Three image errors of three-dimensional space

In the previous chapters, the word "error" was constantly used. It was about errors that occur during the transfer of the geometric shape of the perceptual space visible by a person. The general meaning of the reasoning was mostly clear, but it was time to consider the issue of image errors in more detail.

A few words about the last type of errors, more precisely, just one example: the image of the spatial angle - the top of the cube or the corner of the room where two walls and ceiling meet. It turns out that it is fundamentally impossible to convey visible angles of faces converging at the apex on the plane of the picture, since their sum will be less than 360 °. As previously agreed, the mathematical reasons for this will not be explained here; they can be found in the previous book. It is enough to note here that errors of this type, since they cannot be eliminated, will not be considered anywhere else: people are used to them and therefore do not notice them.

We turn to the consideration of removable errors - length transmission errors (width, height, visible removal, etc.). First of all, it must be recalled that the error is the deviation from the apparent value, therefore, it is necessary to be able to determine it. Today, based on the developed theory, this is not difficult to do, and therefore it is possible not only to detect, but even determine the numerical value of the error. This immediately raises the question of the number of errors that need to be relatively calculated, because a picture can contain many lines of very different lengths and directions. It turns out that a fairly complete picture of image errors and their structure can be obtained by calculating three errors that can naturally be called the main ones, all the rest will be their consequences - this is due to the fact that three-dimensional space is depicted in the picture and in the end it comes down to three errors in the transmission of height, width and depth. Mathematics teaches, however, that instead of the three errors mentioned, one can take the main three others. Having left mathematics again, we will give illustrative examples of the three errors that are accepted as fundamental in this book.

We tried to define them so that they were as close as possible to the practice of artistic creation. Let the first of the considered errors be the error of depth transfer. Let us explain its essence, referring to the scheme. Here the surface of the earth is shown up to the horizon, which is underlined by a conventional image of clouds and is indicated by SS. The surface of the earth is given at the bottom of the pattern diagram - from the base of the AA pattern to the horizon. Suppose that in reality the black flag is located so that in visual perception it is visible at equal distances from the base of the picture and from the horizon. In the diagram, this corresponds to the distance AB from the base of the picture. When depicting the position of a flag in a picture, it may turn out that for one reason or another, the artist showed it closer or further (white flags). This will be a “mistake”, because in the visual perception of space it is visible at a distance corresponding to the black flag. This gives the opportunity not only to indicate an error, but, by measuring the position of the flags in the picture, indicate, for example, that the artist transferred the depth with an error of 15% to the side of increase (he depicted the flag further than necessary from the base of the painting by 15%).

We attribute the second type of basic error to the one that we will call the scale transfer error. Its essence is as follows. Suppose that a rectangular object is to be depicted, the visible size of which, when it is at the base of the picture AA, is shown as a black square B. Let’s move this object away, then its apparent size will decrease. In the diagram, it is also shown in the form of a black square, marked D.

It may happen that the artist, as before, for one reason or another, depict it larger or smaller (white squares). Obviously, then the correct ratio of the scales between the two plans shown will be violated. We agree to judge the scale by how the width of the object is conveyed. Here, the opportunity again opens up not only to find that the artist has violated the correct ratio of scales, but also to clarify how much by indicating the deviation allowed by him from the visible width of the rectangle in percent and which way it is produced (towards increase or decrease). The name of this error is a scale transfer error due to the fact that the correct ratio of the image scales located on different planes of the black squares B and D is violated here.

The third main error will be considered the transmission of similarities. Let on some plane it is necessary to convey an object that has a visible square shape (black square in the diagram). If the artist on his canvas, while maintaining the correct width, shows it not as a square, but as a rectangle elongated horizontally or vertically (white rectangles in the diagram), then a geometric error will appear in his picture: the configuration of the shown object will contradict the natural visual perception, the image not shown will be like visible. As before, here you can not only indicate the error, but also calculate its value in percent.

To illustrate the use of the introduced terminology, it is useful to return to the earlier drawings of the interior, executed in different versions of a single system of scientific perspective.

About pic. we can say that there are no errors in the transmission of depth and scale, therefore, gender is given in full accordance with the visual perception of a person. Inevitable errors concentrated on the transmission of verticals - they are noticeably increased. So there were errors in the transmission of similarities: the apparent relationship between the width and height of the room is broken.

In fig. It is shown what the attempt to reduce the transmission error of scales leads to while maintaining the correct transmission of similarities. This makes inevitable an increase in depth transfer error.

Somewhat vague words that some errors increase, and some decrease or even disappear completely, you can give a more specific character: they can now be calculated and indicate their exact value. On this path, it becomes possible to numerically compare various options for the scientific perspective system. We will evaluate the general correctness of the interior transfer by summing all three possible errors. Suppose, for example, for a certain picture, the error in transmitting depth will be 21%, scale - 37%, similarity - 0% (that is, absent). Then the total estimate of image errors will be 21 + 37 + 0 = 58%.

Continuing to discuss the assessment of previously shown images of the interior, we agree to determine their correctness according to the boundaries of the interior: according to the correctness of the image of the floor, ceiling and walls, while ignoring the issue of the correctness of the image of objects inside the interior. This will allow you to judge the quality of the image as a whole. The issue of the depiction of individual objects will be discussed later, when analyzing the transmission of their visual perception in landscape painting, where this is more appropriate.

Let us now try to find the mathematically best interior transfer system on the plane of the picture. Obviously, this will be the version of a unified system of scientific perspective, which is characterized by the lowest value of the sum of three errors. Calculations for specific types of interiors yielded an unexpected result: the sum of the errors for all spatial transfer options, examples of which were given above, turned out to be almost the same. This suggests that from the point of view of mathematics, all the above methods of depicting the interior are equivalent, which allows us to formulate a peculiar law of conservation of errors, or the law of conservation of distortion in the fine arts, according to which inevitable errors can be shifted from one element to another, but cannot be changed, in particular reduce the total amount of errors.

Until now, it was believed that the scientific system of perspective is absolute in nature, independent of the problem being solved, since the laws of mathematics, optics, and the work of the eye and brain are objective. But the discovered equivalence of various options for a unified scientific system of perspective (the same amount of errors) made the problem of choosing an appropriate option an aesthetic problem.

Aesthetics invaded a seemingly strictly mathematical field from an unexpected angle. It determines the choice of a suitable option for promising constructions. It is aesthetic considerations that help to select from the innumerable options offered by mathematics the one that is most suitable for the solvable artistic task. Unsurprisingly, in search of the best way to convey spatiality, artists may prefer different options.

The above can be illustrated by referring to two paintings with the goal of depicting completely different interiors. In fig. The picture of an unknown artist of the middle of the XIX century is given. "In the evening in the rooms." Here the artist sought to show the appearance of the room illuminated by the lamp, not highlighting any of its elements as the main one. The floor, ceiling and walls seem equally significant, and therefore the preferred image of one at the expense of the other would not make the slightest sense. In addition, the desire to convey the atmosphere of calm everyday life, a kind of silence required equally calm transmission and familiar configurations (in particular, the far wall). Therefore, it was necessary to preserve the similarity in the image. All this predetermined the choice made by the artist. He undoubtedly painted a picture, being inside the shown interior (whether he painted from nature or from memory - it does not matter). A prospective analysis of the image (here it is omitted) shows that if the artist painted a picture based on the laws of the Renaissance system, then the inherent errors of scale transfer would unacceptably reduce the far wall and the person standing near it. Therefore, the artist considered it necessary, preserving the semblance, to correct the scale at which the far part of the room was conveyed. This suggests that he actually used a non-Renaissance version of promising buildings.

Of course, the artist of the XIX century. had no idea about the work of the brain during visual reproduction of space and about possible options for promising constructions, but used conditional techniques developed by then. Artists have long noticed that a reduction in the number of transmission errors of scale in the Renaissance perspective system can be achieved if, following its formal rules, mentally alienate the point of view from the image space - to paint a picture as if from a distance. It can be shown that the image created in this way will be very close to the scientifically accurate one shown in Fig. 10 and obtained, of course, without any unnatural transferences of point of view. Thus, the existing practice now has a scientific explanation - it becomes clear why what was written “incorrectly” (from the perspective of the Renaissance perspective system), from a biased point of view, is perceived by the viewer as more accurately transmitting natural visual perception. If we return to the picture under discussion, it turns out that the artist built the image of the interior according to the formal laws of the Renaissance scientific perspective, but as if through a glass wall from a distance of 3.5 meters from it. However, the viewer thinks that the artist painted a picture while remaining in the room. This is due to the fact that the promising design underlying the image shown in Fig. 10, to which the artist actually resorts, does not at all imply any displacements of the point of view.

Another example is the painting by V. D. Polenov, written during his journey through the Holy Land, "Church of St. Helena" (1882). Here, the artist faced a difficult task: to convey the interior of a small temple, while inside. If we compare the temple with the previously shown room, then it is obvious that in this case, the impeccable transmission of verticals (arches, columns) becomes important, and the distortion of an absolutely inexpressive floor is perfectly acceptable. Therefore, an appeal to that version of the perspective system would be justified here. An analysis of the promising constructions used by V.D. Polenov shows that he did so.

Before turning to a discussion of the geometry of the picture in question, one remark should be made. A mathematical analysis of the geometry of visual perception (which is also omitted here) showed that in many cases a person sees straight lines in a surrounding objective space as curved lines. Artists noticed this a long time ago and often used it in their paintings. However, now this feature of living visual perception has been proved mathematically and it has become possible not only to explain it, but also to calculate (if this, of course, is necessary) the degree of curvature of the lines needed to accurately transmit direct lines of the objective space in the picture.

In the light of what has been said, it becomes clear that the straight lines that exist objectively in the church — the mental line connecting the tops of the three arches in which the fixtures are mounted, and the mental line connecting the cornices supporting the near and far arches — turned out to be curved in V. D. Polenov (white lines in the figure). The horizon line shown here also made it possible to note the vanishing point known from the theory of perspective - a point on the horizon at which the images of straight lines converge, similar to those shown in the figure, if they are mentally continued to infinity.

The two objective lines shown in the figure are shown by curves whose convexities are directed upwards. This is exactly how it should be if we follow a variant of promising constructions that correctly convey vertical lines. Corresponding to this is the “expanded” image of the floor. Moreover, what is shown in the picture not only qualitatively resembles the interior image design scheme, but quantitative coincidences of the theoretical design with perspective designs of V. D. Polenov can also be noted. (The corresponding calculations are not given here.)

Of course, both artists whose works were discussed here, departed from the strict formal rules of the Renaissance system of perspective intuitively, realizing that it would not allow them to show what was important to them. However, striving to realistically convey the essential and distort only the insignificant, they, without violating (as they might have thought) the laws of a scientific perspective, switched from an option that was not optimal for their task to a more suitable, just as scientifically substantiated and legal, even having the same amount mistakes that strict adherence to the rules would give.

The foregoing illustrates well the obvious truth: mathematical methods will always have only an auxiliary value in art. And yet they seem unconditionally useful, as they make it more clear what are the objective possibilities available to the artist who wants to follow his visual perception, and point to insurmountable obstacles along the way.

The interior, cited as an example in the previous chapter in six promising options containing the same amount of errors, will give rise to the following thoughts. From the point of view of mathematics, these options are completely equivalent, but how different their appearance is! And it is clear that none of them can be taken as a model of absolute correctness - such an option simply does not exist. But this diversity of equally correct (and at the same time equally erroneous) images clearly demonstrates the freedom that an artist has, even if he wants to be as close to nature as possible. The foregoing does not mean at all that the poor man must now sit down in a course of mathematics. He simply has to trust his eye and clearly understand that image errors are inevitable and that they can be shifted from one image element to another. Perhaps the above illustrations will attract the attention of architects who prefer to build images "according to all the rules." Now they have the opportunity to choose a version of the perspective system that will emphasize what they consider the main thing.