Theory of Perspective and Photography
viewers usually do not carry rulers around."
Boris Rauschenbach is a well-known Russian scientist and engineer who stood behind many space exploration projects. Also he spent a lot of time to investigate the peculiarities of perspective. He studied many works by famous artists to understand how depth and distance can be shown on a flat surface. As a result, he amended the classical theory of perspective that was discovered during the renaissance. As far back as 1986, he wrote a fundamental book entitled The Systems of Perspective in Art. General Theory of Perspective, in which he described his ideas from both mathematical and aesthetical points of view. As far as I know, this book has not been translated into English so far. In this brief article, I would like to popularize his ideas among English-speaking audience and to apply his theory to photography.
Of course, within such a small text I cannot explain the entire theory to you. You will find neither mathematical formulas, nor scientific problems here. I would like to keep things simple and give you just some basic information.
You should also keep in mind that visual perception is a very complex subject. To understand it, we must know how brain processes information, how our eyes form images, as well as many other related issues. This article does not deal with visual perception directly. It just explains some facts connected with the perception of space.
Any general purpose photographic lens renders images in accordance with the classical linear theory of perspective. Thus, first of all we have to understand methods of the classical theory of art.
The picture below can be found in many books on art. It shows how a room with the dimensions of 4 by 5 yards and three men in it can be shown on a flat surface of a picture.
All four cases represent the perspective of the room correctly. Still, the impression is different. A 50mm lens is considered as normal because it produces most natural perception of depth.
Unfortunately, many people are firmly convinced that all lenses reproduce perspective in the same way. Their reasoning is simple. They suggest taking several pictures with the help of various lenses while standing on the same spot. If we allow for the natural differences in content, we will see that perspective will be practically the same in all such photos. With the help of the same arguments, they demonstrate that perspective cannot depend on the size of the film frame (or digital sensor). “If we simply cut out the central part of the frame, why should its perspective change?” they ask. But when saying this, they just demonstrate that they do not understand the kernel of the problem. Both the focal distance of a lens and the size of the film frame affect the perspective through the same mechanism, namely through the change of the distance between the camera and the foreground. To see the changes in perspective, a photographer has to move closer to (or farther from) the foreground in order to allow the foreground to fill the same share of the image. When we say that different lenses show perspective differently, we imply that we have to change the distance between the camera and foreground to make an adjustment for the changed viewing angle. Only under such a condition, perspective will be completely determined by the viewing angle of the “lens plus frame” system. The picture above demostrates exactly this case.
However some specialists, including Rauschenbach, say it is a rough approach. Thus, let us consider the subtle details.
What is wrong with the classical theory?
Actually among the properties of the visual perception there is such an important thing like constancy. The constancy can be of shape, of color, and of size. Due to the constancy of size, our perception makes close objects smaller, and distant objects larger. According to the theory, for any object and circumstances, there is the optimal distance of natural visual perception.
The photographs taken with a 35mm lens often show distant mountains as unnaturally small parts of the image. At the same time, very close items are shown as huge objects. This can be corrected a little, if we switch to a 50mm lens. It will produce a more natural image as far as the mountains are concerned. They will be enlarged and rendered closer. However, the 50mm lens will fail to decrease the size of the items in the foreground. Thus, there are no lenses with natural perspective. All of them are unnatural to a certain extent, and a photographers task is to minimize deviations from the natural visual perception.
scale, and depth
According to Rauschenbach, there are three main parameters that must be shown correctly in a picture. Those are similarity, scale, and depth.
The following example demonstrates how to treat all of them. Suppose we have to show two identical triangles in our picture. Also suppose we know the distances at which both triangles are located. Of course, the distant triangle should be smaller. The question is what are the other rules for us to follow?
The classical theory treats all such things in a similar way, but it does not take into account the fact that the correct scale optimum does not coincide with the correct depth optimum. The corresponding errors cannot be minimized simultaneously. If we minimize depth errors, the scale errors will remain big enough, and vice versa. This is a fundamental difference between Rauschenbachs approach and the classical theory of perspective.
To a photographer, this means there is no universal focal length. The lenses should be chosen depending on the particular purposes and situations. Still, some lenses are more universal than the others. The explanation to this will be given below.
law of conservation of distortion
This is another crucial point in the general theory of perspective. According to this law, the sum of all representation errors (errors of similarity, scale, and depth) remains constant in all reasonable cases. When the law does not hold, the cumulative error is unusually large.
In photography, the law holds for the lenses with focal lengths of 24mm to 150mm, according to my personal estimations. It does not mean other lenses cannot be used to produce decent images. Even an unnatural perspective can be very artistic!
This law is very important for practical purposes, because it says there is always a tradeoff between similarity, scale, and depth. It is a photographers task to choose what is important and what is not.
Main types of perspective
Rauschenbachs main types of perspective are described in the table below. Actually, the optimal observation distance L depends on the object size, its surroundings and many other factors as well. It is even different for the width and height. However, such subtle deviations are not taken into account in this text.
Those were only the main types of perspective. You may create your own scheme if necessary.
In his book, Rauschenbach said nothing about focal distances. I draw the analogy based on my experience and common sense. Of course, it is not exact. If you are unhappy with it, develop your own conformity approach. Here I just want to demonstrate some general ideas to you.
The main recommendation for photographers
The variety of perspective types demonstrates that there cannot be the universal focal distance in photography. Each time a photographer wants to take a shot, he should choose the optimal focal distance that corresponds to his tasks and circumstances. One should also avoid composing frames with the help of zooming. Once the suitable focal length is selected, it is always better to change the composition by changing the distance to the object.
The 50mm prime lens is still the best!
Both classical books on art and Rauschenbachs theory agree that painters and photographers prefer to show natural depth in their pictures. This task corresponds to Type B in the table, i.e. to a 50mm lens.
As I said above, a 50mm lens is also good for landscapes, because it shows natural depth from the middle area to the background. Of course, close objects are distorted. But one can easily arrange such a composition that minimizes errors of scale. Even close portraits can be made with such a lens. The example of a natural close shot made with a 50mm lens can be found at the end of this article.
Wide-angle and super telephoto lenses
Now that you know the general principles, you may be astonished at the fact that super telephoto lenses are capable of showing objects naturally.
It is easy to demonstrate mathematically that small objects can be naturally shown in an axonometric system. Many painters do use it in practice. Only under such conditions, super telephoto lenses can show objects naturally. One should apply them for other purposes with great care.
The same can be said about wide-angle lenses. They cannot render natural perspective. Like telephoto lenses, they should be used only if the photographer deals with the special artistic task or cannot use any other lens in the given circumstances.
Is it a good theory to follow?
Any scientific theory has its limitations and drawbacks. It does not mean theories are bad. When applying them, you should only remember to observe a number of conditions. As far as art and photography are concerned, your taste and common sense can also assist you a lot.
also look at four my photos that I selected to illustrate this article.
Only one of them complies with the requirements of the theory. Feel free
to draw your own conclusions.
Example 1. Cityscape made with a telephoto lens
example demonstrates how a telephoto lens compresses the depth of the
image. The grey building looks unnaturally close to the trees. Actually,
it is located on the other bank of the river and is 400 meters away from
us. The building in the background also looks too close to us (it is located
at the distance of 2.5 km).
Example 2. Cityscape made with a 50mm prime lens
This shot complies with all the recommendation. Both scales and depth are shown naturally. The only drawback of a 50mm lens in this case is a slightly unnatural rendering of the water surface of the river. Of course, your opinion may not coincide with both my words and the theory.
Example 3. Sketch of a boy made with a telephoto lens
It would have been better to take this shot with a lens with a slightly wider angle of view. The boy looks a little bit flat. Only blurred pavement shows the depth of the picture. However, scales are natural, and the proportions are perfect.
When taking this shot, I had no control over the situation. I also could not move closer in order not to attract the boys attention.
Example 4. Portrait made with a 50mm lens
This picture contradicts both classical recommendations and Rauschenbachs theory. Still, I consider it rather a good image. I did my best to arrange such a composition, when natural perception of scales was not important. Moreover, the portrait obtained natural depth. I admit it is not a masterpiece but quite a decent picture.
The example shows that any theory and any recommendations must be applied with care. Under special circumstances, a 50mm prime lens can be a good portrait lens.
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