What is the centre of perspective in a photograph, and why does it matter?

What is the centre of perspective of a photograph?

The centre of perspective of a photograph is located at the camera lens or at the pinhole for a pinhole camera. It is simplest to visualise with a pinhole camera: Light rays are projected through the centre of perspective to form the image. The basic law of perspective follows from the diagram: $$\textit{Image size}=\textit{Object size}\times\frac{\textit{Image distance}}{\textit{Object distance}}$$

For a camera with a lens, the image distance is approximately equal to the focal length, \$f\$. So the centre of perspective is located at a distance \$f\$ from the image on the sensor (or on the film).

Normally this image is enlarged to a more comfortable size for viewing. If it is enlarged by the factor \$m\$, then the distance to the centre of perspective must be enlarged by the same factor. So the centre of perspective is at a distance \$m\times f\$ from the enlarged image.

Why is it useful?

It is useful to know the location of the centre of perspective if you want to view the image with its correct perspective, i.e. the perspective that you would have seen if you were viewing the scene from the camera position when the shot was taken.

Mathematically, correct perspective can be defined as the perspective in which the angle between any two points in the image (as seen by the viewer of the photograph) is the same as the angle between the corresponding two points in the original scene (viewed from the camera position).

Correct perspective will be seen when an image is viewed from its centre of perspective.

Usually, we don't care whether or not we view a photograph from the "correct" viewing position. In normal circumstances we think that the picture looks the same from wherever we view it; we do not notice any difference in perspective. However, in some special circumstances it may be important to obtain the correct perspective when viewing the image.

An example of a circumstance in which we may want to ensure correct perspective for the viewer is in Virtual Reality photography. If the user of a VR headset is to see the most realistic image, then it is important to ensure that the user's eye is located close to the centre of perspective of the image. If the perspective seen by the user is not correct, that will make it very difficult for the user to judge distances in the scene. That will, in turn, make it much harder for the user to coordinate their body movements correctly with what they see on the headset.

In other circumstances it can be a serious disadvantage to view the image from the centre of perspective. This is the case for extreme telephoto shots (i.e. shots with a narrow angle of view). For example, a 9" x 6" image taken with an 800mm lens on a full-frame camera has its centre of perspective nearly 17 feet from the image! For images taken with a narrow angle of view, it makes sense to view them from a convenient viewing distance much closer than the centre of perspective. Of course, they will then not be seen with the correct perspective, but may be subject to noticeable perspective distortion ("telephoto compression"), depending on the subject matter.

Zeeman's Paradox

Zeeman's Paradox states that although a perspective drawing (or photograph) can be viewed from only one point (the centre of perspective) to be seen with its correct perspective, it nevertheless appears realistic when viewed from most angles and distances.

Whenever we look at a two-dimensional image (whether on screen, on paper or in another way) we see immediately that it is a 2-D object and it behaves as we expect a 2-D object to behave when we view it from different angles and distances. We know we are looking at a 2-D image not at a 3-D scene. It is only in some special circumstances that we are forced to imagine we are looking at the 3-D scene captured in the image.

One such situation is when driving a vehicle. Some vehicles have rear cameras that display an image on a screen visible to the driver. They normally use an ultra-wide-angle lens and the displayed image is seen by the driver at much below life-size (because it is being viewed from much further away than the centre of perspective). The result is that anything seen in the rear camera appears to be much further away (from the camera) than it actually is. Drivers are usually warned about this and to make appropriate allowance when judging distances.

A similar effect occurs with the side mirrors on many vehicles which show a reduced size image (to allow a wider angle of view). Such mirrors are required in some countries to have a warning to the effect: "Objects in mirror are closer than they appear".

References

The Manual of Photography by Jacobson et al. (7th edition 1978) Chapter 4 on "The geometry of image formation" contains a good introduction to perspective. The 9th edition (2000) contains a briefer and more formal introduction to perspective (less easy to understand in my opinion).

Elements of Perspective by Bruce MacEvoy is a comprehensive treatment of classical perspective, although written for artists rather than photographers.

The book written to accompany the Royal Institution 1978 Christmas Lecture on "Geometry and Perspective" by Christopher Zeeman.