Parallel projection. Projecting onto three projection planes Projecting onto three projection planes

Let's consider the projections of points onto two planes, for which we take two perpendicular planes(Fig. 4), which we will call horizontal frontal and planes. The line of intersection of these planes is called the projection axis. We project one point A onto the considered planes using a plane projection. To do this, it is necessary to lower the perpendiculars Aa and A from a given point onto the considered planes.

The projection onto the horizontal plane is called horizontal projection points A, and the projection A? on the frontal plane is called frontal projection.

Points to be projected are usually denoted in descriptive geometry using capital letters A, B, C. Small letters are used to indicate horizontal projections of points a, b, c... Frontal projections are indicated in small letters with a stroke at the top a?, b?, c?…

Points are also designated by Roman numerals I, II,... and for their projections - by Arabic numerals 1, 2... and 1?, 2?...

By rotating the horizontal plane by 90°, you can get a drawing in which both planes are in the same plane (Fig. 5). This picture called diagram of a point.

Through perpendicular lines Ahh And Huh? Let's draw a plane (Fig. 4). The resulting plane is perpendicular to the frontal and horizontal planes because it contains perpendiculars to these planes. Therefore, this plane is perpendicular to the line of intersection of the planes. The resulting straight line intersects the horizontal plane in a straight line ahh x, and the frontal plane - in a straight line a?a X. Straight aahs and a?a x are perpendicular to the axis of intersection of the planes. That is Aahaha? is a rectangle.

When combining horizontal and frontal projection planes A And A? will lie on the same perpendicular to the axis of intersection of the planes, since when the horizontal plane rotates, the perpendicularity of the segments ahh x and a?a x will not be broken.

We get that on the projection diagram A And A? some point A always lie on the same perpendicular to the axis of intersection of the planes.

Two projections a and A? of a certain point A can unambiguously determine its position in space (Fig. 4). This is confirmed by the fact that when constructing a perpendicular from projection a to the horizontal plane, it will pass through point A. In the same way, a perpendicular from projection A? to the frontal plane will pass through the point A, i.e. point A is simultaneously on two specific straight lines. Point A is their point of intersection, i.e. it is definite.

Consider a rectangle Aaa X A?(Fig. 5), for which the following statements are true:

1) Point distance A from the frontal plane is equal to the distance of its horizontal projection a from the axis of intersection of the planes, i.e.

Huh? = ahh X;

2) point distance A from the horizontal plane of projections is equal to the distance of its frontal projection A? from the axis of intersection of the planes, i.e.

Ahh = a?a X.

In other words, even without the point itself on the diagram, using only its two projections, you can find out at what distance a given point is located from each of the projection planes.

The intersection of two projection planes divides space into four parts, which are called in quarters(Fig. 6).

The axis of intersection of the planes divides the horizontal plane into two quarters - the front and rear, and the frontal plane - into the upper and lower quarters. Upper part the frontal plane and the anterior part of the horizontal plane are considered as the boundaries of the first quarter.

When receiving the diagram, the horizontal plane rotates and is aligned with the frontal plane (Fig. 7). In this case, the front part of the horizontal plane will coincide with the bottom part of the frontal plane, and the back part of the horizontal plane will coincide with the top part of the frontal plane.

Figures 8-11 show points A, B, C, D, located in different quarters of space. Point A is located in the first quarter, point B is in the second, point C is in the third and point D is in the fourth.

When the points are located in the first or fourth quarters of them horizontal projections are on the front part of the horizontal plane, and on the diagram they will lie below the axis of intersection of the planes. When a point is located in the second or third quarter, its horizontal projection will lie on the back of the horizontal plane, and on the diagram it will be located above the axis of intersection of the planes.

Frontal projections points that are located in the first or second quarters will lie on the upper part of the frontal plane, and on the diagram they will be located above the axis of intersection of the planes. When a point is located in the third or fourth quarter, its frontal projection is below the axis of intersection of the planes.

Most often, in real constructions, the figure is placed in the first quarter of space.

In some special cases, the point ( E) can lie on a horizontal plane (Fig. 12). In this case, its horizontal projection e and the point itself will coincide. The frontal projection of such a point will be located on the axis of intersection of the planes.

In the case when the point TO lies on the frontal plane (Fig. 13), its horizontal projection k lies on the axis of intersection of the planes, and the frontal k? shows the actual location of this point.

For such points, a sign that it lies on one of the projection planes is that one of its projections is on the axis of intersection of the planes.

If a point lies on the axis of intersection of the projection planes, it and both of its projections coincide.

When a point does not lie on the projection planes, it is called point of general position. In what follows, if there are no special marks, the point in question is a point in general position.

To explain how to obtain projections of a point on a model perpendicular to the projection plane (Fig. 4), it is necessary to take a piece of thick paper in the shape of an elongated rectangle. It needs to be bent between projections. The fold line will represent the axis of intersection of the planes. If after this the bent piece of paper is straightened again, we will get a diagram similar to the one shown in the figure.

By combining two projection planes with the drawing plane, it is possible not to show the fold line, i.e., not to draw the axis of intersection of the planes on the diagram.

When plotting on a diagram, you should always place projections A And A? point A on one vertical line (Fig. 14), which is perpendicular to the axis of intersection of the planes. Therefore, even if the position of the axis of intersection of the planes remains uncertain, but its direction is determined, the axis of intersection of the planes can only be located on the diagram perpendicular to the straight line huh?.

If there is no projection axis on the diagram of a point, as in the first Figure 14 a, you can imagine the position of this point in space. To do this, draw anywhere perpendicular to the straight line huh? projection axis, as in the second figure (Fig. 14) and bend the drawing along this axis. If we restore perpendiculars at points A And A? before they intersect, you can get a point A. When changing the position of the projection axis, different positions of the point relative to the projection planes are obtained, but the uncertainty of the position of the projection axis does not affect the relative position of several points or figures in space.

Let's consider the profile plane of projections. Projections onto two perpendicular planes usually determine the position of a figure and make it possible to find out its real size and shape. But there are times when two projections are not enough. Then the construction of the third projection is used.

The third projection plane is drawn so that it is perpendicular to both projection planes simultaneously (Fig. 15). The third plane is usually called profile.

In such constructions, the common straight line of the horizontal and frontal planes is called axis X , the common straight line of the horizontal and profile planes – axis at , and the common straight line of the frontal and profile planes is axis z . Dot ABOUT, which belongs to all three planes, is called the origin point.

Figure 15a shows the point A and three of its projections. Projection onto the profile plane ( A??) are called profile projection and denote A??.

To obtain a diagram of point A, which consists of three projections a, a, a, it is necessary to cut the trihedron formed by all the planes along the y-axis (Fig. 15b) and combine all these planes with the plane of the frontal projection. The horizontal plane must be rotated about the axis X, and the profile plane is about the axis z in the direction indicated by the arrow in Figure 15.

Figure 16 shows the position of the projections huh, huh? And A?? points A, obtained by combining all three planes with the drawing plane.

As a result of the cut, the y-axis appears in two different places on the diagram. On the horizontal plane (Fig. 16) it takes vertical position(perpendicular to the axis X), and on the profile plane – horizontal (perpendicular to the axis z).

There are three projections in Figure 16 huh, huh? And A?? points A have a strictly defined position on the diagram and are subject to unambiguous conditions:

A And A? should always be located on the same vertical line, perpendicular to the axis X;

A? And A?? should always be located on the same horizontal straight line, perpendicular to the axis z;

3) when carried out through a horizontal projection and a horizontal straight line, and through a profile projection A??– a vertical straight line, the constructed straight lines will necessarily intersect on the bisector of the angle between the projection axes, since the figure Oa at A 0 A n – square.

When constructing three projections of a point, you need to check whether all three conditions are met for each point.

The position of a point in space can be determined using three numbers called its coordinates. Each coordinate corresponds to the distance of a point from some projection plane.

Determined point distance A to the profile plane is the coordinate X, wherein X = huh?Huh(Fig. 15), the distance to the frontal plane is coordinate y, and y = huh?Huh, and the distance to the horizontal plane is the coordinate z, wherein z = aA.

In Figure 15, point A occupies the width of a rectangular parallelepiped, and the measurements of this parallelepiped correspond to the coordinates of this point, i.e., each of the coordinates is represented in Figure 15 four times, i.e.:

x = a?A = Oa x = a y a = a z a?;

y = а?А = Оа y = а x а = а z а?;

z = aA = Oa z = a x a? = a y a?.

In the diagram (Fig. 16), the x and z coordinates appear three times:

x = a z a?= Oa x = a y a,

z = a x a? = Oa z = a y a?.

All segments that correspond to the coordinate X(or z), are parallel to each other. Coordinate at represented twice by an axis located vertically:

y = Oa y = a x a

and twice – located horizontally:

y = Oa y = a z a?.

This difference appears due to the fact that the y-axis is present on the diagram in two different positions.

It should be taken into account that the position of each projection is determined on the diagram by only two coordinates, namely:

1) horizontal – coordinates X And at,

2) frontal – coordinates x And z,

3) profile – coordinates at And z.

Using coordinates x, y And z, you can construct projections of a point on a diagram.

If point A is given by coordinates, their recording is defined as follows: A ( X; y; z).

When constructing point projections A the following conditions must be checked:

1) horizontal and frontal projections A And A? X X;

2) frontal and profile projections A? And A? must be located at the same perpendicular to the axis z, since they have a common coordinate z;

3) horizontal projection and also removed from the axis X, like profile projection A away from the axis z, since projections ah? and eh? have a common coordinate at.

If a point lies in any of the projection planes, then one of its coordinates is equal to zero.

When a point lies on the projection axis, two of its coordinates are equal to zero.

If a point lies at the origin, all three of its coordinates are zero.

Projecting onto one projection plane. As you already know, to construct a projection of an object, projection rays are first mentally drawn through all its points. Then the points of intersection of these rays with the projection plane are marked and connected with straight or curved lines.

Let's place the object in front of the projection plane so that when projected, three of its sides are visible on the resulting image (Fig. 36). Looking at these images, it is easy to imagine a spatial image of the object.
Such projection in drawing is used to construct visual images.
Visual images can be obtained as a result of both rectangular and oblique parallel projection
However, in visual images, objects receive large distortions. For example. round parts are projected into elliptical ones, right angles into obtuse and acute ones. Some dimensions of the object also change. Therefore, such images are rarely used in practice.

Let's place the object in front of the projection plane so that in the image it is visible only from one side (Fig. 37), and construct its rectangular projection. Now the dimensions of the length and width of the object do not change, the angles between straight lines will not be distorted, round hole will be represented by a circle.
However, it does not have a third dimension - height. To make such an image suitable for use in practice, it is supplemented with an indication of the height of the object. The height can be indicated conventionally in the drawing. This is done if the depicted object does not have protrusions, depressions, etc.

In Fig. Figure 38 shows a drawing of a part called a “gasket”. The drawing contains one rectangular projection. The drawing shows that the length of the part is 30 mm and the width is 24 mm. The part has one round through hole 0 16 mm. From the entry made in the drawing, we learn that the thickness (i.e. height) of the depicted part is 4 mm (s 4). You saw examples of drawings containing one rectangular projection in Fig. 31 and 32.
In a drawing obtained by rectangular projection onto one plane, you can indicate the height of not only the object as a whole, but also each of its parts, for example, each point (vertex). In this case, there is no need to write down the word “height” or “thickness” each time. It is enough to put a number next to the projection of one or another part of the object indicating its height.
Projections on which the height of parts of objects are indicated by a number are called projections with numerical marks.
You have already encountered projections with numerical marks in geography.

Projecting onto two projection planes.

In Fig. 41 shows the process of projecting multiple objects. As you can see, they all have the same projections. Therefore, from a drawing containing one projection, it is not always possible to accurately judge the geometric shape of an object (parallelepiped, cylinder or other body). In addition, in such a drawing the object is visible only from one side; it does not reflect the height of the object. All these shortcomings can be eliminated if you build not one, but two projections of the object. For this purpose, it is necessary to take two projection planes in space (Fig. 42), located perpendicular to each other.

One of the projection planes is placed horizontally. It is called the horizontal plane of projections and is designated H (the Latin letter ash). The projection of an object onto this plane is called a horizontal projection.

The second projection plane V (reads “ve”) is placed vertically. There can be several vertical planes, so the projection plane located in front of the viewer is called frontal (from the French word “frontal”, which means “facing the viewer”). The projection of an object obtained onto this plane is called frontal. Note that the hole in the part was projected onto the front projection plane as invisible, so it is shown as dashed lines.

The projections constructed in this way turn out to be located in space in different planes(horizontal and vertical). A drawing of an object is built on one sheet, i.e. in one plane. Therefore, to obtain a drawing of an object, both planes are brought (combined) into one. This process can be easily traced if we imagine the projection planes intersecting each other along the x line, which is called the projection axis (Fig. 42, b). If we now turn the horizontal plane of the projections down by 90° so that it coincides with the vertical plane, both projections will be located in the same plane (Fig. 43).
The boundary of the projection planes may not be shown in the drawing (Fig. 43, b). Projecting rays and the line of intersection of projection planes, i.e., the projection axis, are not drawn on the drawing if this is not necessary.
In order to see that the projections shown in the drawing represent images of the same object, they are placed in strict order, one below the other.
In Fig. 43 horizontal projection is located under the frontal one. This rule for placing projections, accepted in drawing, cannot be violated. An example of a drawing containing two rectangular projections - frontal and horizontal. The method of rectangular projection onto two mutually perpendicular planes was developed by the French geometrician Gaspard Monge at the end of the 18th century. Therefore, this method is sometimes called the Monge method.
G. Monge laid the foundation for the development of a new science of depicting objects - descriptive geometry.

Projection onto three projection planes.

Using two projections of an object, it is also not always possible to accurately represent the spatial image of the object. Images in Fig. 45, but can be projections of the objects shown in Fig. 45, b, fig. 45, in, etc. In addition, in practice it is often necessary to construct drawings that are very complex subjects, where two projections are not enough to identify the geometric shape and size of the depicted object.

To obtain such a drawing, from which it is possible to establish a single image of the depicted object, sometimes it is necessary to use not two, but three projection planes (Fig. 46).
The third plane of projections W (read “double ve”) is called profile, and the projection obtained on it is called the profile projection of the object (from the French word “profile”, which means “side view”).
The profile plane of projections is vertical. To construct a drawing of an object, it is positioned so that it is simultaneously perpendicular to the horizontal and frontal planes of projections. At the intersection with the H plane, it forms the y-axis, and with the V plane, the z-axis.
To obtain a drawing, the W plane is rotated 90° to the right, and the H plane is rotated down. The drawing obtained in this way (Fig. 46) contains three rectangular projections of the object. (Projection axes and projecting rays are not shown in the drawing.) In the drawing, the profile projection is always placed at the same height as the frontal one, to the right of it. We will call such a drawing a drawing in a system of rectangular projections.

In some cases, projection onto three projection planes is required if, for example, the geometric object has a complex structure.

Let us introduce a third projection plane into the system of two projection planes – the profile plane W (Figure 1.4). A geometric object in a system of three projection planes is projected onto the H, V and W planes and three projections of one point are obtained - horizontal, frontal and profile.

If all three projection planes are continued in geometric space in all directions, then it will be divided by three planes into eight parts, called octants (Figure 1.5). Octants are characterized by different signs of coordinates along the 0X, 0Y and 0Z axes.

Figure 1.3 – Projecting a point onto two planes

The signs of the coordinates of a point in various octants are presented in the table.

Coordinate signs in octants

Figure 1.6 shows the transformation of the spatial model of the first octant along with the projections of the point into diagrams:

a) Remove the geometric object, but retain its projections along with the communication lines (see Figure 1.6b);

b) Mentally “cut” the octant along the 0Y axis and unfold the H and W planes as shown in Figure 1.6c;

c) Obtain a planar system of three projection planes with axes, connection lines and point projections (see Figure 1.6d);

d) The projection planes are removed and only the axes are retained. As a result of the transformations we get complex drawing points or Monge diagrams on three projection planes (Figure 1.6d). It should be noted that two 0Y axes were formed on the diagram: one axis refers to the H plane, the other, marked with an asterisk*, refers to the W plane.

The diagram of a point in three projections is the basis of descriptive geometry and technical drawing. Let us consider the properties of the Monge diagram, which follow from the spatial drawing of the orthogonal projection onto three projection planes and the diagram:

1) The horizontal projection of point A is determined by the X and Y coordinates, and to construct it, the Y coordinate is plotted along the vertical 0Y axis;

2) The frontal projection of point A is determined by the X and Z coordinates;

3) The profile projection of point A is determined by the Z and Y coordinates,

wherein the Y coordinate is plotted along the horizontal axis 0Y*;

4) The horizontal and frontal projections of the point are on the same connection line, perpendicular to the 0X axis;

5) The frontal and profile projections of the point are on the same connection line, perpendicular to the 0Z axis;

6) The segments on the communication lines AхA/= AzA///are equal to the same Y coordinate. The same conclusion follows from consideration of the spatial layout;

7) From the previous property follows the fundamental property of the Monge diagram - from two projections of a point you can construct a third one.

What was discussed above applied to a point located in the octant at general situation. However, a point may belong to projection planes or axes. This position of the point is called a particular position.

From Figure 1.7 it is clear that if a point belongs to any projection plane, then its two projections will be on the axes (Figure 1.7a, b). If a point belongs to any projection axis, then two of its projections will be on the axes, and the third projection will be at point 0 (Figure 1.7c).

Goals and objectives of the lesson:

educational: show students how to use the rectangular projection method when making a drawing;

The need to use three projection planes;

Create conditions for the formation of skills to project an object onto three projection planes;

developing: develop spatial concepts, spatial thinking, cognitive interest and Creative skills students;

educating: responsible attitude towards drawing, to cultivate a culture of graphic work.

Methods and techniques of teaching: explanation, conversation, problem situations, research, exercises, frontal work with the class, creative work.

Material support: computers, presentation “Rectangular projection”, tasks, exercises, exercise cards, presentation for self-test.

Lesson type: lesson to consolidate knowledge.

Vocabulary work: horizontal plane, projection, projection, profile, research, project.

During the classes

I. Organizational part.

State the topic and purpose of the lesson.

Let's carry out lesson-competition, for each task you will receive a certain number of points. Depending on the points scored, a grade for the lesson will be assigned.

II. Repetition of projection and its types.

Projection is the mental process of constructing images of objects on a plane.

Repetition is carried out using presentation.

1. Students are asked problematic situation . (Presentation 1)

Analyze geometric shape details on the front projection and find this detail among the visual images.

From this situation it is concluded that all 6 parts have the same frontal projection. This means that one projection does not always give a complete picture of the shape and design of the part.

What is the way out of this situation? (Look at the part from the other side).

2. There was a need to use another projection plane. (Horizontal projection).

3. The need for a third projection arises when two projections are not enough to determine the shape of an object.

Sizing:

• on the frontal projection – length and height;
• on a horizontal projection – lenght and width;
• on profile projection – width and height.
  

Conclusion: this means that in order to learn how to make drawings, you need to be able to project objects onto a plane.

Exercise 1

Fill in the missing words in the definition text.

1. There are _______________ and ______________ projection.

2. If ______________ rays come out from one point, projection is called ______________.

3. If ______________ rays are directed parallel, projection is called _____________.

4. If ______________ rays are directed parallel to each other and at an angle of 90 ° to the projection plane, then the projection is called ______________.
5. Natural image of an object on the projection plane is obtained only with ______________ projection.

6. The projections are located relative to each other______________________________.

7. The founder of the rectangular projection method is _______________

Task 2. Research project

Match the main types indicated by numbers with the parts indicated by letters and write the answer in your notebook.

Fig.4

Task 3

An exercise to review knowledge of geometric bodies.

By verbal description find a visual image of the part.

Description text.

The base of the part has the shape of a rectangular parallelepiped, the smaller faces of which have grooves in the shape of a regular quadrangular prism. In the center of the upper face of the parallelepiped there is a truncated cone, along the axis of which there is a through cylindrical hole.

Rice. 5

Answer: part No. 3 (1 point)

Task 4

Find the correspondence between the technical drawings of the parts and their frontal projections (the direction of projection is marked with an arrow). Based on the scattered images of the drawing, make a drawing of each part, consisting of three images. Write your answer in the table (Fig. 129).

Rice. 6

III. Practical work.

Task No. 1. Research project

Find the frontal and horizontal projections for this visual image. Write the answer in your notebook.

Assessment of work in the lesson. Self-test. (Presentation 2)

The points for grading the first part of the work are written on the board:

23-26 points “5”

19-22 points “4”

15 -18 points “3”

Task No. 2. Creative work and checking its implementation
(creative project)

Draw the frontal projection into your workbook.
Draw a horizontal projection, changing the shape of the part in order to reduce its mass.
If necessary, make changes to the front projection.
To check the completion of the task, call one or two students to the board to explain their solution to the problem.

(10 points)

IV. Summing up the lesson.

1. Assessment of work in the lesson. (Checking the practical part of the work)

V. Homework assignment.

1. Research project.

Work according to the table: determine which drawing, designated by a number, corresponds to the drawing, designated by a letter.

The process of obtaining an image on a plane is called projection. How are projections made?

Let's take an arbitrary point in space A and some kind of plane N. Let's draw through the point A straight line until it intersects with the plane N, the resulting point A there are intersections of line and plane projection points A. The plane on which the projection is obtained is called projection plane. Straight Ahh called projecting beam(Fig. 35).

Rice. 35. Projecting a beam onto a plane

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. Word projection- Latin, translated into Russian means “throw forward.”

The points taken on the object indicate in capital letters A, B, C, and their projections are lowercase a, b, c.

If the projecting rays come from one point, then projection called central. The point S from which the rays emanate is called central (Fig. 36).

Rice. 36. Central projection

Examples of central projection are photographs, film frames, and shadows cast from an object by the rays of an electric light bulb.

If the projecting rays are parallel to each other, then projection called parallel, and the resulting projection – parallel. An example of parallel projection can be considered the sun's shadows from objects.

With parallel projection, all rays fall on the projection plane at the same angle. If it's anyone sharp corner, then the projection is called oblique(Fig. 37).

Rice. 37. Parallel projection

In the case when the projecting rays are perpendicular to the projection plane, projection called rectangular. The resulting projection is called rectangular (Fig. 38).

Rice. 38. Rectangular projection

Of all the projection methods considered, the basis for constructing an image is rectangular projection method, since the resulting image is projected onto the plane without distortion.

In space, the projection plane can be located anywhere: vertical, horizontal, oblique.

To obtain a projection of an object on a plane, it is placed parallel to this plane and rays are drawn through each vertex perpendicular to this projection plane.

Let's consider constructing a projection of the object shown in Fig. 39 per plane.

Rice. 39. Projection onto the frontal plane of projections

Let's choose a vertical projection plane located in front of the viewer. This plane is called frontal(from the French word « frontal», what does it mean « facing the viewer» and denoted by the letter V(ve).

Mentally consider the object parallel to the frontal plane and draw projecting rays through all points perpendicular to plane V. Mark the points of intersection of the rays with the plane and connect them with straight lines, and the points of the circle with a curved line. We get a projection of the object on a plane, which is called frontal projection(Fig. 40).

Rice. 40. Frontal projection

Based on the resulting projection, one can judge only two dimensions - height, length and diameter of the hole.

What is the width of the object? Using the resulting projection, we cannot say this. This means that one projection does not reveal the third dimension of an object; in addition, one projection does not always determine geometric the shape of the object (Fig. 41).

Rice. 41. Ambiguity in identifying the shape of an object with one projection:

A– frontal projection; b, c– possible shape of an object

Frontal projection shown in Fig. 42, matches all details.

Rice. 42. Projections on the frontal and horizontal planes of projections

In order to determine the shape of an object, it is necessary to construct a second projection onto the plane, which is called horizontal plane and is designated by the letter N (ash). The projection of an object onto this plane is called horizontal projection.

The horizontal plane is located at an angle of 90 0 to the frontal one. The V and H planes intersect along the OX axis (O is the point of intersection of the axes), which is called the projection axis. From the horizontal projection you can determine the length and width of the part.

Images of an object are made in one plane, therefore, to obtain a drawing of an object, both planes are combined into one, rotating the horizontal plane around the OX axis downward by 90 0 so that it coincides with the frontal plane (see Fig. 42).

The boundaries of the plane are not shown in the drawing, as well as the axis of projections, if this is not necessary (Fig. 43).

Rice. 43. Location of frontal and horizontal projections in the drawing

The horizontal projection is located strictly under the frontal projection. The location between the projections is chosen arbitrarily, while providing space for applying dimensions.

2.2. Projection onto three projection planes. Kinds.
Arrangement of views in the drawing

Often, even two projections of a part do not give a complete picture of its geometric shape (Fig. 44).

Rice. 44. Examples of ambiguous identification of the shape of a part using two projections

This drawing several parts correspond, so it becomes necessary to construct a third projection onto the plane. This plane is positioned perpendicular to the projection plane V and H.

The third projection plane is called profile, and the projection obtained on it is profile projection subject.

The profile plane is designated by the letter W (double - ve). The profile plane of projections is vertical; at the intersection with the H plane it forms the OY axis, and with the V plane it forms the OZ axis. The profile projection is located to the right of the frontal projection at the same height
(Fig. 45 A, b) Planes V,H,W form triangular angle. We place the projected object in the space of a trihedral angle and draw projecting rays through all points of the object until they intersect with the projection planes. Let us connect the intersection points with straight or curved lines, the resulting figures will be projections of the object onto planes V,H,W(Fig. 45, b).

Rice. 45. Projections of an object onto three planes of projections V, H, W

The projected object is placed in the space of a trihedral angle A) projections of an object on planes V, H, W.

To obtain a drawing of an item planes V,H,W combined into one plane, turning the W plane 90 0 to the right, and H – 90 0 down (Fig. 46, b). The boundaries of the planes, projection axes and projecting rays are not shown in the drawing (Fig. 46, c, d).

Rice. 46. ​​Location of projection planes and axes on the plane:

A– trihedral angle formed by planes V, H, W; b– process of combining planes
3-sided angle with the plane of the drawing sheet; V- location of projection planes on the plane of the drawing sheet; G– location of the axes on the plane of the drawing sheet

Having examined the process of projection onto three projection planes, we can conclude that projection is carried out in the following sequence:

Object in the system of projection planes V, H, W;

The projecting rays are perpendicular to V and directed from the front, resulting in a frontal projection;

The rays are perpendicular to H and directed from above, resulting in a horizontal projection;

The rays are perpendicular to W and directed from the left, resulting in a profile projection;

We combine V, H, W into one plane.

A drawing consisting of several rectangular projections is called complex drawing or a drawing in a system of rectangular projections.

If a drawing is constructed with coordinate axes, it is called main drawing, and if without axes, it is called axleless. All projections in the drawing are in a projection connection, which is carried out through communication lines(Fig. 47).

Rice. 47. Construction of a profile projection of an object based on two data

You already know that the rules for the design and construction of drawings are established by ESKD standards. One of the standards of this system sets rules for depicting objects on the drawings, it provides definitions of the various images used in the execution of the drawings.

In technical drawings, projections on planes are called species.

View - This is an image of the visible part of an object facing the observer. The same standard states that the object is positioned relative to the frontal plane so that the image on it gives the most complete idea of ​​the shape and size of the object. Therefore, the image on the frontal plane is called main view or front view.

The image on the horizontal plane is called top view.

The image on the profile plane is called left view(Fig. 48).

Rice. 48. Location of part views on projection planes

The top view is located below the main view, and to the right of the main view and at the same height as it is the left view.

Invisible parts of an object in views are shown with dashed lines.

The number of views in the drawing should be minimal, but sufficient to understand the shape of the depicted object. Views, like projections, are located in the same projection relationship with each other.

2.3. Geometric bodies and their projections.
Projections of vertices, edges, faces on a plane.
Projections of a group of geometric bodies

The shapes of parts found in technology are a combination of different geometric bodies or their parts.

To learn how to represent the shape of an object from a drawing, you need to know how geometric bodies are depicted in drawings.

Geometric body- this is a closed part of space, limited by planes or curved surfaces.

All geometric bodies are divided into polyhedra(cube, parallelepiped, prisms, pyramids) and bodies of revolution(cylinder, ball, cone).

Geometric bodies consist of certain elements - vertices, edges, faces(Fig. 49).

Rice. 49. Elements of geometric bodies

Edges located perpendicular to the projection planes are projected onto them in point.

Edges located parallel to the projection planes are projected onto them in natural size.

Faces perpendicular to the projection planes are projected into straight segments.

Faces parallel to projection planes are projected real size.

Faces and edges inclined to projection planes are projected onto them with distortion.

When constructing a drawing, you need to clearly imagine how each vertex, edge and face of the object will be depicted on it. It should be remembered that each view is an image of the entire object, and not just one side of it. The only difference is that some faces are projected into a true figure, others into straight segments (Fig. 50).

Rice. 50. Projecting faces and edges of geometric bodies onto projection planes

The projections of geometric bodies are flat geometric figures.

Let's consider the basic geometric bodies and their projections.

Projections Cuba are three equal squares, prisms– two rectangles and a polygon; pyramids- two triangles and a polygon; truncated pyramid– two trapezoids and a polygon; cone– two triangles and a circle; truncated cone- two trapezoids and a circle; ball– three circles, a cylinder – two rectangles and a circle (Fig. 51).

A- tetrahedral prism b- triangular prism V- tetrahedral pyramid

G- 4-sided truncated pyramid d- cone

e- cone and- ball

Rice. 51. Projections of geometric bodies on projection planes

Let's consider a drawing of a group of geometric bodies (Fig. 52).

Rice. 52. Projection of a group of geometric bodies onto three projection planes

The group consists of three geometric bodies. The first geometric body on the planes V and W is depicted as a triangle, and on the plane N – all around. Such projections are only cone. The second geometric body on the H and W planes is represented by two rectangles, and on the frontal plane - circumference. Such projections have cylinder. The third geometric body on all planes is represented by rectangles, which means parallelepiped.

Thus, we can conclude that the drawing represents a group geometric bodies, consisting of cone, cylinder And parallelepiped. To determine which of the geometric bodies is closer to us, we need to consider view from above. Based on the analysis, we come to the conclusion that there are closer to us parallelepiped And cylinder.

2.4. Analysis of the geometric shape of an object.
Projections of points lying on the surface of geometric bodies and objects

You already know that the objects around us, parts of machines and mechanisms have the shape of geometric bodies or their combinations.

Let's look at Fig. 53. Various details are depicted here, some of simple shapes, others of more complex shapes.

How to determine the shape of an object from a drawing? For this purpose, a complex-shaped part mentally dismember into separate parts shaped like geometric bodies.

Rice. 53. Parts consisting of a combination of simple geometric bodies

For example in Fig. 54. An image of the part is given. It is made up of parallelepiped, two half cylinders And truncated cone. The details include cylindrical hole.

Rice. 54. Analysis of the geometric shape of the support:

A– image of the support; b- components of the support

The mental division of an object into its constituent geometric bodies is called geometric shape analysis.

Any point on the image of geometric bodies is a projection of one or another element - vertices, edges, faces, curved surfaces.

This means that the image of any geometric body is reduced to the image of its vertices, edges, faces and curved surfaces.

Let's consider the process of constructing projections of points on drawings of geometric bodies and parts.

The work is carried out in the following sequence:

Set the face of the polyhedron or part of the surface of revolution on which the projection of the point is specified, and determine the visibility of this part of the geometric body in all views (Fig. 55, A);

Through a given projection of a point, draw a projection of an auxiliary straight line, construct it and the projection of the point in the view where the projection of the geometric body is combined with the projection of its base (Fig. 55, b);

Construct a projection of the auxiliary line and find on it the desired projection of the given point (Fig. 55, V).

Rice. 55. An example of constructing a projection of a point on a given surface of geometric bodies

If you need to construct projections of points on the surface of an object represented by a drawing, then:

Analyze the geometric shape;

Set up geometric bodies on the surface of which points are specified;

Determine the projection of points one by one on each geometric body.

On the part, the points are indicated in capitals letters A, B, C, and their projections are lowercase, for example projections point A on planes N-a, V-а ′, W-а″, invisible points are included in brackets, for example, V-(a′), H-(a), W-(a″).

2.5. The procedure for reading and constructing a drawing of a part.
Construction of the third type based on two given

To get acquainted with the structure of any product, you need to read its drawing.

The drawing is read in the following sequence:

Determine what types of parts are given in the drawing;

Determine the geometric shape of the part;

Determine the overall dimensions of the part and its elements;

Let's look at an example of reading a drawing of a part (Fig. 56).

Rice. 56. Guide drawing

Questions about the drawing

1. What is the name of the part?

2. What material is it made from?

3. At what scale is the drawing made?

4. What types are shown in the drawing?

5. The combination of what geometric bodies determines the shape of the part?

6. What are the overall dimensions?

Answers on questions

1. The part is called a “guide”.

2. A part is made of steel.

3. Scale 1:1.

4. The drawing shows two views; main view and left view.

5. Having selected the parts of the part, we consider them from left to right, comparing both views.

Extreme left side in the main view it has the shape of a rectangle, and in the left view it has the shape of a circle. So it's a cylinder.

The second part from the left in the main view is a trapezoid, in the left view it is two o circles, this frustum. The third part is shown as a rectangle in the main view, and in the left view - circle, that means cylinder. The fourth part on the main view – rectangle, and in the left view – hexagon, Means this is a hexagonal prism. The leftmost part in the main view is rectangle, and in the view on the left - circle, This cylinder. Dashed lines on the main view and circle ø 20 in the left view indicates that the part has through cylindrical hole.

6. dimensions details 160x90x90.

Many technical details have a variety of technological and design elements that have their own names (Fig. 57).

Holes

Rice. 57. Title structural elements details

Hole– a through or blind element of a part, having the shape of a geometric body.

Groove- a narrow slot or recess.

Cutout– removal of part of a part with two or big amount planes.

Slice– removal of part of a part using one plane.

Rib (stiffening rib)– a thin wall designed to enhance the rigidity of the structure.

Before you start constructing images, you need to clearly imagine the geometric shape of the part.

Let's consider the sequence of constructing views in the drawing (Fig. 58).

Rice. 58. Visual representation of the support

The general shape of the object shown in Fig. 58 – parallelepiped. It has rectangular cutouts and a triangular prism cutout. Let's start depicting the detail with it general form– parallelepiped (Fig. 59).

Rice. 59. An example of the sequence of constructing views of a part:

A- image common types details; b– construction of cutouts; V– drawing dimensions

By projecting the parallelepiped onto the planes V,H,W, we obtain rectangles on all three planes (Fig. 59, A).

All constructions are done first with thin lines. Since the part is symmetrical, we will plot the axes of symmetry in the main view and top view.

Now let's show the cutouts. It makes more sense to show them first in the main view.

To do this, you need to set aside 12 mm to the left and to the right from the axis of symmetry and draw vertical lines through the resulting points. Then, at a distance of 14 mm from the upper border, we draw segments of horizontal straight lines (Fig. 59, b).

Let's construct projections of these cutouts on other views. This can be done using communication lines. After this, in the top and left views you need to show the segments that limit the projections of the views.

In conclusion, the drawing is outlined and dimensions are applied (Fig. 59, V).

In drawing, quite often there are tasks related to the construction of a third one using two given types.

Let us consider the sequence of construction of the third type based on two given ones (Fig. 60).

Rice. 60. Drawing of a block with a cutout

In Fig. 60 you see an image of a block with a cutout. Two views are given: front and top; you need to build a view on the left. To do this, you must first imagine the shape of the depicted part. Having compared the types, we determine that the block has the shape of a parallelepiped measuring 10x35x20 mm. A rectangular cutout measuring 12x12x10 mm is made in the parallelepiped.

In the front view, using communication lines, we draw two horizontal lines, one at the level of the lower base of the parallelepiped, the other at the level of the upper base. These lines limit the height of the view on the left. Draw a vertical line anywhere between the horizontal lines (Fig. 61).

Rice. 61. Sequence of constructing the third projection

It will be a projection of the back face of the block onto the profile plane of projections (Fig. 61, A). From it to the right we will set aside a segment equal to 20 mm, i.e. the width of the block, and draw another vertical line - the projection of the front edge (Fig. 61, b).

Let us now show the cutout in the part in the left view. To do this, put a 12 mm segment to the left of the right vertical line, which is the projection of the front edge of the block, and draw another vertical line (Fig. 61, V).

After this, we delete all auxiliary construction lines and outline the drawing (Fig. 61, G).