In 1986, Academician Boris Rauschenbach published a book called “The General Theory Perspectives. In this work he suggested that a successful mathematical model that allows to create different display systems surround the world on the plane pattern. His theory also allowed to quantify the deviation from natural visual perception for a particular method of perspective construction. Rauschenbach with mathematical rigor to substantiate the fact that the classical theory, known since the Renaissance, is not always possible to create natural images on planar images.
This article attempts to popular exposition of the basic principles of the theory Raushenbaha for photos. The material contained in brief as possible and accessible. Those who are not afraid of scientific style of presentation, are encouraged to review the text of this book in the original.
In reading this material is very important to understand the following:
- Here, nothing will be said about what lenses draw the prospect of true and what is not. In a sense, all they can get the correct image.
- Also there will be nothing said about the fact that some lenses are capable of seeing reality like a man. The sight of the man – a single major issue, which in this article deals in a very limited extent.
- Without a doubt, and an artist, photographer and solve the problem first and foremost art. In this sense, the beautiful illusion of a better reality, and not always worth striving for display in a work of art naturally perceived image of the world. Nevertheless, knowledge of the laws of natural visual perception significantly helps the artist in solving creative tasks. In other words, follow the laws of perception is not necessary, but it is useful to know them.
All quotations are from: Rauschenbach, BV Systems perspective in the visual arts. General theory of perspective. Nauka, Moscow, 1986.
To begin with we recall the classical
Any camera operates in the linear system perspective. The properties of such systems are well known and described in detail in books on all types of fine art, as well as projective geometry. Before discussing the theory Raushenbaha, recall briefly the main classical situation and their relationship to photography.
What is meant by “undistorted perspective” in the classical theory?
Consider the image, taken me out of a textbook on painting. It shows how it will look promising for the construction of different distances between the eye of the artist and the image plane (in the photos, these cases correspond to different focal length lens). In all four cases we are talking about the image of three people in the same room measuring 4 x 5 m.
a. The artist was too close to the picture plane (the case of wide-angle lens). The distance between the faces of a man and seems far more true of 4 meters. The window was taken very wide. Beside him is clearly able to rise a few people. The depth of the room is passed unnaturally exaggerated.
b. and c. Here we used the distance to the image plane, which is recommended by most guidelines. Depth is transmitted almost naturally.
d. The eye of the artist is far from the picture plane (the case of a telephoto lens). The room does not seem to us so deep. It looks as if it had a depth of two meters, not four. Far man approached us. The window looks very narrow. It seems that near him may get only one person.
Note that in all four cases, the prospect of being built is absolutely right! You can not argue that some of these constructs geometrically correct. All lenses draw the prospect’s true! But the audience is perceived differently. And in painting, and photography!
Unfortunately, quite often be found judging that the transfer of Outlook does not depend on the focal length lens. Supporters of this idea reinforce its position by the following argument: “Do not move, make two shots: one – full-time lens, and the other – a telephoto lens. If the first shot to make vykadrovku corresponding visual angle telephoto lens, it is easy to find that the prospect of equal size prints will look the same. Consequently, all lenses transmit equally perspective. In the same way they “prove” the independence of the picture perspective on the magnitude of the diagonal frame. “Is that something can fundamentally change what we cut out of the whole frame of its central part?” – They exclaim. However, proponents of this position, only show their lack of understanding of the mechanism of different ways to transfer perspectives in painting and photography. But to understand the causes and consequences – very important! Of course, if the photographer is in one place, the change in focal length will only lead to a change in scale. The reasons for different transmission perspective is quite different!
Supporters of the above arguments overlook one simple thing: lens focal length (as well as the value of the diagonal frame) affects the prospects of drawing indirectly: through the mechanism of changing the distance to the object, not the other way! In order to see changes in the transmission perspective, the photographer is simply obliged to move, making sure that the foreground took full frame. The above image is just that and shows. Thus, to talk about the differences in the display of the prospects for a fixed frame format, different lenses can only be provided ravnomasshtabnoy shooting foreground: f / d = const (f – lens focal length; d – the distance from the camera to the foreground!) (See also, eg, Fig. 100 of Chapter 19 in the book of VP Mikulina “25 lessons pictures.”) It is in this sense (and only in this sense) can talk about different perspectives of the transfer lenses with different focal length .
Generally speaking, the transfer of prospects determined by the angle of view of “lens + recording medium”. Equal angles correspond to the same display perspective. Different angles – different perspective building, while in the pictures is understood that the shooting at different angles is accompanied by a change in the distance between the camera and the foreground.
To quote the textbook (School of Fine Arts: Issue 1, the manual.. M, depict. Art, 1986): “Most artists depict nature, at a distance equal to one and a half or two diagonals of the picture plane, that is, the angle of view equal to elect 28 ° or 37 ° ». Classical theory asserts that it is in this range of viewing angles and photographic images, and paintings will be perceived as natural. The word “natural” should be understood as forming the viewer to be accurate picture of the change in the scope and the distance between objects along the axis directed from the beholder (ie, deep). Recommended normal (not to be confused with the “right”, because they are all “correct”) image in our case the image «b».
For those who do not overpower reading this article until the end 
I will say at once that the theory Raushenbaha (like many other theories) is different from the classical approach, mainly to the fact that states: it is impossible to achieve a natural display of scale and natural transmission of depth at the same time. Using the more correct wording, it can be argued that the mathematical model Raushenbaha generally proves impossible without distorting the reality show on the plane. However, it is not surprising. This view dominates among professionals involved in the visual perception of man.
So, we remember why lenses with a normal focal lengths naturally convey perspective in the classical approach. Now we proceed to discuss the finer things.
What is wrong in the classical approach?
The visual perception of humans is characterized by the same property as constant.
Constancy of form makes the artist to depict the ball round, in some place where the picture he was not, despite the fact that the projective geometry requires portray it in some cases, as ovals.
Due to the constancy of color people will always see the white chalk and carbon black, even when the lighting conditions on the chalk reflects less light than coal (Cretaceous – in the shadows, and coal – in the sun).
Constant quantity (size) explains why the size of the observed objects does not change the laws of classical linear perspective. Indeed, very close objects in perception is automatically reduced by the brain, while increasing somewhat remote. According to the theory, for each object and the environment there is an optimum viewing distance, providing the most natural visual perception.
The classical theory of perspective, as, indeed, fotooptika does not take mechanisms constancy values in the calculation. If we take pictures multifaceted landscape lens with a focal length of 35 mm, that the picture we find that the long-range plan looks too small, and short-range – unnaturally large. You can slightly improve matters, if slightly larger focal length (50-70 mm). This would bring together long-range and medium plans almost to the level of natural perception. If this was to reduce the foreground, then everything would be fine. But, alas, the lenses with such properties does not exist. And in this sense, no lens is not able to form an image closer to the natural human visual perception.
On the role of mathematics in the theory Raushenbaha
At first glance at the text of the monograph Raushenbaha may be the misperception that all the arguments and conclusions with mathematical precision result. However, in this case should not seek logical (mathematical) relationship between initial conditions and conclusions. Theory Raushenbaha is not solely a logical (mathematical) consequence of a certain set of assumptions.
As stated in the previous section, the theory is based Raushenbaha lie constancy mechanisms, which are almost impossible to describe within a single mathematical model of perception.
But if the results can not be obtained purely mathematical way, what is the role of mathematics in this theory? The answer is: mathematics in this case is used as an auxiliary tool in the evaluation of distortion.
Therefore, in theory Raushenbaha should not seek the procedure mathematically rigorous results. Perception – Psychophysical and not mathematical problem.
For this reason and in this article do not have to try to identify the logical flaws and strong arguments leading to the only possible conclusion. It will contain only some results, with an extremely truncated form.
The scale, similarity and depth
Mimicking the real world on the plane, an artist or photographer is always striving to natural transmission (ie, without noticeable distortion) similarities, the scope and depth. Alas, says Rauschenbach, can not properly convey these three parameters simultaneously. And since this is so, the author of a work should always seek a reasonable compromise, depending on the subject, camera angle, the ratio of plans at various distances, etc.
Elucidate this small illustration. Suppose we need to represent two similar triangles are from us at different distances (see figure). We also assume that the distance to the triangles are known.
1. Correct display of similarities implies respect equality a1 / b1 = a2 / b2. All the traditional system of linear perspective to comply with this condition. All general-purpose photographic lenses also draw the prospect of no errors displaying similarity. In any case, their manufacturers are doing everything possible to minimize such errors. (Specialty lenses, such as type of “fish-ai” are not considered here.)
2. Portraying the far smaller triangle, we must choose the scale, ie, the ratio a1 / a2 = b1 / b2. According Raushenbahu, there is some scale mapping, which corresponds to the natural perception. This is the reference value of the ratio a1 / a2 in the general case depends on the plot of the image, the distance between us and the first triangle, as well as on many other factors. However, in the first rough approximation we can assume that there is some absolute standard of correct display size. Photographer can change the scale, removing the plot lenses with different focal length.
3. To correctly display the depth, the author of works should choose the optimal value of the ratio a1 / d. As is the case with the choice of the correct scale, the desired optimum is determined by many factors. But as a first approximation again, we assume that there is some universal standard of natural mapping depth. As in the second case, the photographer can select a value ratio, which characterizes the depth by changing the focal length of your lens.
With all this new and surprising? Why is the author of the theory needed to complement the classical theory of perspective? The answer is simple: according to Raushenbaha (and I see no reason not agree with him) model of the natural mapping the extent and pattern of natural transmission does not match the depth and force the artist to depict a different perspective. Is crucial place in the whole theory Raushenbaha. If the application of these two standards would lead to two identical images, then it could be limited to the classical theory. But, alas! Reduced to zero error mapping scale, we can not get rid of the error to the depth. If we achieve the depth of natural images, the error magnitude remains nonzero.
For the photographer this means one thing: there is no focal length lens, Godea for all occasions. Select the focal length should be determined by the task facing the photographer.