# Some tips for analyzing the geometry of paintings Automatic translate

In art history analysis, sometimes it becomes necessary to evaluate the perspective constructions used by the artist. Here it turns out to be obligatory to refer to such concepts as the horizon line, vanishing point, height of the point of view, and similar ones. What will change in connection with the expansion of the concept of “scientific perspective” and the inclusion of a multivariate perceptual system of perspective in it? In particular, should one not reconsider old, established opinions, as was shown by the example of Cezanne’s landscape painting, which was unreasonably (as it became clear from the perspective of the perceptual perspective system) attributed to the desire to use an elevated horizon?

Particularly acute are the problems of evaluating promising constructions in the transfer of interiors. It has already been repeatedly emphasized above that close areas of space are the focus of various large and difficult to eliminate errors (we are, of course, not talking about an isolated image of a separate object - the realm of axonometry). That is why the analysis of promising constructions in the image of the interior often attracted the attention of researchers. At the same time, they invariably used the renaissance version of the scientific system of perspective as a standard, which could lead (and led) to erroneous conclusions. Indeed, in an effort to bring the image closer to the visual perception of a person, to make it more natural, the artist sometimes intuitively approached one or another version of the perceptual perspective system, and if you evaluate this artist’s work from the perspective of the Renaissance perspective system, nothing but confusion will arise.

Consideration of the problems arising in the analysis of promising constructions, we begin with the task of finding the horizon. In fig. The scheme of the image of the interior is shown. Two rectangles, shown in bold lines, mean "entrance" into the interior and its distant wall. Suppose that it seemed important to the artist to correctly depict the floor and ceiling. If you mentally make the room infinitely deep, that is, remove the far wall of the room to infinity until it turns into point A (this will be the vanishing point of the left and right borders of the floor), then the floor of such an infinitely deep room would turn out as it is shown by a dashed line, - limited by curved lines (this is very important!) Directed outward by the bulge (VA curves). Obviously, the horizontal line going through the vanishing point A will be the horizon line. The ceiling would naturally have a similar configuration. As for the walls, they, in accordance with the already mentioned theorem on the inevitability of errors, would have been conveyed incorrectly. An external sign of such an error is that the convexity of the curved wall contours is directed inward (as was agreed, statements of this kind in this book are given without evidence, which, of course, exists). If you look at the vanishing points of the ceiling and walls, then they all coincide with point A.

Let’s return to the usual image of the interior. Both VA curves reaching the vanishing point, of course, are not shown completely. It is quite natural to limit ourselves to those segments of these curves that reach the far wall, that is, segments of the sun. Due to the relative smallness of such segments, the VS curves are often practically indistinguishable from straight lines, and therefore artists usually depict them in this way. Of course, all that has been said applies both to the ceiling and to the walls. As a result, the art historian deals with the image schematically shown in Fig. 50 to the right (solid lines). On this diagram, both the true vanishing point A and the true horizon line (which the art critic does not know about) from the left diagram are placed.

Based on the theory of the Renaissance perspective system (which has nothing to do with the discussed image), the researcher will look for the vanishing point and horizon line, matching his search with the Renaissance rules. Continuing the boundaries of the AF floor with straight lines (rather than curves, as necessary in the example under consideration), he will find the erroneous vanishing point D lying above true A and the erroneous horizon going through it. If constructions of this kind are continued, then the vanishing point of the ceiling boundaries will lie at the lower point D, through which its own horizon line will pass.

Faced with the fact that two vanishing points D and two horizons appeared, the art critic, who does not doubt the absolute truth of the Renaissance doctrine of perspective, will argue that the artist painted the floor with an increased, and the ceiling with a lowered point of view, and think about the reasons that made artist to do it. If an art critic is unfamiliar with the perceptual system of perspective, it would never occur to him that the interior was actually written from one point of view, has one horizon and no special principles (except the desire to accurately convey the visual perception of the main thing) the artist was not guided by. Is there an easy way to find this true horizon? It turns out that it exists, and it is very simple: the vanishing points of the side walls (point E), although they do not give the true vanishing point A, nevertheless lie on the true horizon. This is clear from the diagram. From here follows a simple rule: the vanishing points of vertical structures (walls on the diagram), found according to the Renaissance rules, determine the true horizon (according to a non-true vanishing point). The true vanishing point can sometimes be found as the intersection of the line of the true horizon EE with the line connecting the points D. To find the true horizon, vanishing points of horizontal structures (floor and ceiling) cannot be used, although this is often done.

The formal (or, if you wish, fictitious) four vanishing points — two points D and two points E — may provide important information about the type of perceptual perspective system that the artist adhered to. Here and below, formal vanishing points will be called points obtained according to the rules of the Renaissance perspective system, regardless of the system actually used by the artist. If such symmetrical vanishing points of horizontal planes (points D) lie with "overlap" (that is, the ceiling point D lies below the floor point D), this indicates the artist’s desire to convey horizontal structures, consistent with the person’s natural visual perception. The fact that the formal vanishing points of the symmetrical vertical structures (points E) do not reach each other, suggests that the artist sacrificed the correct transfer of the geometry of the walls for the sake of an unmistakable image of the floor and ceiling.

In fig. the same diagrams are shown as in the previous figure, but created on the assumption that the main thing for the artist is the correct transfer of vertical structures, such as columns, and in our example, walls. Now the curves bounding the walls are directed outward by the convexity, which indicates the artist’s desire to convey them undistortedly, while the borders of the floor and ceiling are shown by the convexity inward, which, as already mentioned in the discussion of the previous figure, indicates the artist’s willingness to sacrifice the correctness of their image. It should be recognized as remarkable that the true horizon is again determined by the points E, that is, the formal vanishing points (found by the Renaissance rules) of vertical structures (walls on the diagram). As in the previous example, the true vanishing point A is at the intersection of lines DD and EE. The mutual arrangement of the four formal vanishing points (pairs of points D and E) allows the researcher to establish which version of the perceptual perspective system the artist gravitated to. The fact that the symmetric vanishing points of the floor and ceiling D in the circuit under consideration do not reach each other, indicates his desire to correctly convey the appearance of the walls.

Having obtained a graphical analysis of the image of pairs of points D and E and having determined from them a variant of the perceptual perspective system that was intuitively used by the artist, one can recreate the correct schemes of the perspective design. These four points determine the true horizon and the true vanishing point, which allows a better understanding of the artist. In fig. the eruption of Paolo Veronese’s “Feast at Leviticus”, borrowed from the perspective course of Professor N. A. Rynin [Rynin, 1918], is given. It is shown in the drawing that three horizons are used in the picture, and the vanishing points of the objectively parallel lines lie not only at different levels, but also shift horizontally. To explain this from the perspective of the Renaissance rules, it must be assumed that the artist painted different areas of the interior from different points of view. And all these assumptions about the artist’s movements, which he most likely did not make at all, are needed only to squeeze the version of the scientific system of perspective used by him into the Procrustean bed of the Renaissance version! Built according to the Renaissance rules and an intricate perspective scheme, it nevertheless allows one to reveal the true horizon and the true vanishing point. The first is determined by the straight line SS - the straight line on which the formal vanishing points of vertical structures (associated with the image of columns) lie, and the true vanishing point is defined as the intersection of the straight line SS and the straight line DD, on which the formal vanishing points of horizontal structures (floor and foot of arches) lie.

If we now transfer the true horizon of the SS and the true vanishing point A to the eruption of the picture, then we can restore the true perspective pattern of the picture. The fact that the formal vanishing points of horizontal structures in Fig. overlapped, and the SS points in Fig. do not reach each other, says that Paolo Veronese sought to flawlessly transfer the floor, not the columns. The fact that the space of the floor attracted special attention of the artist is quite natural - after all, it is here that the feast unfolds. Veronese’s departure from the strict rules of the Renaissance perspective system (which art historians have long noticed) had one of the motivating reasons for the desire to approach natural vision.

What has been said here before does not at all mean that artists have never used several points of view when writing a picture. This was even inevitable in ancient and medieval art, when separate objects were depicted (each from its own point of view), and not space as a whole. Later, there was a desire to write everything from a single point of view, but it was not always carried out absolutely sequentially. An art historian should be able to distinguish between cases when the artist really uses different points of view from those cases when he is credited with a plurality of points of view, trying to describe in terms of the Renaissance version of the perspective system his desire to follow natural visual perception.

The above schemes indicate that the relative position of the formal pairs of vanishing points allows much to be clarified in the analysis of the perspective scheme of the picture. Moreover, if, for example, vanishing points D lie with overlap, which indicates the artist’s special attention to the transfer of sex, then by the degree of overlap one can judge how fully he followed the visual perception of the floor and, accordingly, distorted the appearance of the walls. Often, the artist reduces the accuracy of the transmission of the floor (preserving, however, its priority) and thereby increases the accuracy of the transmission of walls. In this case, the overlap of points D decreases, both points D and points E converge.

It may happen that points D and E merge into one point A. What does this mean? Mathematical analysis shows that in this case, the artist primarily sought to correctly convey similarities. After all, absolutely correct transfer of the floor can lead to such a distorted transfer of walls that the artist will be forced to mask them, for example, breaking off the image of the walls. This issue has already been discussed in connection with the painting by Y. Pimenov, "Before entering the stage." The preservation of similarities (the correct ratio between width and height) is sometimes more important than the correct transfer of the floor or walls.

As already noted, the Renaissance perspective system has the property of preserving similarities, so if the artist strictly followed its rules, then his pairs of points D and E also merge with point A. However, such a merger does not mean that the picture in question is necessarily written according to the Renaissance rules. The class of perspective systems preserving similarity is much wider, and the Renaissance system is just one of the special cases.

How to distinguish a perspective scheme, which is based on the Renaissance perspective system, from a scheme based on one of the variants of the perceptual perspective system that preserves similarity? The answer to this question is not so simple, it cannot be obtained by resorting to elementary geometric techniques, as was done above. The main difference (and disadvantage) of the renaissance version of the scientific system of perspective is often unacceptably large scale transmission errors. The background objects are too small, and the foreground objects are too large. A typical example of a fairly strict adherence to the Renaissance rules is a painting by G.V. Soroka, “Cabinet at home in Ostrovok, estate of N. P. Milyukov”. If we compare the sizes of objects lying on the desk in the foreground and the sizes of paintings on the wall in the background, then their ratio can in no way be called natural for visual perception. In such a small room, a person, although he sees deleted objects reduced, but not to such an exaggerated degree.

In the picture “In the Evening in the Rooms,” the discrepancy between the scale of the foreground and background is noticeably mitigated (in comparison with the canvas by G. V. Soroki), and this indicates that the author of the picture, considering it important to preserve the correctness of the transfer of similarities, at the same time considered it necessary to mitigate errors transmission scale. Therefore, he evaded strict adherence to the rules of the Renaissance system of perspective and intuitively chose the version of the perceptual system in which the transmission of similarities is not associated with strongly disturbed transmission of the aspect ratios of objects in the near and distant planes. Technically, this is done through the formal use of the Renaissance rules, but by artificially removing the point of view from the depicted area of space. We are talking about formal use, because the rules of any system of scientific perspective do not allow any biases in point of view. The reception in question is capable of producing the desired effect, since the artist, using a biased point of view, paints a picture so that the viewer does not feel this bias, but believes that the transmission errors of the scale are simply reduced. For an art critic, the discovery of the removal of a point of view of this type is a sure sign of a transition from a Renaissance to a perceptual system of perspective, to the option shown, for example.

Along with the problem of evaluating promising constructions used by the artist, which may be useful in trying to understand his motives, there are other problems associated with promising constructions. It is widely believed that the artist very often places a semantically important element of the image at the vanishing point that characterizes the interior shown. A classic example of this kind is Leonardo da Vinci’s Last Supper fresco. On this fresco, the vanishing point of the walls, ceiling, and carpets lies on the image of the head of Christ, which is not only the geometric, but also the semantic center of the fresco. Straight lines that convey the boundaries of these elements of the image are directed towards the head of Christ, as if directing the viewer’s eyes to the center of the composition.

With all the persuasiveness of examples of this kind, they can mislead the art historian, since they always speak only of a formal (purely geometric) vanishing point, while in reality there can be two vanishing points: formal and visually perceptible, which can to lie at different points in the picture. The fresco of Leonardo da Vinci is that special case in which both of these points coincide, and therefore the above considerations are true. In other cases, if such a coincidence is absent, this can lead to erroneous conclusions. However, before giving relevant examples, it should be clarified what exactly is meant by a visually perceived vanishing point and why it can differ from the formal one.

When chapter 8 spoke of visual illusions and, in particular, the reproduction of signs of depth that enhance the sense of spatiality, the following important question was ignored. If an artist succeeds in transmitting spatiality and the viewer, looking at the picture, sees deep space, in other words, if the artist was able to “deceive” the viewer to a certain extent, will his visual system not again subconsciously process the retinal image that arose from contemplation her?In other words, does the person who always looks at the picture see its geometry as it objectively exists in the picture, or does he see it differently (recall that the word “sees” here also means the result of the joint work of the eye and brain).

Without examining here in detail the laws of psychology of visual perception, we only recall that due to the mechanism of constant magnitude (underlying the developed perceptual perspective system), the sizes of distant objects seem to increase. To show that this mechanism continues (albeit in a weakened form) to act when looking at the image, we turn to the figure, which shows an elongated parallelepiped in a perspective view. When you look at the image above, it seems that the parallelepiped is transmitted in a weak reverse perspective, while in fact its edges are strictly parallel. So, the brain is able to transform the retinal image that arose from the picture.

If the picture depicts an interior, then transformations similar to those illustrated by the example of a parallelepiped can lead to completely different effects. The corresponding schemes are shown in Fig. 56. Scheme A is the source. The reduction of the far wall in comparison with the "entrance" to the interior indicates its remoteness. The subconscious processes taking place in the system of visual perception will force the beholder to see this distant wall somewhat larger compared to its formal geometric size. But then the narrowing of the floor, ceiling and walls depicted in the diagram will also be somewhat weakened in visual perception in comparison with the formal-geometric narrowing in the diagram. In the above discussion, all planes that limit the interior are considered equivalent,therefore, the removed wall will expand equally in all directions (dashed rectangle in the diagram). As a result, the formal vanishing point O and the visually perceptible vanishing point (for the dashed rectangle) coincide. This takes place on a fresco by Leonardo da Vinci, which was discussed above.

However, there may be cases when horizontal planes (floor, ceiling) and vertical (side walls) are not equivalent image elements, as in diagram A. In diagrams B and C, the initial geometry of diagram A is shown by dashed lines. If for some reason the viewer’s main attention is concentrated on horizontal planes, then their illusory expansion as the image depth increases will lead to scheme B, where the illusory expansion of the floor and ceiling is shown by solid lines (it occurs due to walls). Mentally continuing the illusory boundaries of the floor and ceiling to the intersection, we immediately get two (dissimilar) visually perceptible vanishing points of the ceiling and floor (not shown in the diagram), and they, of course, will not coincide with the formal vanishing point indicated by O in diagram A.Quite similar considerations can be made for the case when the main attention of the viewer is attracted to the walls. This case is reflected in diagram C. Here, both the formal vanishing point O and the visually perceptible vanishing point of the upper boundaries of the walls F.

The last example allows us to understand one feature of the central part of the fresco of Rafael Santi "School of Athens". It is immediately evident here that the feeling of the depth of space directly connected with Plato and Aristotle is created, as already mentioned, by the “walls”. If you try by eye, without applying a ruler to the image, find the vanishing point of the upper parts of the “walls” (more precisely, the cornices that form the base of the arches of two arches), then it will lie somewhere in the area of point A. As you can easily see, the vanishing point lies noticeably below. If the vanishing point should somehow attract the viewer’s attention to the semantically important element of the image, then this vanishing point can only be visually perceived, since only it is given in a visual impression, and no one senses the formal vanishing point, its position can only be found using a ruler.

The position of the visually perceived vanishing point between the heads of arguing philosophers is quite justified, because the main thing that happens here is the argument of Plato and Aristotle. The formal vanishing point falling into Plato’s lowered hand is clearly meaningless (although there are works known in which they tried to attach some special significance to it) and, moreover, is not visible to anyone. The foregoing indicates that Raphael deliberately used the visual illusion that was discussed here to enhance expressiveness. Rafael looked at his work, and did not apply a ruler to it, he believed more in his perception than in boring formal geometry, and art historians should probably follow his example.