Displaying figures on a plane (definition)

2024 Author: Leah Sherlock | [email protected]. Last modified: 2023-12-17 05:25

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

But it is also important for a person who is far from these spheres to learn how to correctly perceive and display the world around him. This helps to understand much more deeply all its versatility. If you do not have a sufficient idea of how to do this correctly, then you most likely will not succeed in a project, picture or drawing of any invention. That is, this skill is important both for solving simple, everyday tasks, and for those of global, universal significance.

A bit of history

Since ancient times, people have tried to depict what they saw around them: other people, some primitive buildings of those times, an amazingly beautiful world of plants and animals, majestic mountains, and simply things, household items. That is, the world in all its diversity and grandeur.

But then they still had no idea how it could be done accurately and competently, so that the display of various three-dimensional objects on a plane was really realistic, alive. The person did not have the relevant knowledge and, moreover, there were no special skills, except perhaps for the most elementary ones.

It is said in earlier sources that the first painting in the world consisted of just one line that went along the shadow of a person cast by the sun on the wall. That is, nature itself suggested in which direction it is worth moving in search of the right solution to this issue.

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

Pattern geometry

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

But the question of how correctly the display of the figure on the plane and how much they correspond to the volume of the original objects, once became very relevant. In some way, a new science calledgeometry. More precisely, its section is descriptive geometry.

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

Conversion methods

An important feature in the visual arts is the display of figures on the image plane. After all, in fact, this is the capture of three-dimensional spatial objects in two-dimensionality. Namely: the complex must be converted into a simple one, that is, an object that has a length, width, height must be transferred to a plane.

And descriptive geometry makes such "transitions" thanks to some methods. There are about six of them in total. Here are the three main ones and the most popular around the world:

• perspective (when the depicted object is removed in space);
• orthogonal projection (projection in parallel, where the rays are perpendicular to the plane);
• oblique projection (projection in parallel, where the rays are tilted relative to the plane).

The depicted object appears quite clearly in axonometric projection (which includes orthogonal and oblique). But it is projected most clearly and truly when it is shown in perspective. And it is the above methods that largely solve the question of how to display figures on a plane.

Perspective

Perspective takes the most honorable place among other methods of image. Because the human eyelike a camera lens, it sees the surrounding space in a similar way. Things that are further away from the observer appear smaller in size, and sometimes much smaller, than when they are close.

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, what is most interesting, they should not just converge at one point, but have a single intersection point.

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

And the world famous genius - Leonardo da Vinci - first argued the concept of aerial perspective. This includes a change in the color, tone of the object, changes in its contrast characteristics (decreasing as the object moves away).

Orthographic projection

Orthogonal is called parallel projection, which is directed to a straight line, which 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 is, as it were, decomposed into three views: from the side, from the front and from above. And looking at all this at the same time, you can adda representation of what an object looks like in volume. At the same time, the dimensions of the figure remain unchanged both in the three-dimensional image and in the two-dimensional one.

Oblique projection

This projection is subdivided into several more subspecies, namely:

• isometric view;
• dimetric projection;
• trimetric projection.

Isometric has distortion coefficients in all 3 axes (length, width, height). That is, the angles between pairwise taken axes are equal to 120 degrees. In the dimetric one, the distortions along the 2 axes are equal, and the third one is different. And in the trimetric projection, all distortion coefficients (that is, along all 3 axes) are different.

Shapes of rotation

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

It follows that by rotating the plane along some axis, we get the so-called figures of rotation.

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

But the task becomes even more difficult if, when displayingspatial figures on the plane, they have an inclined axis. In this case, it is important that the contours of the bodies of revolution are equidistant from the axis of the latter.

A little about chiaroscuro

Chiaroscuro plays an important role in displaying figures on a plane. Because the volume of the depicted object is created not only due to 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, determining their sizes, the features of the correct overlay of lightness and dark spots are quite possible thanks to the above methods. And, most importantly, these are really proven methods in practice, which are used by leading experts of our time.