Mapping figures on a plane (definition)

The ability to correctly display different shapes on a plane Sheet, canvas and any other surface is a fairly significant skill. And above all, it is important for people of art: painters, sculptors, graphic artists, designers (interior spaces of buildings and architectural environments), and for people of science: mathematicians, physicists, designers, inventors.

But to a person far from these spheres, learning to correctly perceive and reflect the surrounding world is also important. This helps to understand much of its versatility much deeper. If there is not enough understanding of how to do it competently, then the project, picture or drawing of any invention is unlikely to succeed. That is, this skill is important both for solving simple, everyday tasks, and for having a global, universal significance.

A bit of history

Since ancient times people have tried to depict what they saw around themselves: other people, some primitive structures of those times, a surprisingly beautiful world of plants and animals, majestic mountains, and just things, household items. That is, peace in all its diversity and grandeur.

But then they did not yet have an idea of how this can be done accurately and correctly, so that the mapping of different volumetric objects on the plane was really realistic, alive. There was no adequate knowledge for a man, and especially lack of special skills, apart from, perhaps, the most elementary.

It is said in earlier sources that the first picture in the world consisted of only one line that went along the shadow of a man thrown by the sun on the wall. That is, nature itself suggested in which direction it is worth to move in search of the correct solution of this issue.

And this question worried the man of that time also for this reason: he did not want to simply admire the voluminous living silhouette, the original, so to speak, but tried to capture the spatial object on the plane. And he did this so that it could either decorate his home or a place sacred for him, or take a package with a drawing and carry it to any distance.

Drawing geometry

And whatever you say, but the years went by, centuries passed and somehow, as civilization developed, people gradually learned to display complex figures in two-dimensional space, that is, on a plane. Only here the accuracy of the sizes and proportions of the depicted objects began to seem very approximate.

But the question of how correctly the mapping of the figure on the plane and how much they correspond to the voluminous original objects became one day very relevant. In a way, a new science, called geometry, helped to solve this problem. More precisely, its section is descriptive geometry.

Here she is just studying shapes and planes, straight lines and points, and also their relationship to each other - both in three-dimensional and two-dimensional space.

Conversion Methods

An important feature in the visual arts is the representation of figures on the image plane. After all, in fact, this is the imprint of three-dimensional spatial objects in two-dimensionality. Namely: the complex must be transformed into a simple, that is, an object that has a length, width, height, you need to translate into a plane.

And descriptive geometry performs such "transitions", thanks to some methods. In all there are about six. Here are the three main ones and the most popular in the whole world:

• Perspective (when the displayed object is deleted in space);
• Orthogonal projection (projection in parallel, where the rays are perpendicular to the plane);
• Oblique projection (projection in parallel, where the rays are inclined relative to the plane).

The depicted object appears fairly clearly under the axonometric projection (to which the orthogonal and oblique angles are referred). But most clearly and truly he is projected when portrayed in perspective. And it is the above methods that largely solve the problem of how to make the mapping of figures on a plane.

Perspective

The perspective among other ways of image takes the most honorable place. Because the human eye, like the camera lens, sees the surrounding space in a similar way. Things that are farther from the observer, in size, look smaller, and sometimes much less than when they are near.

For example, take the image of a cube in space. If, in fact, all its edges are parallel to each other, then when you look at this object in the distance, it may seem that the edges converge (or should converge) at one point. And, most interestingly, it is not just necessary to converge at one point, but have a single point of intersection.

Thanks to the masters of the Renaissance: Albrecht Dürer, Piero Della Francesca, Andrea Mantegna, Leon Baptiste Alberti, modern painting knows what a direct linear perspective is, how to determine the height of the horizon and the points of descent.

A world-famous genius - Leonardo da Vinci - for the first time argued the concept of aerial perspective. This is the change in color, the tone of the object, changes in its contrast characteristics (decrease as the object is removed).

Orthogonal projection

Orthogonal refers to parallel design, which is directed to a straight line that is perpendicular to the plane. In the process of its application, the dimensions of the contours of the object remain unchanged. That is, the object is displayed without distortion.

The projected three-dimensional object as it decomposes into three types: side, front and top. And looking at all this at the same time, you can add a representation of how the object looks in volume. In this case, the dimensions of the figure remain unchanged both in the three-dimensional image and in the two-dimensional image.

Oblique projection

This projection is divided into several subspecies, namely:

• Isometric projection;
• Dimetric projection;
• Trimetric projection.

The isometric distortion coefficients in all 3 axes (along the length, width, height). That is, the angles between the pairwise taken axes are 120 degrees. In the case of dimetric distortion, the 2-axis distortion is equal, and the third is different. And in the trimetric projection, all the distortion coefficients (i.e., over all 3-axes) are different.

Shapes of rotation

When a rectangular triangle rotates along the axis of one of the two legs, its third side (hypotenuse) will describe a new shape, called a cone. And if you rotate a rectangle (square) on one of its sides, you get a cylinder. When the semicircle rotates, a sphere will come out.

Hence it follows that rotating the plane along some axis, we get the so-called rotation figures.

These figures have an axis of rotation. The way they look in the plane depends on their placement relative to the eye level. For example, the upper and lower sides of the cylinder, in fact, are circles. And if you look at them in a plane, they look like ellipses.

But the problem becomes even more complicated if, when displaying the spatial figures on the plane, they have an inclined axis. It is important that the contours of the bodies of rotation are equidistant from the axis of the latter.

A bit about chiaroscuro

An important role in the representation of figures on the plane is chiaroscuro. Because the volume of the depicted object is created not only through lines, but also due to the correct distribution of light and shadow on its sides. And then it looks quite voluminous in the plane of a two-dimensional surface.

Thus, the display of figures on the plane, the determination of their dimensions, the features of the correct superposition of lightness and dark spots, is quite possible due to the above methods. And, most importantly, these are actually tested in practice methods that are used by leading experts of our time.