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The secrets of vision and the science of geometry

Секреты зрения и наука геометрия

The visual system is a binocular (stereoscopic) optical system of a biological nature that has evolved in animals and is capable of perceiving electromagnetic radiation from the visible spectrum (light), creating a sense of the position of objects in space. The visual system provides the function of sight.

Human vision (visual perception) is a process of psychophysiological image processing of objects of the surrounding world, carried out by the visual system, and allows you to get an idea of ​​the size, shape (perspective) and color of objects, their relative position and distance between them. According to various sources, from 80% to more than 90% of the information a person receives through vision.

Geometry (from ancient Greek, earth and measured) is a branch of mathematics that studies spatial structures and relations, as well as their generalizations.

We are accustomed to trusting our eyes and never ask ourselves the question: why does the same object look closer and shallow? Or why do different sized objects sometimes appear the same size? The mechanisms of vision are very complex, but some of its features can already be explained on the basis of geometric representations.

What is the angle of view

Секреты зрения и наука геометрия
The angular size of an object is the angle of view, under which the entire object is visible (in this case, the angle ABC).

Every object has linear dimensions: length, width and height. But as soon as he gets into our field of vision, he acquires another size - the angular one. Let's see what this means. When we look at an object, then through each of its points a beam, called the line of sight, can be drawn from the eye. It is clear that they will be infinitely many. Any two lines of sight form an angle of view. The angle of view, under which the object is visible in its entirety, is usually called the angular size of the object. Like any flat angle, it is measured in degrees, minutes, seconds, or radians.

The concept of angular size is used in geometric optics, geodesy, astronomy. It is also found in geometry, but here it is customary to talk about the angle of view, under which from a specified point this segment is “visible” - the height of the figure, its diameter, etc.

The angular size depends on the choice of the observation point, which is easily verified by measuring it from two points located at different distances from the object. Depending on the nature of the object, the angle of view under which it is visible is determined using special instruments, for example, a theodolite is used for measurements on the ground, a sextant is used to determine the height of celestial objects above the horizon, etc.

Секреты зрения и наука геометрия
Measuring the height of the luminaries using the staff of Jacob.

In ancient times, more primitive tools were used for the same purpose. One of them is the staff of Jacob, the forerunner of the modern sextant. It was a rod through which the transverse rail slid; divisions corresponding to certain angles were made on the rod (they were previously measured with a protractor). The observer brought one end of the staff to the eye, the other pointed in the direction of the object being measured and then moved the staff until it “touches” the horizon line with one end and the celestial object with the other. After that, it was necessary only to “take readings” - to see what division on the rod corresponds to the rake. This convenient and simple tool is easy to make yourself; it is quite suitable for approximate measurement of angles in any plane.

Секреты зрения и наука геометрия
The hand is a natural protractor.

Finally, one can literally estimate the angular size of the object with “bare hands”. A protractor will serve as a hand, unless, of course, know some angles. For example, the nail of the index finger of the arm extended in front of us is seen at an angle of approximately 1 degree, the fist is at an angle of 10 degrees, and the gap between the ends of the big toe and the little finger is at an angle of 22 degrees.

Angle size and distance

The angular size of an object is not constant and depends on the distance of the object from the eye: the farther the object, the smaller the angle of view under which it is visible.

Секреты зрения и наука геометрия
The same object can be visually of different sizes depending on the distance from the observer's eye.

To understand the cause of this phenomenon, we recall that on the retina of the eye, the image of an object is reversed and reduced. When the subject is removed, its image on the retina becomes smaller, which is why it seems to us to be decreasing. When reducing the distance, the image, on the contrary, increases and the object seems to be increasing. In the language of geometry, this means that the magnitude of the angle of view is inversely proportional to the distance to the object.

Секреты зрения и наука геометрия
The image of the object on the retina is turned upside down (reverse) and reduced.

This feature of vision helps to understand some of our actions and phenomena around us. Why, for example, to view the details of a picture hanging on a wall or a small print on a book’s page, you have to come closer to the canvas or bring the text to your eyes. The answer is simple: we need to enlarge the image on the retina, and for this we need to increase the angle of view, which we do, reducing the distance to the subject.

Another example. Imagine two parallel “runaway” lines (railroad tracks, the edges of a straight highway). They seem to "converge" at one point. The same impression is created by the rows of telegraph poles or trees along the road. Vision seems to be trying to convince us that, contrary to the laws of geometry, parallel lines intersect. But this is only an illusion, which arises due to the apparent decrease in the distance between the straight lines as they are removed.

From one angle

Often we have to face another situation. If we consider objects of the same shape, but different linear dimensions from the same angle of view, it seems that their dimensions are equal. This confirms the simple experience. Line up several dolls in height and look at them from the side of the smallest figure, and then slowly step back, without changing the direction of the look. You will see how the dolls begin to "merge", blocking one another. Finally, when you move a certain distance away, only one matryoshka will be visible - the one closest to you. If you now move the figures to the sides so that they are all fully visible, then the nested dolls will visually appear the same size.

Секреты зрения и наука геометрия
The geometry of the total solar eclipse.

A similar phenomenon can be observed in nature. For example, during a total solar eclipse, the lunar disk exactly obscures the solar. At this moment, an observer from Earth sees both celestial bodies from one angle of view. To see such a unique phenomenon would be impossible if the linear dimensions of the Sun and the Moon, as well as the distance from them to the Earth, were not in a certain mathematical relationship.

Секреты зрения и наука геометрия
From one angle of view, the apparent linear dimensions of objects appear to be the same.

From the point of view of geometry, in both cases we are dealing with the similarity of figures, more precisely, with homothety, with the center coinciding with the eye of the observer. Therefore, if two objects of similar shape are visible from one angle of view, then their linear dimensions differ by as many times as the distances to objects differ. Thus, the diameters of the Sun and the Moon (D and d) and the distances from these bodies to the Earth (L and l) are related by a simple formula:

Секреты зрения и наука геометрия

We have not revealed all the secrets of sight. Features of view, when a person looks with two eyes, an explanation of some visual illusions, the creation of visual effects in architecture and painting.

Candidates of pedagogical sciences Marina EGUPOVA and Natalya KARPUSHINA via