print. The magnification applied will be about 8x.

The correct viewing distance is computed, 8 x 100 = 800mm = 31 ½ inches. This is about how we would view an 8x10 portrait print sitting on the mantel.

In short, an image will appear correct as to perspective if viewed from a distance about equal to the focal length multiplied by magnification used to make the displayed print.

Post Script: The only effect of focal length on perspective is to change the size of the image and the distance from which the displayed image is to be viewed.

The human perspective is simply what we see with our eye/brain combination. Standing before a glass window, you can draw, with wax pencil, the outline of objects you are observing. The perspective you observe is thus duplicated by the drawing. We are talking about the spatial relationships of the objects.

If a camera is used to image this vista, it is placed at about the same distance from the glass as your eye. The image it makes corresponds to the image drawn on the glass provided this image is viewed at zero magnification from a distance equal to the focal length of the taking lens.

If the viewed image is an enlargement, the viewing distance, to see this vista in correct perspective becomes the focal length multiplied by the degree of enlargement.