Which projection is called parallel. Projection: Projecting onto one projection plane

The images in the drawing are performed according to the rules of projection. By projection is the process of obtaining an image of an object on a plane - paper, screen, chalkboard, etc. The resulting image is called projection .

« Projection" is a Latin word. Translated into Russian it means “ throw (throw) forward».

The rules for constructing images in a drawing are based on the projection method. Projection method - mapping a geometric figure onto a plane by projecting its (figure) points.

To construct an image of an object using the projection method, it is necessary to draw imaginary rays through points on the object (for example, through its vertices) until they meet the plane. Rays that project an object onto a plane are called projecting .

The plane on which an image of an object is obtained is called projection plane .

Rice. 1. Concepts of projection.

Methods of depicting objects differ from each other, both in methods of projection and in the conditions of their construction. Some methods provide a more visual image and are easy to construct; others are less visual, but easier to construct.

To find out what the projection method is, let's look at examples.

Let's place an object in front of the light bulb. The shadow received on the wall can be mistaken for the projection of an object. Place a flat object on the paper and trace it with a pencil. You will receive an image corresponding to the projection of this object.

Let's look at the process of obtaining projections of the geometric figures that make up road signs(Fig. 2, 5, 8). To construct images of these geometric figures, the projection method was used.

In Figure 2, b, the projection of the point A there will be a point A, i.e. point of intersection of the projection beam Oa with the projection plane. Projection of a point IN there will be a point b etc. If we now connect these points on the plane with straight lines, we will get a projection of the depicted figure, for example a triangle.

Rice. 2 . Center projection

In the images, the points are in kind, i.e. points on an object, we will denote by large ( in capitals) letters of the Latin alphabet. Projections of these points on the plane are denoted by the same, but small ( lowercase) letters.

The considered example of constructing images constitutes the essence projection method.

If the projecting rays, with the help of which the image of an object is constructed, diverge from one point, projection is called central (Fig. 2). The point from which the rays emerge ( ABOUT), called projection center. The resulting image of the object is called central projection .

Rice. 3. Central projection on the plane.

The magnitude of the projection depends on the position of the object in relation to the picture plane, as well as on its distance to this plane and to the center of projection. In Fig. 3, and the object is located between center ABOUT And picture plane TO and therefore its image is enlarged. If the item is placed behind the plane TO(Fig. 3, b), the image will be reduced.

Central projections are often called perspective. Mutually parallel lines of an object, not parallel to the picture plane, are projected as a group of lines converging at one point (Fig. 4).

Rice. 4. Perspective

Projections of each group parallel lines have their own vanishing point O1 And O2. The vanishing points of the projections of all groups of parallel lines are located on one straight line, called the horizon line. The object shown in Fig. 4, is located in relation to the picture plane so that none of its faces are parallel to this plane. This central projection is called angular perspective.

The image produced by central projection is similar to a photograph in that it appears approximately as the human eye sees it. Also examples of central projection are film frames, shadows cast from an object by rays light bulb, etc. The central projection method is used in architecture, construction, as well as in academic drawing - drawing from life.

If the projecting rays are parallel to each other, then the projection is called parallel , and the resulting image is parallel projection . An example of a parallel projection is solar shadows (Fig. 5, 8).

Fig.5. Parallel projection

With parallel projection, all rays fall on the projection plane at the same angle.

If it is any angle other than a right angle, then the projection is called oblique (Fig. 6). In an oblique projection, as in the central one, the shape and size of the object are distorted. However, constructing an object in a parallel oblique projection is easier than in a central one.

Fig.6. Parallel oblique projection on planes.

In technical drawing, such projections are used to construct visual images(Fig. 7).

Rice. 7. The process of teaching with a visual image.

In the case when the projecting rays are perpendicular to the projection plane (Fig. 8), i.e. make an angle of 90° with it. projection is called rectangular . The resulting image is called rectangular projection of an object.

Fig. 8. Parallel rectangular projection.

Projection drawing has great value for the development of spatial understanding, without which it is impossible to consciously read drawings, much less carry them out (Figure 9).

Rectangular projections are also called orthogonal . Word " orthogonal"comes from Greek words" orthos" - straight and " gonia" - corner.

Fig.9. Parallel rectangular projection on a plane

The rectangular projection method is main in drawing. It is used to construct images on drawings and visual images of objects, since they are quite visual and easier to perform than central ones.

Drawings in the system of rectangular projections provide fairly complete information about the shape and size of an object, since the object is depicted from several sides.

To depict objects on a plane, “Descriptive Geometry” uses projection method. It consists in the fact that a certain ray characterizes the direction of a straight line (it itself is infinite in space).

Rice. 4

Depending on the location of straight lines in relation to the projection planes, straight lines are distinguished general And private situation.

General lines are neither parallel nor perpendicular to any of the projection planes (Fig. 4). Direct quotients are divided into straight level And projecting. The first are parallel to one of the projection planes, and the second are perpendicular to one of them, which explains their names. Horizontal- a straight line parallel to the horizontal projection plane (horizontal level straight line) . Frontal- straight line parallel to the frontal plane (frontal level straight line). Finally, profile straight parallel to the third projection plane. In the drawing, these straight lines look as shown in Fig. 5 (h 2 - frontal projection of the horizontal; h 1 - horizontal projection of the horizontal; f 2 - frontal projection of the front; f 1 - horizontal projection of the front; p 1, p 2 - corresponding projections of the profile line).

Rice. 5

Because the projecting line is perpendicular to any of the projection planes, its projection onto this plane degenerates into a point and is called main. There are frontal (Fig. 6 a), horizontal (Fig. 6 b) and profile-projecting (Fig. 6 c) straight lines.

Rice. 6

Note that the projecting lines are also level lines. Thus, the frontally projecting straight line is both horizontal and profile, since it is parallel to both the horizontal and profile planes of projections. For the same reason, a horizontally projecting line is both frontal and profile, and a profile-projecting line is both horizontal and frontal. So, the projecting straight lines are simultaneously twice level straight lines.

In space, lines can intersect, cross or be parallel. Complex drawings of such cases of the location of straight lines are presented in Fig. 7.

Rice. 7

Curved lines are most often defined by their projections (Fig. 8).

Rice. 8

Of greatest interest is the image of circles located in planes parallel to one of the projection planes. In Fig. Figure 9 shows two-projection complex drawings of circles, the planes of which are parallel to the frontal (Fig. 9 a), horizontal (Fig. 9 b) and profile (Fig. 9 c) projection planes.

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Goals:

• give students the concept of projection, types of projection;
• introduce the elements of rectangular projection;
• teach to project an object onto a projection plane;
• develop spatial understanding and spatial thinking;
• cultivate accuracy in graphic representations.

Methods: conversation, explanation, exercises.

Equipment: textbook, educational presentation "Projection", drawing tools, workbook on a printed basis for the textbook “Drawing” by A.D. Botvinnikov, author V.I. Vyshnepolsky.

Lesson type: learning new material.

Lesson structure:

1. Organizational moment: message of the topic /writing it in a notebook in drawing font/, goals, objectives of the lesson and motivation for learning activities, collecting completed work homework in printed workbooks – 3-5 minutes.
2. Repetition of what has been learned: completing the test on a printed basis (Task 2, 10 options, “Drawing Task Cards” edited by V.V. Stepakova. Enlightenment) – 5-7 minutes.
3. New material– 20 minutes.
4. Consolidation: performing an oral exercise – 10 minutes.
5. Final part: summing up, evaluating those who worked well, giving homework - 3-5 minutes.

PROGRESS OF THE LESSON

1. Organizational moment

Communicating the topic, purpose, objectives of the lesson, collecting completed homework in printed workbooks.

2. Repetition of what has been covered

Teacher: you have test cards on your tables. (Task 2, 10 options, “Cards-tasks for drawing” edited by V.V. Stepakova, ed. Education - print cards according to the number of students).
I will ask you to answer questions within 5 minutes. And pass the cards to the first desk.
The topic of today's lesson is “Projection. Projecting onto one projection plane". Write it down in your notebook in blueprint (the topic is displayed on the board, written in the presentation in blueprint). (Slide 1)

3. New material

The image of objects in the drawings is obtained by projection. (Slide 2) Projection is the process of constructing an image of an object on a plane. The resulting image is called a projection of the object. The word projection comes from the Latin projection - throwing forward. In this case, we look (take a glance) and display what we see on the plane of the sheet.
How are projections made? Consider this example. Let us take an arbitrary point A and some plane H in space (Slide 3). Let us draw a straight line through point A so that it intersects the plane H at some point a. Then point a will be the projection of point A. The plane on which the projection is obtained is called the projection plane. The straight line Aa is called the projecting ray. With its help, point A is projected onto plane H. Using this method, projections of all points of any spatial figure can be constructed.
Consequently, in order to construct a projection of a figure on a plane, it is necessary to draw imaginary projecting rays through the points of this figure until they intersect with the plane. The projections of all points of a figure form the projection of a given figure. In the future, we will denote points taken on an object by capital letters, and their projections by lowercase letters.
Now let’s write down what we call projections. (Slide 4)

• Projection is the process of constructing a projection of an object.
• Projection plane – the plane on which the projection is obtained.
• The projecting ray is a straight line with the help of which the projection of vertices, faces, and edges is constructed.

Depending on the relative placement of the projecting rays in space, there are central And parallel projection ( Slide 5). Parallel projection is divided into two types: rectangular and oblique.

Consider the central projection (Slide 6). Let's write down the definition:

• If the projecting rays come from one point, then such a projection is called central.
• The point from which the projection emerges is the center of projection.

Teacher: (Students' answers)

Example: photographs and film footage, shadows cast from an object by the rays of an electric light bulb.
Feature: the projection is larger than the original figure.

Teacher: Let's get acquainted with parallel projection (Slide 7).
Let's write down the definition:

• If the projecting rays are parallel to each other, then such projection is called parallel.

Teacher: Try to give examples of this type of projection yourself. (Students' answers)

Teacher: An example of a parallel projection can be considered the sun's shadows of objects, as well as streams of rain.
Parallel projection, as we have already said, can be rectangular and oblique (Slide 8).
Let's consider how projections on a plane are obtained with these types of projection and write down the definition:

• Oblique projection - the projecting rays are parallel and fall on the projection plane at an acute angle.
• Rectangular projection - the projecting rays are parallel and fall on the projection plane at an angle of 90 degrees.

Conclusion: In science, technology, and production, parallel projections are used, since they are quite visual.
The theoretical foundations of the rectangular projection method were developed at the end of the 18th century by the French scientist Gaspard Monge.

Projecting onto one projection plane

Let's consider the question of obtaining a rectangular projection of an object, i.e. projecting an object onto one projection plane (Slide 9).
Let's choose a vertical plane of projections and denote it with the letter V. Such a plane located in front of the audience is called frontal (from the French word frontal, which means facing the viewer). Let's place the object in front of the plane so that its edge is parallel to the frontal plane of projections, because then, with rectangular projection, the dimensions of the width and height of the object will not change, and the angles between straight lines will not be distorted. As a result, on the frontal plane of projections we received a frontal projection of the object.
Let's write down the definition:

• The plane located in front of the viewer is called frontal, and is designated by the letter V.
• The object is placed in front of the plane so that its two surfaces are parallel to this plane and projected without distortion.

Summary: Based on the resulting projection, we can judge only two dimensions of the object - height and length, and the diameter of the hole.
What is the thickness of the object? (Question to students).
Using the resulting projection, we cannot say this. In order to judge the shape of the part from such a drawing, it is sometimes supplemented with an indication of the thickness (S). (Slide 10).

4. Fixing the material

Let's look at the images on the slide. (Slide 11).
Tell me, what kind of “projection” did the water jets give in each case?

• Central
• Parallel rectangular

Teacher: We've covered all the lesson material, let's check ourselves to see how we've mastered it.
(Slide 12). On the slide you see a table in which new concepts are given. Your task is to correctly distribute the concepts and their definitions.
Let's check your answers (by clicking the mouse on the slide, the correct answers appear in the cells).

No.	New concepts	Definition
1	Projection.	Image on a plane.
2	Projection plane.	The plane on which the projection is obtained.
3	Projection beam.	A straight line with which an object is projected onto a plane.
4	Central projection.	Projection in which the projecting rays emerge from a single point.
5	Parallel projection.	Projection in which the projecting rays are parallel to each other.
6	Rectangular projection.	Projection in which the projecting rays fall on the projection plane at right angles.
7	Oblique projection.	Projection in which the projecting rays do not fall on the projection plane at right angles.
Projection beam, central projection, projection, oblique projection, plane projection, parallel projection, rectangular projection.

5. Final part(1 min.)

Teacher: We achieved our goals and objectives. (Evaluation of those who worked well) Write down your homework. (Slide 13)

6. Homework: textbook pages 32-37.

Teacher: Lesson is over, thank you, goodbye.

The projection of point A onto the projection plane π 1 is the point A 1 of intersection of the projecting line with the projection plane π 1 passing through point A (Fig. 1.1):

The projection of any geometric figure is the set of projections of all its points. The direction of the projecting straight lines and the position of the π 1 planes determine the projection apparatus.

Central projection is a projection in which all projecting rays emanate from one point S - the center of projection (Fig. 1.2).

Parallel projection is a projection in which all projecting lines are parallel to a given direction S (Fig. 1.3).

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Rice. 1.1. Projection of point A onto the projection plane π 1

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Rice. 1.2. Example of central projection

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Rice. 1.3. Parallel Projection Example

Parallel projection is special case central projection, when point S is at an infinitely large distance from the projection plane π 1.

With a given projection apparatus, each point in space corresponds to one and only one point on the projection plane.

One projection of a point does not determine the position of this point in space. Indeed, the projection A 1 can correspond to an infinite number of points A ’, A ’’, ... located on the projecting line (Fig. 1.4).

To determine the position of a point in space with any projection apparatus, it is necessary to have two of its projections, obtained with two different projection directions (or with two different projection centers).

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Rice. 1.4. An example of the location of a set of points on a projecting line

So, from Fig. 1.5 it is clear that two projections of point A (A 1 and A 2), obtained with two directions of projection S 1 and S 2, uniquely determine the position of point A itself in space - as the intersection of projecting lines 1 and 2 drawn from projections A 1 and A 2 parallel to the projection directions S 1 and S 2 .

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Rice. 1.5. Determining the position of point A in space

We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"

1. Center projection(Fig. 1.1). It is believed that projection is carried out using rectilinear rays emanating from one point in space - projection center.

Such a projection is irreversible: a point in space determines the position of its projection, while the projection of a point does not determine the position of this point in space, since the projection can simultaneously belong to many points located on the projecting ray.

2. Parallel projection. Projection is carried out using parallel rays. This implies that the projection plane can make any angle with the projecting rays. This type of projection is also irreversible.

3. Rectangular projection. This method is a special case of parallel projection, when the projecting rays are perpendicular to the projection plane. This type of projection is adopted in mechanical engineering to construct images on a drawing. However, the irreversibility of projection remains.

1.5. Properties of Orthographic Projection

1. Any point in space has a single projection on a given plane.

2. The projection of a straight line onto a plane is a straight line.

3. If a certain point belongs to a certain line, then the projection of a given point also belongs to the projection of a given line.

4. If a point in space divides a segment in in this regard, then the projection of this point divides the projection of a given segment in the same ratio.

5. Projections of parallel straight lines are parallel.

6. When transferring projection planes (or a figure) in parallel, the projection of the figure does not change.

7. The point of intersection of the projections of intersecting lines is the projection of the point of intersection of these lines.

8. If at least one of the parties right angle parallel to a given plane of projections, then it is projected onto this plane without distortion.

9. The length of a segment, in general, is greater than the length of its projection.

10.If the plane of a circle is not parallel to the plane of projections, then the projection of this circle is an ellipse.

11. A geometric figure is called projecting if one of its projections has a dimension that is one less. For example, a straight line perpendicular to the projection plane is projected onto it in the form of a point (Fig. 1.2).

1.6. Types of graphics tasks

All graphic tasks encountered when constructing and reading images can be divided into the following groups.

PZ - positional tasks, which are associated with determining the relative position of geometric shapes and their elements (points and lines) from the drawing:

PZ.1- a type of positional tasks associated with determining from a drawing the order of the relative position of projection objects: to the left, to the right, further, closer, higher, lower.

PZ.2 - tasks related to determining accessory from a drawing geometric shapes their elements: points or lines.

PZ.3 - tasks related to determining from a drawing the results of the mutual intersection of geometric shapes. These tasks are called: main positional tasks (GPZ).

MOH - metric problems, which are associated with determining the dimensional characteristics of projected objects from the drawing (length, distances, angles, areas).

All the diversity MOH is solved using two basic problems called basic metric tasks (OMZ):

OMZ.1-tasks to determine the length of a segment from a drawing.

OMZ.2-tasks to determine from a drawing the perpendicularity of straight lines to each other.

KomZ - complex tasks, containing several tasks, both positional and metric.

KonZ - constructive tasks, which are associated with the construction of a drawing of geometric figures and their elements that meet certain specified design conditions (for example, to construct a drawing of a surface, all points of which would be equidistant from a given straight line).