Images on the plane receive projection method. The projection apparatus is shown in Figure 1.

Figure 1. Projection apparatus

Projection object - point BUT. through the dot BUT passes projecting beam i with the direction to the picture plane, called projection plane. The point of intersection of the projecting beam with the projection plane is called point projection. The projection designation of a point must contain the index of the projection plane. For example, when projecting onto a plane P n the projection of the point will be denoted − A n .

Projection types

Distinguish central and parallel projection. In the first case, the source of the rays is located in the foreseeable space - the point S is proper, in the second - the source of the rays is located at infinity. Schemes of central and parallel projection are shown in Figures 2 and 3, respectively. The central projection model is a pyramid (Figure 4) or a cone; parallel projection model - a prism (Figure 5) or a cylinder.

Figure 2. Central projection scheme

By projecting onto one projection plane, an image is obtained that does not unambiguously determine the shape and size of the object. In figure 1, the projection of the point A - An does not determine the position of the point itself in space, since it is impossible to determine the distance at which the point is from the plane from one projection P. The presence of only one projection creates uncertainty in the image. In such cases, when it is impossible to reproduce the spatial image (original) of the object, they speak of the irreversibility of the drawing.

Figure 3. Diagram of parallel projection

Figure 4. Central projection model (pyramid)

Figure 5. Parallel projection model (prism)

To eliminate uncertainty, objects are projected onto two, three or more projection planes. Orthogonal projection onto two planes was proposed by the French geometer Gaspard Monge (XVIII century). The Monge method is shown in Figure 6, a, b, c (a is a visual image of a point in a dihedral angle, b is a complex drawing of a point, c is the restoration of an object, point A, in space according to its projections).

Figure 6. Point projection:
a - formation of projections of the spatial point A;
b - drawing of point A;
c - restoration of the spatial image of the point A according to the projections A1 and A2

Invariant properties of parallel projections:

• the projection of a point is a point;
• the projection of a straight line is generally straight;
• projections of mutually parallel lines in the general case are parallel lines;
• projections of intersecting lines - intersecting lines, while the points of intersection of the projections of the lines lie on the same perpendicular to the projection axis;
• if flat figure occupies a position parallel to the plane of projections, then it is projected onto this plane into a congruent figure.

There are oblique and rectangular parallel projections. If the projecting rays are directed to the projection plane at an angle other than a direct one, then the projections are called oblique. If the projecting rays are perpendicular to the projection plane, then the resulting projections are called rectangular. For rectangular projections, the term orthogonal is used from the Greek ortos - straight.

In orthogonal projection, two or three mutually perpendicular planes are introduced into space, which are assigned the following names and designations:

• horizontal plane of projections - П1
• frontal projection plane - P2
• profile projection plane - П3

The projection planes are infinite and, intersecting, divide the space into eight parts - octants, as shown in Figure 7.

Figure 7. Three mutually perpendicular projection planes P1, P2 and P3 divide the space into eight parts (octants)

In the practice of constructing images, the first octant is most often used, which will be referred to as the trihedral angle. A visual representation of a trihedral angle is shown in Figure 8.

Figure 8. Trihedral angle, first octant

When the projection planes intersect, straight lines are formed - the projection axes:

X-axis (x) - abscissa axis Y (y) - ordinate axis Z-axis (z) - applicate axis

If the axes are graduated, then we get a coordinate system in which it is easy to build an object according to the given coordinates. The system of rectangular coordinates was proposed by Descartes (XVIII century). Orthogonal projections have all the properties of parallel projections. Figure 9 shows the transformation of a trihedral angle and the formation of a complex drawing of a point BUT.

Figure 9. Transformation of a trihedral angle and the formation of a drawing of a point in three projections
a - visual image, b - development of a trihedral angle, c - drawing of a point

Figure 10 shows a complex drawing of a direct circular cone, the point is marked S is the apex of the cone. Projection axes X, Y, Z not shown, which is often used in drawing practice.

Currently, the most common display devices that synthesize images on a plane - a display screen or paper. Devices that create truly three-dimensional images are still quite rare. But more and more information about such developments appears, for example, about three-dimensional displays or even about three-dimensional printers.

When using any graphic devices, projections are usually used. A projection specifies how objects are displayed on a graphics device. We will consider only projections onto a plane.

World and screen coordinates

When displaying features on a screen or on a piece of paper using a printer, you need to know the coordinates of the features. We will consider two coordinate systems. First - world coordinates, which describe true position objects in space with a given accuracy. The other is the coordinate system of the image device in which the image of objects is displayed in a given projection.

Let world coordinates be 3D Cartesian coordinates. Where the center of coordinates should be placed, and what will be the units of measurement along each axis, is not very important for us yet. What matters is that for images, we will know some numerical values ​​of the coordinates of the displayed objects.

To obtain an image in a certain projection, it is necessary to calculate the projection coordinates. From them you can get the coordinates for the graphics device - let's call them screen coordinates. To synthesize an image on a plane, a two-dimensional coordinate system is sufficient. However, some rendering algorithms use 3D screen coordinates, such as the Z-buffer algorithm.

Main types of projections

Most common in computer graphics parallel and central projections (Fig. 2.15).

For central projection (also called promising) projection rays emanate from a single point located at a finite distance from objects and the projection plane. For a parallel projection, the projection rays are parallel.

Axonometric projection

Axonometric projection is a type of parallel projection. For it, all projection rays are located at right angles to the projection plane (Fig. 2.16).

[Let's set the positions of the projection plane using two angles - α and β , Position the camera so that the axis projection z on the projection plane, X0Y would be a vertical line (parallel to the y-axis).

Rice. 2.16. Axonometric projection

In order to find the relationship between the coordinates (x, y,z) and (X, Y, Z) for any point in three-dimensional space, consider transformations of the coordinate system ( X, y,z) into the system (X, Y, Z). We define such a transformation in two steps.

1st step. Rotate a coordinate system about an axis z to the angle α. Such a rotation of the axes is described by the matrix

2nd step. Rotate the coordinate system (x′ , y",z") about the axis X" angle β - get coordinates (X, Y, Z). Rotation matrix

Coordinate transformations are expressed by the product of matrices B*A:

Let's write down
transformation for projection coordinates in the form of formulas:

Do you think the same projection will be obtained if the coordinate transformations are described in the same two steps, but in a different sequence - first, the rotation of the coordinate system relative to the x axis by cornerβ , and then rotate the coordinate system about the axis z" angle α? And will there be vertical lines in the coordinate system (x, y, z) be drawn also by verticals in the coordinate system (X, U, Z)? In other words, does A*B - B*A?Reverse coordinate transformation axonometric projection. In order for the projection coordinates (X, Y, Z) convert to world coordinates (x, y,z), you need to do the reverse sequence of turns. First rotate through the angle -β and then - rotate through the angle - α . We write the inverse transformation in matrix form

Rotation matrices:

Multiplying matrices A -1 and B -1 , we get the inverse transformation matrix:

Let us write the inverse transformation also in the form of formulas

perspective projection

Perspective projection (Fig. 2.17) will first be considered with a vertical camera position, when a=β= 0. Such a projection can be imagined as an image on glass, through which the observer is looking, located from above at the point (x, y,z) = (0, 0, zk). Here the projection plane is parallel to the plane (hoo).

Based on the similarity of triangles, we write the following proportions:

Given also the Z coordinate:

In matrix form, coordinate transformations can be written as follows:

Rice. 2.17. perspective projection

Note that here the coefficients of the matrix depend on the coordinate z (in the denominator of fractions). This means that the coordinate transformation is non-linear (more precisely, linear-fractional), it belongs to the class projective transformations.

Now consider the general case - for arbitrary camera tilt angles (a and R) the same as for the parallel axonometric projection. Let (x", y",z 1 ) - coordinates for the coordinate system rotated relative to the initial system (x, y,z) at angles α and β .

We write the transformations of the perspective projection coordinates in the form:

The coordinate transformation sequence can be described as follows:

The transformation is generally non-linear. It cannot be described by one matrix of constant coefficients for all scene objects (although the matrix form can also be used to transform coordinates).

For such a perspective projection, the projection plane is perpendicular to the beam emanating from the center (x, y,z)= (0, 0, 0) and inclined at an angle α , β . If the camera is moved away from the center of coordinates, then the central projection changes. When the camera is at infinity, the central projection degenerates into a parallel projection.

Let us indicate the main properties of the perspective transformation. In the central

projections:

□ the ratio of lengths and areas is not preserved;

□ straight lines are shown as straight lines;

□ Parallel lines are shown converging at one point.

The latter property is widely used in descriptive geometry for hand drawing on paper. Let's illustrate this with the example of a house frame (Fig. 2.18).

There are other perspective projections that differ in the position of the projection plane and the location of the point of convergence of the projection rays. In addition, the projection can be carried out not on a plane, but, for example, on a spherical or cylindrical surface.

Consider an oblique projection, for which the projection rays are not perpendicular to the projection plane. The main idea of ​​​​such a projection is that the camera is raised to a height h while maintaining the vertical position of the design plane (Fig. 2.19).

Rice. 2.18. Parallel lines are depicted in the central projection converging at one point.

Rice. 2.19. oblique projection

You can get such a projection in the following way:

1. Perform rotation around the axis z on the corner a.

2. We replace z" on the -y", a.y" to z".

3. Shift the coordinate system up to the height of the camera h

4. In plane (x", y", 0) we build a perspective projection by the method already considered above (the vanishing point of the rays on the axis z).

Coordinate transformation can be described in this way. First, (x", y",z′ ).

And then a perspective transformation is performed

This projection has the advantage of keeping the vertical lines parallel, which is sometimes useful when depicting houses in architectural computer systems.

Examples of images in various projections. Let us give examples of images of identical objects in different projections. The objects will be cubes of the same size. The position of the camera is determined by the tilt angles α = 27°, β = 70°.

An example of an axonometric projection is shown in fig. 2.20.

Rice. 2.20.Axonometric projection

Now consider examples for perspective projection. Unlike parallel projection, the image in perspective projection is highly dependent on the position of the projection plane and the distance to the camera.

In optical systems, the concept is known focal length. The longer the focal length of the lens, the less the perception of perspective (Fig. 2.21 "and vice versa, for short-focus lenses, the perspective is greatest (Fig. 2.22). You probably already noticed this effect if you were shooting with a video camera or camera. In our examples, you can observe some correspondence between the distance from the camera to the projection plane { z k – z pl ) and focal length of the lens. This correspondence, however, is conditional; the analogy with optical systems is incomplete here.

For the examples below (Fig. 2.21, 2.22) z pl = 700. Camera tilt angles α = 27°, β = 70°.

Rice. 2.21.Perspective projection for a telephoto camera( z K = 2000)

Rice. 2.22.Perspective projection for a short throw camera( z K = 1200)

In the case of a short-focus camera (z K = 1200) Perspective perception is most noticeable for cubes that are closest to the camera. Vertical lines of objects are not projected verticals (objects fall apart").

Let's see examples of oblique projection (Fig. 2.23, 2.24). For it, the vertical lines of objects retain their vertical position on the projection. The position of the camera (point of convergence of projection rays) is described turning angle α = 27° and lift height h = 500. The projection plane is parallel to the plane (x "Oy") and located at a distance z pl = 700.

Rice.2.23. Oblique perspective projection for a telephoto camera( z K = 2000)

Rice. 2.24.Oblique perspective projection for a short throw camera( z K = 1200)

Consider another example of a central projection image, a Star Wars-style tag:

Display in window

As we have already discussed above, the mapping onto the projection plane corresponds to some coordinate transformation. This coordinate transformation is different for different types of projection, but one way or another, the transition to a new coordinate system - the projection coordinates - is carried out. The projection coordinates can be used to generate an image with an output graphics device. However, this may require additional transformations, since the coordinate system in the projection plane may not match the coordinate system of the display device. For example, objects measured in kilometers should be displayed, and in a bitmap display, the unit of measurement is a pixel. How to express kilometers in pixels?

In addition, you have probably seen that on a computer screen you can show an enlarged, reduced image of objects, as well as move them. How it's done?

Let us introduce notation. Let (Hey, woo,Ze) are the screen coordinates of the objects in the graphic display device. Note that the word "screen" should not be taken as if we are talking only about displays - all of the following can be attributed to any other devices using the Cartesian coordinate system. We denote the projection coordinates here as (X, Y, Z).

Let's call window rectangular output area with screen coordinates

XuhminWetp) - (Hetah Wetah) - Usually You Should Display To Window OR ALL

scene, or a separate part of it (Fig. 2.25).

Rice. 2.25. Scene projection display

a - scene boundaries in projection coordinates;b - in the window part of the scene, in - the whole scene with the preservation of proportions is inscribed in the window

The transformation of projection coordinates to screen coordinates can be specified as stretch/shrink and shear:

X E = CH +dx, ; Y E = KY+ dy; Ze =KZ.

This transformation preserves the proportions of objects due to the same stretch / shrink factor. (TO) for all coordinates. Note that for a planar display, you can discard the Z coordinate. Consider how you can calculate TO,dxanddy. For example, you need to fit the entire image of the scene into the window given dimensions. The entry condition can be defined as follows:

If we add (1) to (3), we get:

From inequalities (2) and (4) it follows:

The solution of system (1)-(4) for K will be: To≤ min (Kx, Ku) = K min .

If the value To X or value K Y equals infinity, then it must be discarded. If both - then the value To min you can set equal to one. Dga| in order for the image in the window to have the largest size, select To= To min Now you can find dx. From inequality (1):

From inequality (3): I

Because the dx1 < dx2, then the value dx can be chosen from the interval I dx1 ≤dx≤ dx2. Let's choose the central location in the window: I

Similarly, we find dy:

With such values dxanddy the center of the scene will be in the center of the window.

In other cases, when it is necessary to show only part of the scene with the appropriate scale in the window, you can directly set the numerical values ​​of the scale (TO) and shift coordinates (dx, dy). At when designing a graphics system interface, it is desirable to limit the choice TO,dx, dy range of valid values.

Graphics systems use a variety of ways to set the display layout and define the boundaries of the scene to display in the viewport. For example, scrolling sliders are often used to shift. Also "south point to a point on the stage with the cursor, and then this point becomes the center point of the window. Or you can outline a rectangle, highlighting the boundaries of a fragment of the scene - then this fragment will then be inscribed in the window. And so on. All these display methods are based on stretching and compression (scaling), as well as shift, and are described by an affine transformation of coordinates.

Lecture: PROJECTION DRAWING AND BASIC DRAWING TYPES

ELEMENTS OF DESCRIPTIVE GEOMETRY

DIMENSIONS PUT ON THE DRAWING DETAILS

1. PROJECTION DRAWING 2

2. METHODS OF OBTAINING GRAPHIC IMAGES 2

3.CENTRAL AND PARALLEL PROJECTION 3

4.ORTHOGONAL PROJECTIONS AND MAIN VIEWS OF THE DRAWING 6

5. PROJECTIONS OF POINT 10

6. PROJECTION DIRECT 17

7. METHODS OF DEFINITION OF THE PLANE ON THE PLATE 24

8. RELATIONSHIP OF A LINE, A POINT AND A PLANE 29

9. INTERCECTION OF A LINE WITH A PLANE AND INTERCECTION OF TWO PLANES 33

10. SECTIONS, SECTIONS AND VIEWS 40

11. DIMENSIONS TO BE PUT ON THE DRAWING OF THE PART 43

1. projection drawing

descriptive geometry studies ways of constructing images of spatial figures on a plane and solving spatial problems in a drawing.

projection drawing considers the practical issues of constructing drawings and solves problems in the ways considered in descriptive geometry, first on the drawings of geometric bodies, and then on the drawings of models and technical details.

1. Methods for obtaining graphic images

The shape of any object can be considered as a combination of individual simple geometric bodies. And to depict geometric bodies, you need to be able to depict their individual elements: vertices (points), edges (straight lines), faces (planes).

At the heart of the construction of images is the method of projection. To get an image of an object means to project it onto the plane of the drawing, i.e. project its individual elements. Since the simplest element of any figure is a point, the study of projection begins with the projection of a point.

To obtain an image of point A on the plane P (Fig. 4.1), a projecting beam Aa is drawn through point A. The point of intersection of the projecting beam with the plane P will be the image of point A on the plane P (point a), i.e. its projection onto the plane P.

This process of obtaining an image (projection) is called projection. The P plane is the projection plane. An image (projection) of an object, in this case a point, is obtained on it.

The principle of projection is easy to understand by the example of obtaining the shadow of an object on a wall or a sheet of paper. On fig. 4.1 shows the shadow of a pencil illuminated by a lamp, and in fig. 4.2 - the shadow of a pencil illuminated by sunlight. If we represent the light rays as straight lines, that is, as projecting rays, and the shadow as a projection (image) of an object on a plane, then it is easy to imagine the projection mechanism.

Depending on the relative position of the projecting rays, the projection is divided into central and parallel.

1. Center and side projection

central projection - obtaining projections using projecting rays passing through the point S, which is called the center of projection (Fig. 4.3). If we consider the lamp as a point source of illumination, then the projecting rays come out from one point, therefore, the central projection of the pencil is obtained on the plane P (Fig. 4.1).

An example of central projection is the projection of movie frames or slides onto a screen, where the frame is the object of projection, the image on the screen is the projection of the frame, and the focus of the lens is the center of projection.

The images obtained by the central projection method are similar to the images on the retina of our eye. They are clear, understandable for us, as they show us the objects of the surrounding reality as we are used to seeing them. But the distortion of the size of objects and the complexity of building images with a central projection do not allow it to be used for making drawings.

Central projections are widely used only where clarity in images is needed, for example, in architectural and construction drawings when depicting the perspectives of buildings, streets, squares, etc.

Parallel projection . If the center of projection - point S is removed to infinity, then the projecting rays will become parallel to each other. On fig. 4.4 shows the obtaining of parallel projections of points A and B on the plane P.

Depending on the direction of the projecting rays with respect to the projection plane, parallel projections are divided into oblique and rectangular.

At oblique projection the angle of inclination of the projecting rays to the projection plane is not equal to 90 o (Fig. 4.5).

With rectangular projection, the projecting rays are perpendicular to the projection plane (Fig. 4.6).

The projection methods considered above do not establish a one-to-one correspondence between an object (point A) and its image (projection). For a given direction of the projecting rays on the projection plane, only one projection of the point is always obtained, but it is impossible to judge the position of the point in space by its one projection, since on the same projecting ray Aa (Fig. 4.7) the point can occupy different positions, being above or below the given point A, and what position of the point in space corresponds to the image (projection) a, it is impossible to determine.

Rice. 4.4. Rice. 4.5. Rice. 4.6.

In order to determine its position in space from the image of a point, it is necessary to have at least two projections of this point. In this case, the relative position of the projection planes and the direction of projection must be known. Then, having two images of point A, it will be possible to imagine how the point is located in space.

The simplest and most convenient is projection onto mutually perpendicular projection planes using projecting beams perpendicular to the projection planes.

Such projection is called orthogonal projection, and the resulting images are called orthogonal projections.

Image projection occurs whenever flat image displayed on a curved surface or vice versa, and projections in particular are ubiquitous in panoramic photography. A projection is made when a cartographer displays a spherical globe of the Earth on a flat sheet of paper, for example. Because the the total field of view around us can be thought of as the surface of a sphere(for all viewing angles), photographs to be displayed on a flat monitor or printed require a similar projection of the sphere into a plane.

For small viewing angles, displaying an image on a flat sheet of paper is relatively easy, since the sector being viewed is almost flat. When displaying a spherical image on a flat surface, some distortion is unavoidable, so each projection type attempts to minimize one type of distortion at the expense of the others. As the viewing angle widens, the sectors in question become more and more curved, and consequently, the difference between the types of panoramic projections becomes more pronounced. The timing of each projection depends primarily on the subject being depicted and the application; here we will focus on a few types of projections that are most common in digital photography. Many of the projection types discussed in this chapter can be used as an output format by several panorama assembly software packages; PTAAssembler allows you to use all of the listed projections.

Types of Image Projection in Photography

If all these types of image projections seem a little daunting, try reading and understanding the difference between rectangular and cylindrical projections (highlighted) first, as they are the most widely used in digital panorama assembling.

Equidistant projection maps the latitude and longitude coordinates of a spherical globe directly to the horizontal and vertical grid coordinates, where the grid is approximately twice as wide more height. The horizontal stretch, as a consequence, increases towards the poles, so that the north and south poles are stretched to the entire upper and lower boundaries of the flat grid, respectively. Equidistant projections can show full vertical and horizontal angles up to 360 degrees.

Cylindrical the projection of the image is similar to the equidistant one, except that as you approach the north and south poles, the objects also stretch vertically so that infinite vertical stretch is achieved at the poles (so there is no horizontal line at the top and bottom of the flat grid). It is for this reason that cylindrical projections are unsuitable for images with a large vertical angle of view. Cylindrical projections are also the standard type displayed by traditional panoramic film cameras with a rotating lens. Cylindrical projections retain more accurate relative sizes of objects than rectangular projections, but this is achieved due to the curvature of the lines, parallel lines vision (which would otherwise remain straight).

Rectangular Image projection has the main advantage of mapping straight lines in 3D space to straight lines on a flat 2D grid. This type of projection corresponds to that produced by most conventional wide angle lenses, so it is probably the most understandable. Its main disadvantage is that it can greatly exaggerate the perspective as the viewing angle increases, resulting in a visible blockage of objects towards the edges of the frame. It is for this reason that rectangular projections are generally not recommended for viewing angles that are substantially greater than 120 degrees.

Fish eye - is an image projection, the purpose of which is to create a flat grid, where the distance from the center of the grid is approximately proportional to the actual angle of view; it forms an image that looks like a reflection from a metal sphere. Typically, this projection is not used as the output format of a panoramic photograph, but instead it can represent the original images if a fisheye lens was used to capture them. This projection is further limited to a vertical and horizontal viewing angle of 180 degrees or less, producing an image that fits into a circle. It is characterized by the increasing curvature of lines (which would otherwise be straight) as they move away from the center of the image. A camera with a fisheye lens is extremely useful for creating panoramas that cover the entire field of view, since it will be enough to collect a small number of shots.

Projection Mercator most closely correlates with cylindrical and equidistant projections; it is a compromise between the two types, providing less vertical stretch and a wider usable angle of view than cylindrical projection, but with more line curvature. This projection is probably the most recognizable as it is used in flat maps of the world. Note also that an alternative form of this projection (transverse Mercator) can be used for high altitude vertical panoramas.

sinusoidal the projection of the image tries to keep equal areas in all parts of the grid. If you unfold the globe into a plane, you can imagine that such a projection can be rolled back to form a sphere that will be identical to the original in shape and surface area. Characteristic equal area useful because if you record a flat projection of a spherical image, it will keep the same horizontal and vertical resolution throughout the image. This projection is similar to fisheye and stereographic, minus that it preserves absolutely horizontal lines from the original sphere.

stereographic projection is very similar to a fisheye, but it retains a better sense of perspective by increasing the stretch of objects as they move away from the perspective point. This perspective-enhancing characteristic is somewhat similar to a rectangular projection, although it is less pronounced here.

Examples: wide horizontal field of view

How do all these image projections actually affect panoramic photo? The following series of images is used to visually demonstrate the differences between the two types of projection that are most often found in panorama builders: rectangular and cylindrical. The shots have been chosen to show only the distortion differences for a wide horizontal angle of view; vertical panoramas are selected below to illustrate the difference in vertical distortion between other types of projections.

The first example demonstrates how a rectangular projection could display a photo panorama of the three shots shown above.

Notice the significant distortion at the edges of the angle of view, in addition to the dramatic loss of resolution due to image stretching. The following image shows what the heavily distorted image shown above would look like if it were cropped at a horizontal angle of view of just 120 degrees.

As can be seen, such a cropped rectangular projection produces quite pleasant impression, since all straight architectural lines in the assembly remain straight. On the other hand, this is achieved at the expense of the relative size of objects within the angle of view; objects at the edges of the angle of view (left and right edges) are significantly enlarged compared to objects in the center (tower with an entrance at the bottom).

The following example shows how the assembly result would look like using a cylindrical projection. Its advantage lies in the relatively even distribution of resolution, and in addition, it requires minimal framing. In addition, the difference between cylindrical and equirectangular projections is negligible for photographs that do not have an exceptionally large vertical angle of view (as in the following example).

Examples: high vertical field of view

The following examples illustrate the difference between projection types for a vertical panorama (large vertical field of view). It gives a chance to show the difference between the equidistant, cylindrical and Mercator projections, which in the previous example would look almost the same (for a wide horizontal angle of view).

Note: The perspective point for this panorama is set at the base of the tower, and as a result, the actual vertical angle of view appears as if the FOV were 140 degrees (as if the perspective point were at half height).

	Transverse Mercator

Such a large vertical angle of view allows us to clearly see how each of the selected projections of the image differs in the degree of vertical stretching/compression. Equidistant projection compresses the vertical perspective so much that it loses the sense of great height that the direct observer has. For this reason, equidistant projection is recommended only when it is absolutely necessary (such as for panoramas with the widest field vision both vertically and horizontally).

All three projections shown are intended to maintain nearly straight vertical lines; The transverse Mercator on the right introduces some rounding in order to maintain a more realistic (subjective) perspective. This type of projection is often used for extremely large vertical viewing angles. Let's also note how well this projection preserves the original appearance of each of the original images.

The difference between the rectangular and cylindrical projections for such a narrow horizontal angle of view is barely perceptible, so the rectangular projection is omitted.

Panoramic field of view calculators

The following calculator can be used to estimate the horizontal and vertical angles of view of your camera when using lenses of different focal lengths, which can help you determine the correct projection type.

Wrote on March 4th, 2015

In this post I will talk about the principles of creating 3d drawings on asphalt and not only on it. The word asphalt means a horizontal plane on which we walk every day, it can be concrete and a wooden base, glass and even sand, yes, yes, now there is such 3d drawing on the sand. It just so happened that we began to call it "on the pavement", apparently because in childhood we said: "Drawing with chalk on asphalt", although they were often painted more on concrete, it is possible that the word concrete does not sound. Abroad in literal translation- 3d street painting in English. 3d street painting.

Many of you who are reading this article are already familiar with this kind of street art from photographs that were found on the Internet, or maybe even one of you saw 3d drawings live, and maybe even tried to create it with his own hands, and for sure the majority wondered, but how street artists seek 3d effect?
I am sure that some of you have already exclaimed: "Chu, what's the secret here!? ... This is an elementary projection of an image onto a plane!" And they will be right. I would clarify that this is a projection + perspective, although of course the concept projections cannot be separated from perspectives are interacting concepts.

So where does work start? 3d drawing? And the work begins, like with all artists, with the definition of the plot and the development of a sketch, which depends on the size of the site on which it will be performed picture. You ask how the plot depends on the size of the site?

To do this, you need to understand that the drawing on the pavement is a projection onto a plane that is at an angle to us and has its own perspective contraction, and if you decide to depict an object that is larger than human growth, suppose an adult bear attacking a person, which will be the person being photographed, then such picture we will stretch for many meters, this is provided that the height at the point of inspection, from which a person looks at the drawing, is equal to the average height of a person. Therefore, sometimes artists can use a combination of a plane under their feet and a wall, or even two walls, in which three and four planes (floor, ceiling and two walls) are involved - the corner part of the room.

In this image, you can see how the image dimensions change during projection onto a plane by the line of sight. And the sharper the angle of the line of sight to the asphalt plane, the more elongated the drawing will be.
Yes, we knew all this without you, let's move on!

After you have decided on the sketch, you need to transfer it to the plane in our case, asphalt. How to do it?
Some of you have already exclaimed, yes, with the help of a projector! Yes, I will answer, it is possible with the help of a projector, but there is one small condition,pictureyou need to complete within one daylight hours how this can happenfestival, at which the process of using the projector becomes impossible - the projected image is simply not visible in bright light. So how!?...

To do this, I will introduce you a little bit into the course of the subject. perspective and the method of constructing geometric objects in space architect's method. Why geometric? Because first we will need to build a grid in space. This method is more familiar artists and architects relevant educational institutions, although someone came across the basics in the subject of drawing.

From the point of view 3d drawing should look exactly like your sketch.

At the same time, on asphalt, the drawing of an apple will look like this (top view). You can see how the pattern is deformed on a plane, so on 3d drawing or whatever else they call it anamorphic Drawing, not to be confused with amorphous! :) you need to look only from one point.
The diagram shows the field of view in humans is approx. 120° .

The point of view for the viewer is indicated by such a sign (which I use) or by any other sign that makes it clear to the person that you need to be and shoot exactly here and in this direction. So you need to look for such a sign for a high-quality photo.

A couple of photos to understand how the picture changes in size.
On this a photo through the camera lens from the designated viewing point.

That's how picture transforming (rear view)
The drawn sewer manhole, which looks from the point of view (where the tripod stands) as a round lying pancake, the width of which is almost twice as long, actually has the shape of an oval elongated in length, which has opposite values ​​- the length is greater than the width.

An example of using two planes for 3d drawing

What does the deformation look like? drawing and from another vantage point.

First you need to set the size of the rectangular area that will capture yourdrawing on asphalt and define perspective scale, namely length and width scale. To do this, on a piece of paper you need to outline the horizon and draw a line H , parallel to the horizon, this line is the edge of the picture plane in our drawing, which we will still reach, on asphalt, this line is the edge of a rectangular grid, which will be divided into squares 50x50 cm in size.

This size is set by the artist arbitrarily, depending on the complexity of the image, according to the principle the more details, the smaller the squares - for more exact definition the position of the lines in the drawing.
We all remember that the horizon passes at the level of a person's eyes, provided that the line of sight of a person looking at this figure is at the same height, that is, roughly speaking, if these figures are of the same height. And of course, if someone is higher or lower, our horizon line changes.

Thus, knowing the height of a person (take average height 170 cm) we can set the footage on the picture plane, i.e. on the line H.
Next, we carry out center line, which is at an angle of 90° to the edge of the picture plane, in this case to the line H.

For convenience, I break the meter segments into floors and connect them with a point P on the horizon , thus obtainingvanishing point Pand the scale of the length of the segments, which we have equal to 50 cm.

Now the main thing we need to define width scale or can you say more depth scale piece 50cm long. Simply put, we need to determine how visually we will shrink the grid in perspective, laid on the asphalt. I recommend that you initially stock up on a larger paper format for the drawing.

Set the distance to the main viewing point (from which the public will take pictures3d drawing) i.e. to the edge of your drawing (or rather, to the edge of your future grid on asphalt) I set 2 meters, the artist arbitrarily sets the distance that he needs, but I don’t think that it makes sense to make it less than 1.5 meters.
On the center line of our drawing, from the edge of the picture plane, what is the line H , set aside a distance of 2 meters, as a result, getting a segment C N. This very point N for the further construction of the drawing does not play a role.

Next we need to get the distance point D1 on the horizon, from which the beam will cross the picture plane at an angle of 45 °, at the point c, this will help us determine the vertex of the square. To do this, we set the distance twice the height of the human figure, since the figure is the object from which we are measuring. Why 2 times from the picture plane? Cause in the device human eye, the angle of capture in width is greater than in height. For more or less normal, not distorted perception, we need to be at a distance from the object twice its height)

Thus we get a point Q(we won't need it on the site). From main vanishing point P set aside (you can use a compass) a segment equal to PQ on the horizon line, thus obtaining a point D1 and D2, most often it will go beyond a sheet of paper, so the segment PQ divided by 2 to get a point D½ and by four for the dot D¼. Passing a ray through points D1,C we get a straight line that intersects the plane of the picture at an angle of 45 ° in perspective.

Received point B1 segment BPis the vertex of the square, the segmentB,B1-side 50 cm long in perspective.

As I said above, the remote point D1 goes beyond a sheet of paper, for convenience, a cut D1,P divided into four parts and get a point D¼
Using remote point D¼keep in mind that in this case the rays intersect the side of the square B1,C1 at a different angle (this in prbl. 75° ) to the picture plane. And to find the point of intersection, the segment BC is divided into four equal parts like any other segment on the line of the picture plane, a straight line is drawn from the intersection point to the vanishing point P, from D¼ in FROM- intersection point and will determine the side B1,C1 how does a ray drawn from D1 in FROM.

In such a cunning way, at the intersections of rays from a distant point with rays of contractionsAP, BP, CP, DP, EPwe get a grid measuring 2 by 2 meters in perspective reduction with a size of square sections of 50x50 cm. Voila!

The height of the figure of a person in the picture and the height of the viewer at the viewing point is 170 cm, the distance to the viewing point is 2 meters.
As you can see in the photo below, placing our apple sketch on the resulting grid, 3d drawing from the point of view on the site, it should look exactly the same as on the sketch, i.e. without distortions and deformations.

Now we need to draw a grid without distortion, this is our projection sketch, with which we will work on the site and transfer the image to the asphalt.
Our grid is built on the edge of the picture plane, which is our straight line H, the grid will be parallel to the picture plane and perpendicular to the plane of the base, i.e. "asphalt". The size of the grid squares is still the same - 50 cm, in the drawing, of course, you have it in the scale you have chosen.

Next, watch your hands... Let's number the squares for convenience. Run a beam, I called it " projection beam", from the point of view N, at the point of any intersection of our drawing with the grid that lies in our perspective, I chose the edge of the apple leaf - it is on the line of our grid in perspective (the base of the square C2). Crossing our usual grid, which is parallel to us, the projection beam beats off a point, which is the edge of our apple leaf.

In such a tricky way, we find all the intersection points on our grid. The points that fall on the center line are found by the method of proportional calculation.
To achieve a more accurate result of building parts and lines 3d drawing, the grid is given by a smaller cell step.
We connect all the points with a smooth line, as it was once in kindergarten ...
3d drawing in the projection sketch is ready!
As you can see from the result, the sketch turned out to be deformed. Now it remains to transfer it to the asphalt in kind, where you have already drawn the grid, sit and wait.

By the same principle, the image is built on the walls and ceilings. This is where the fairy tale ends.
And don't forget that 3d drawing this is primarily a drawing that requires skills in drawing, color and composition, otherwise the work may not turn out spectacular.

Although 3d drawing is called a drawing, it can also be done with paint, where, logically, it would be more correct to call it 3d painting on asphalt, but it so happened that we began to call it a drawing, let me remind you that abroad it is most often called 3d street painting - 3d street painting, although sometimes you can see the term 3d drawings like we have.

Taken from Maksiov The secret to creating a 3D drawing. Part 1 and the secret to creating a 3D drawing Part 2

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