In order to get an axonometric projection of an object (Fig. 106), it is necessary mentally: to place the object in a coordinate system; select the axonometric projection plane and place the object in front of it; choose the direction of parallel projecting rays, which should not coincide with any of the axonometric axes; direct projecting rays through all points of the object and coordinate axes until they intersect with the axonometric projection plane, thereby obtaining an image of the projected object and coordinate axes.

On the axonometric projection plane, an image is obtained - an axonometric projection of the object, as well as projections of the axes of coordinate systems, which are called axonometric axes.

An axonometric projection is an image obtained on an axonometric plane as a result of a parallel projection of an object along with a coordinate system, which clearly displays its shape.

The coordinate system consists of three mutually intersecting planes that have a fixed point - the origin of coordinates (point O) and three axes (X, Y, Z) emanating from it and located at right angles to each other. The coordinate system allows you to make measurements along the axes, determining the position of objects in space.

Rice. 106. Obtaining an axonometric (rectangular isometric) projection

You can get many axonometric projections, differently placing the object in front of the plane and choosing at the same time a different direction of the projecting rays (Fig. 107).

The most commonly used is the so-called rectangular isometric projection (hereinafter we will use its abbreviated name - isometric projection). An isometric projection (see Fig. 107, a) is such a projection, in which the coefficients of distortion along all three axes are equal, and the angles between the axonometric axes are 120 °. Isometric projection is obtained using parallel projection.

Rice. 107. Axonometric projections established by GOST 2.317-69:
a - rectangular isometric projection; b - rectangular dimetric projection;
c - oblique frontal isometric projection;
d - oblique frontal dimetric projection

Rice. 107. Continuation: e - oblique horizontal isometric projection

In this case, the projecting rays are perpendicular to the axonometric projection plane, and the coordinate axes are equally inclined to the axonometric projection plane (see Fig. 106). If we compare the linear dimensions of the object and the corresponding dimensions of the axonometric image, we can see that in the image these dimensions are smaller than the actual ones. The values ​​showing the ratio of the dimensions of the projections of line segments to their actual dimensions are called the distortion coefficients. The distortion coefficients (K) along the isometric projection axes are the same and equal to 0.82, however, for the convenience of construction, the so-called practical distortion coefficients are used, which are equal to one (Fig. 108).

Rice. 108. The position of the axes and distortion coefficients of the isometric projection

There are isometric, dimetric and trimetric projections. Isometric projections are those projections that have the same distortion coefficients in all three axes. Dimetric projections are called such projections, in which two distortion coefficients along the axes are the same, and the value of the third one differs from them. Trimetric projections include projections in which all distortion coefficients are different.

For 3D objects and panoramas.

Axonometric projection limitations

Isometric projection in computer games and pixel art

Drawing of a TV set in near-isometric pixel art. The pixel pattern has a 2:1 aspect ratio

Notes

1. According to GOST 2 .317-69 - Unified system for design documentation. Axonometric projections.
2. Here, the horizontal plane is the plane perpendicular to the Z-axis (which is the prototype of the Z-axis").
3. Ingrid Carlbom, Joseph Paciorek. Planar Geometric Projections and Viewing Transformations // ACM Computing Surveys (CSUR): magazine. - ACM, December 1978. - Vol. 10. - No. 4. - S. 465-502. - ISSN 0360-0300. - DOI:10.1145/356744.356750
4. Jeff Green. GameSpot Preview: Arcanum (English) . GameSpot (February 29, 2000). (unavailable link - story) Retrieved September 29, 2008.
5. Steve Butts. SimCity 4: Rush Hour Preview. IGN (September 9, 2003). Archived
6. GDC 2004: The History of Zelda (English). IGN (March 25, 2004). Archived from the original on February 19, 2012. Retrieved September 29, 2008.
7. Dave Greely, Ben Sawyer.

For a visual representation of objects (products or their constituent parts) it is recommended to use axonometric projections, choosing the most suitable one in each individual case.

The essence of the method of axonometric projection lies in the fact that a given object, together with the coordinate system to which it is referred in space, is projected onto a certain plane by a parallel beam of rays. The direction of projection onto the axonometric plane does not coincide with any of the coordinate axes and is not parallel to any of the coordinate planes.

All types of axonometric projections are characterized by two parameters: the direction of the axonometric axes and the distortion coefficients along these axes. Under the distortion coefficient is understood the ratio of the size of the image in the axonometric projection to the size of the image in the orthogonal projection.

Depending on the ratio of the distortion coefficients, axonometric projections are divided into:

Isometric, when all three distortion coefficients are the same (k x =k y =k z);

Dimetric, when the distortion coefficients are the same along two axes, and the third is not equal to them (k x = k z ≠k y);

Trimetric, when all three distortion coefficients are not equal to each other (k x ≠k y ≠k z).

Depending on the direction of the projecting rays, axonometric projections are divided into rectangular and oblique. If the projecting rays are perpendicular to the axonometric projection plane, then such a projection is called rectangular. Rectangular axonometric projections include isometric and dimetric. If the projecting rays are directed at an angle to the axonometric projection plane, then such a projection is called oblique. Oblique axonometric projections include frontal isometric, horizontal isometric and frontal dimetric projections.

AT rectangular isometry the angles between the axes are 120°. The actual coefficient of distortion along the axonometric axes is 0.82, but in practice, for the convenience of construction, the indicator is taken equal to 1. As a result, the axonometric image is enlarged by a factor of 1.

Isometric axes are shown in Figure 57.

Figure 57

The construction of isometric axes can be performed using a compass (Figure 58). To do this, first draw a horizontal line and draw the Z axis perpendicular to it. From the point of intersection of the Z axis with the horizontal line (point O), draw an auxiliary circle with an arbitrary radius that intersects the Z axis at point A. From point A with the same radius, draw a second circle to intersection with the first at points B and C. The resulting point B is connected to the point O - the direction of the X axis is obtained. In the same way, the point C is connected to the point O - the direction of the Y axis is obtained.

Figure 58

The construction of an isometric projection of the hexagon is shown in Figure 59. To do this, it is necessary to plot the radius of the circumscribed circle of the hexagon along the X axis in both directions relative to the origin. Then, along the Y axis, set aside the size of the turnkey, from the obtained points, draw lines parallel to the X axis and set aside the size of the side of the hexagon along them.

Figure 59

Construction of a circle in a rectangular isometric projection

The most difficult flat figure to draw in axonometry is a circle. As you know, a circle in isometry is projected into an ellipse, but building an ellipse is quite difficult, so GOST 2.317-69 recommends using ovals instead of ellipses. There are several ways to construct isometric ovals. Let's look at one of the most common.

The size of the major axis of the ellipse is 1.22d, the minor one is 0.7d, where d is the diameter of the circle whose isometry is being constructed. Figure 60 shows a graphical way to define the major and minor axes of an isometric ellipse. To determine the minor axis of the ellipse, points C and D are connected. From points C and D, as from centers, arcs of radii equal to CD are drawn until they intersect. Segment AB is the major axis of the ellipse.

Figure 60

Having established the direction of the major and minor axes of the oval, depending on which coordinate plane the circle belongs to, two concentric circles are drawn along the dimensions of the major and minor axes, at the intersection of which with the axes they mark the points O 1, O 2, O 3, O 4, which are the centers oval arcs (Figure 61).

To determine the junction points, lines of centers are drawn, connecting O 1, O 2, O 3, O 4. from the obtained centers O 1, O 2, O 3, O 4, arcs are drawn with radii R and R 1. the dimensions of the radii are visible in the drawing.

Figure 61

The direction of the axes of the ellipse or oval depends on the position of the projected circle. There is the following rule: the major axis of the ellipse is always perpendicular to the axonometric axis that is projected onto a given plane to a point, and the minor axis coincides with the direction of this axis (Figure 62).

Figure 62

Hatching and isometric view

The hatching lines of sections in isometric projection, according to GOST 2.317-69, must have a direction parallel either only to the large diagonals of the square, or only to the small ones.

Rectangular dimetry is an axonometric projection with equal distortion indicators along the two axes X and Z, and along the Y axis the distortion indicator is half as much.

According to GOST 2.317-69, the Z-axis is used in rectangular dimetry, located vertically, the X-axis is inclined at an angle of 7 °, and the Y-axis is at an angle of 41 ° to the horizon line. Distortion on the X and Z axes is 0.94, and on the Y axis is 0.47. Usually, the reduced coefficients k x =k z =1, k y =0.5 are used, i.e. along the X and Z axes or in directions parallel to them, the actual dimensions are set aside, and along the Y axis, the dimensions are halved.

To build dimetry axes, use the method indicated in Figure 63, which is as follows:

On a horizontal line passing through the point O, eight equal arbitrary segments are laid in both directions. From the end points of these segments, one such segment is laid down vertically on the left, and seven on the right. The resulting points are connected to the point O and receive the direction of the axonometric axes X and Y in rectangular dimetry.

Figure 63

Construction of a dimetric projection of a hexagon

Consider the construction in dimetry of a regular hexagon located in the P 1 plane (Figure 64).

Figure 64

On the X axis, we set aside a segment equal to the value b, to have it the middle was at point O, and along the Y axis - a segment a, which is halved in size. Through the obtained points 1 and 2, we draw straight lines parallel to the OX axis, on which we set aside segments equal to the side of the hexagon in full size with the middle at points 1 and 2. We connect the resulting vertices. In Figure 65a, a hexagon is shown in dimetry, located parallel to the frontal plane, and in Figure 66b, parallel to the profile plane of the projection.

Figure 65

Construction of a circle in dimetry

In rectangular dimetry, all circles are represented by ellipses,

The length of the major axis for all ellipses is the same and equals 1.06d. The value of the minor axis is different: for the frontal plane it is 0.95d, for the horizontal and profile planes - 0.35d.

In practice, the ellipse is replaced by a four-centered oval. Consider the construction of an oval that replaces the projection of a circle lying in the horizontal and profile planes (Figure 66).

Through the point O - the beginning of the axonometric axes, we draw two mutually perpendicular straight lines and plot the value of the major axis AB=1.06d on the horizontal line, and the value of the minor axis CD=0.35d on the vertical line. Up and down from O vertically we set aside segments OO 1 and OO 2, equal in value to 1.06d. Points O 1 and O 2 are the center of large arcs of the oval. To determine two more centers (O 3 and O 4), we lay off the segments AO 3 and BO 4 on a horizontal line from points A and B, equal to ¼ of the size of the minor axis of the ellipse, that is, d.

Figure 66

Then, from points O1 and O2 we draw arcs, the radius of which is equal to the distance to points C and D, and from points O3 and O4 - with a radius to points A and B (Figure 67).

Figure 67

The construction of an oval that replaces the ellipse, from a circle located in the P 2 plane, we will consider in Figure 68. We draw the axes of dimetry: X, Y, Z. The minor axis of the ellipse coincides with the direction of the Y axis, and the major one is perpendicular to it. On the X and Z axes, we set aside the radius of the circle from the beginning and get the points M, N, K, L, which are the conjugation points of the oval arcs. From points M and N we draw horizontal straight lines, which, at the intersection with the Y axis and perpendicular to it, give points O 1, O 2, O 3, O 4 - the centers of the arcs of the oval (Figure 68).

From the centers O 3 and O 4 they describe an arc with a radius R 2 \u003d O 3 M, and from the centers O 1 and O 2 - an arc with a radius R 1 \u003d O 2 N

Figure 68

Hatching a rectangular dimeter

The hatching lines of cuts and sections in axonometric projections are made parallel to one of the diagonals of the square, the sides of which are located in the corresponding planes parallel to the axonometric axes (Figure 69).

Figure 69

1. What types of axonometric projections do you know?
2. At what angle are the axes in isometry?
3. What figure does an isometric projection of a circle represent?
4. How is the major axis of the ellipse located for a circle belonging to the profile plane of projections?
5. What are the accepted distortion coefficients along the X, Y, Z axes for constructing a dimetric projection?
6. At what angles are the axes in the dimeter?
7. What figure will be the dimetric projection of a square?
8. How to build a dimetric projection of a circle located in the frontal projection space?
9. Basic rules for hatching in axonometric projections.

Image of circles in isometric projection

Consider how circles are depicted in isometric projection. To do this, let's draw a cube with circles inscribed in its faces (Fig. 3.16). Circles located respectively in planes perpendicular to the axes x, y, z are shown in isometry as three identical ellipses.

Rice. 3.16.

To simplify the work, ellipses are replaced by ovals outlined by arcs of circles, they are built like this (Fig. 3.17). A rhombus is drawn, into which an oval should fit, depicting a given circle in an isometric projection. To do this, on the axes lay off from the point O in four directions, segments equal to the radius of the depicted circle (Fig. 3.17, a). Through the received points a, b, c, d draw straight lines forming a rhombus. Its sides are equal to the diameter of the circle being drawn.

Rice. 3.17.

From the vertices of obtuse angles (points BUT and AT) describe between points a and b, as well as with and d arc radius R, equal to the length of the straight lines Wa or Вb(Fig. 3.17, b).

points With and D lying at the intersection of the diagonal of the rhombus with straight lines Wa and Bb, are the centers of small arcs conjugating large ones.

Small arcs are described by radius R, equal to the segment Sa (Db).

Construction of isometric projections of parts

Consider the construction of an isometric projection of a part, two types of which are given in Fig. 3.18, a.

Construction is carried out in the following order. First, draw the original shape of the part - a square. Then they build ovals depicting an arc (Fig. 3.18, b) and circles (Fig. 3.18, c).

Rice. 3.18.

To do this, find a point on a vertically located plane O, through which isometric axes pass X and z. By this construction, a rhombus is obtained, in which half of the oval is inscribed (Fig. 3.18, b). Ovals on parallel planes are built by transferring the centers of arcs to a segment equal to the distance between these planes. Double circles in Fig. 3.18 shows the centers of these arcs.

on the same axes X and z construct a rhombus with a side equal to the diameter of the circle d. An oval is entered into the rhombus (Fig. 3.18, c).

They find the center of the circle on a horizontally located face, draw isometric axes, build a rhombus into which an oval is entered (Fig. 3.18, G).

The concept of a dimetric rectangular projection

The location of the axes of the dimetric projection and the way they are constructed are shown in fig. 3.19. Axis z swipe vertically, axis X- at an angle of about 7 ° to the horizontal, and the axis at forms an angle of approximately 41 ° with the horizontal (Fig. 3.19, a). You can build axes using a ruler and a compass. For this, from the point O lay horizontally to the right and left of eight equal divisions (Fig. 3.19, b). From the extreme points, perpendiculars are restored. Their height is: for the perpendicular to the axis X - one division, for perpendicular to the axis at- seven divisions. extreme points perpendiculars are connected to point O.

Rice. 3.19.

When drawing a dimetric projection, as well as when constructing a frontal one, the dimensions along the axis at are reduced by 2 times, and along the axes X and z put off without cuts.

On fig. 3.20 shows the dimetric projection of a cube with circles inscribed in its faces. As can be seen from this figure, circles in the dimetric projection are depicted as ellipses.

Rice. 3.20.

technical drawing

Technical drawing - this is a visual image, made according to the rules of axonometric projections by hand, by eye. It is used in cases where you need to quickly and clearly show the shape of an object on paper. This is usually necessary in design, invention and rationalization, as well as in teaching the reading of drawings, when, with the help of a technical drawing, it is necessary to explain the shape of the part shown in the drawing.

Performing a technical drawing, they adhere to the rules for constructing axonometric projections: the axes are placed at the same angles, the dimensions along the axes are also reduced, the shape of the ellipses and the sequence of construction are observed.

The standard establishes the following views obtained on the main projection planes (Fig. 1.2): front view (main), top view, left view, right view, bottom view, rear view.

The main view is the one that gives the most complete idea of ​​the shape and size of the object.

The number of images should be the smallest, but providing a complete picture of the shape and size of the subject.

If the main views are located in a projection relationship, then their names are not indicated. For the best use of the drawing field, it is allowed to place views outside the projection connection (Fig. 2.2). In this case, the view image is accompanied by a type designation:

1) the direction of view is indicated

2) a designation is applied above the image of the view BUT, as in fig. 2.1.

Types are indicated by capital letters of the Russian alphabet in a font that is 1 ... 2 sizes larger than the font of dimensional numbers.

Figure 2.1 shows a part that needs four views. If these views are placed in a projection relationship, then they will take up a lot of space on the drawing field. You can arrange the necessary views as shown in Fig. 2.1. The drawing format is reduced, but the projection relationship is broken, so you need to designate the view on the right ().

2.2 Local views.

A local view is an image of a separate limited place on the surface of an object.

It can be limited by a cliff line (Fig. 2.3 a) or not limited (Fig. 2.3 b).

In general, local views are drawn up in the same way as the main views.

2.3. Additional types.

If any part of the object cannot be shown on the main views without distorting the shape and size, then additional views are used.

An additional view is an image of the visible part of the surface of an object, obtained on a plane that is not parallel to any of the main projection planes.

If an additional view is performed in a projection connection with the corresponding image (Fig. 2.4 a), then it is not indicated.

If the image of an additional view is placed in a free space (Fig. 2.4 b), i.e. the projection connection is broken, then the direction of view is indicated by an arrow located perpendicular to the depicted part of the part and is indicated by the letter of the Russian alphabet, and the letter remains parallel to the main inscription of the drawing, and does not turn behind the arrow.

If necessary, the image of an additional view can be rotated, then a letter and a rotation sign are placed above the image (this is a circle 5 ... 6 mm with an arrow, between the leaves of which the angle is 90 °) (Fig. 2.4 c).

An additional view is most often performed as a local one.

3. Cuts.

A cut is an image of an object mentally dissected by one or more planes. The section shows what lies in the cutting plane and what is located behind it.

In this case, the part of the object located between the observer and the cutting plane is mentally removed, as a result of which all the surfaces covered by this part become visible.

3.1. Construction of cuts.

Figure 3.1 shows three types of an object (without a cut). On the main view internal surfaces: Rectangular slot and cylindrical stepped hole shown with dashed lines.

On fig. 3.2, a section is drawn, obtained as follows.

The cutting plane, parallel to the frontal plane of projections, mentally dissected the object along its axis, passing through a rectangular groove and a cylindrical stepped hole located in the center of the object. Then the front half of the object, located between the observer and the cutting plane, was mentally removed. Since the object is symmetrical, it makes no sense to give a full section. It is performed on the right, and the view is left on the left.

The view and section are separated by a dash-dotted line. The section shows what happened in the cutting plane and what is behind it.

Looking at the drawing, you will notice the following:

1) dashed lines, which in the main view indicate a rectangular groove and a cylindrical stepped hole, are circled in the section with solid main lines, since they became visible as a result of the mental dissection of the object;

2) on the section, the solid main line denoting the cut, which ran along the main view, disappeared altogether, since the front half of the object is not depicted. The cut, located on the depicted half of the object, is not indicated, since it is not recommended to show invisible elements of the object with dashed lines on cuts;

3) on the section, a flat figure is highlighted by hatching, located in the cutting plane, hatching is applied only in the place where the cutting plane cuts the material of the object. For this reason, the rear surface of the cylindrical stepped hole is not shaded, as well as the rectangular groove (when the object was mentally dissected, the secant plane of these surfaces was not affected);

4) when depicting a cylindrical stepped hole, a solid main line is drawn, depicting a horizontal plane formed by a change in diameters on the frontal projection plane;

5) the section placed in the place of the main image does not change the top and left view images in any way.

When making cuts in the drawings, the following rules must be followed:

1) perform only useful cuts in the drawing ("useful" are cuts selected for reasons of necessity and sufficiency);

2) previously invisible internal outlines, depicted by dashed lines, outline with solid main lines;

3) hatch the section figure included in the section;

4) mental dissection of an object should refer only to this section and not affect the change in other images of the same object;

5) dashed lines are removed on all images, since the inner contour is well read on the section.

3.2 Designation of cuts

In order to know in which place the object has the shape shown in the cut image, the place where the cutting plane passed and the cut itself are indicated. The line denoting the cutting plane is called the section line. It is shown as a broken line.

In this case, the initial letters of the alphabet are chosen ( A B C D E etc.). Above the cut obtained using this cutting plane, an inscription is made according to the type A-A, i.e. two paired letters through a dash (Fig. 3.3).

The letters at the section lines and the letters denoting the section should be bigger size than the digits of the dimensional numbers on the same drawing (for one or two font numbers)

In cases where the cutting plane coincides with the plane of symmetry of a given object and the corresponding images are located on the same sheet in direct projection connection and are not separated by any other images, it is recommended not to mark the position of the cutting plane and not to accompany the cut image with an inscription.

Figure 3.3 shows a drawing of an object on which two cuts are made.

1. On the main view, the section is made by a plane, the location of which coincides with the plane of symmetry for this object. It runs along the horizontal axis in plan view. Therefore, this section is not marked.

2. Cutting plane A-A does not coincide with the plane of symmetry of this part, so the corresponding section is indicated.

Letter designation secant planes and cuts are placed parallel to the main inscription, regardless of the angle of inclination of the secant plane.

3.3 Hatching of materials in cuts and sections.

In cuts and sections, the figure obtained in the cutting plane is hatched.

GOST 2.306-68 establishes graphic designation various materials (fig.3.4)

Hatching for metals is applied in thin lines at an angle of 45° to the contour lines of the image, or to its axis, or to the lines of the drawing frame, and the distance between the lines must be the same.

Hatching on all cuts and sections for a given object is the same in direction and pitch (distance between strokes).

3.4. Classification of cuts.

Sections have several classifications:

1. Classification, depending on the number of cutting planes;

2. Classification, depending on the position of the cutting plane relative to the projection planes;

3. Classification, depending on the position of the cutting planes relative to each other.

Rice. 3.5

3.4.1 Simple cuts

A simple cut is a cut made by one secant plane.

The position of the cutting plane can be different: vertical, horizontal, inclined. It is chosen depending on the shape of the object, the internal structure of which needs to be shown.

Depending on the position of the cutting plane relative to the horizontal projection plane, the sections are divided into vertical, horizontal and oblique.

A vertical cut is a cut with a secant plane perpendicular to the horizontal plane of projections.

A vertically located cutting plane can be parallel to the frontal plane of projections or profile, thus forming, respectively, frontal (Fig. 3.6) or profile cuts (Fig. 3.7).

A horizontal cut is a cut with a cutting plane parallel to the horizontal projection plane (Fig. 3.8).

An oblique cut is a cut with a secant plane that makes an angle with one of the main projection planes that is different from a straight one (Fig. 3.9).

1. According to the axonometric image of the part and given dimensions draw its three views - main, top and left. Do not overdraw visual image.

7.2. Task 2

2. Make the necessary cuts.

3. Construct lines of intersection of surfaces.

4. Apply dimension lines and put down dimension numbers.

5. Outline the drawing and fill in the title block.

7.3. Task 3

1. Redraw the given two types of the object in size and build the third type.

2. Make the necessary cuts.

3. Construct lines of intersection of surfaces.

4. Apply dimension lines and put down dimension numbers.

5. Outline the drawing and fill in the title block.

For all tasks, views should be drawn only in a projection relationship.

7.1. Task 1.

Consider examples of task execution.

Task1. According to the visual image, build three types of parts and make the necessary cuts.

7.2 Task 2

Task2. Based on two views, construct a third view and make the necessary cuts.

Task 2. III stage.

1. Make the necessary cuts. The number of cuts should be minimal, but sufficient to read the inner contour.

1. Cutting plane BUT opens the inner coaxial surfaces. This plane is parallel to the frontal projection plane, so the cut A-A aligned with the main view.

2. The left side view shows a partial cut showing a Æ32 cylindrical hole.

3. Dimensions are applied on those images where the surface is read better, i.e. diameter, length, etc., for example, Æ52 and length 114.

4. Extension lines should not be crossed if possible. If the main view is selected correctly, then the largest number of dimensions will be on the main view.

Check:

1. So that each element of the part has a sufficient number of dimensions.
2. To ensure that all protrusions and holes are tied with dimensions to other elements of the part (size 55, 46, and 50).
3. Dimensions.
4. Outline the drawing, removing all invisible outline lines. Fill in the title block.

7.3. Task 3.

Build three views of the part and make the necessary cuts.

8. Information about surfaces.

Construction of lines belonging to surfaces.

Surfaces.

In order to build lines of intersection of surfaces, you need to be able to build not only surfaces, but also points located on them. This section covers the most commonly encountered surfaces.

8.1. Prism.

A trihedral prism is set (Fig. 8.1), truncated by a front-projecting plane (2GPZ, 1 algorithm, module No. 3). S Ç L= t (1234)

Since a prism projects relatively P 1, then the horizontal projection of the intersection line is already on the drawing, it coincides with the main projection of the given prism.

Cutting plane projecting relatively P 2, which means that the frontal projection of the intersection line is on the drawing, it coincides with the frontal projection of this plane.

The profile projection of the intersection line is built according to two given projections.

8.2. Pyramid

Given a truncated trihedral pyramid Ф(S,АВС)(fig.8.2).

This pyramid F intersected by planes S, D and G .

2 GPZ, 2 algorithm (Module No. 3).

F Ç S=123

S ^ P 2 Þ S 2 \u003d 1 2 2 2 3 2

1 1 2 1 3 1 and 1 3 2 3 3 3 F .

F Ç D=345

D ^ P 2 Þ = 3 2 4 2 5 2

3 1 4 1 5 1 and 3 3 4 3 5 3 built on belonging to the surface F .

F Ç G = 456

G CH 2 Þ Г 2 = 4 2 5 6

4 1 5 1 6 1 and 4 3 5 3 6 3 built on belonging to the surface F .

8.3. Bodies bounded by surfaces of revolution.

Solids of revolution are geometric figures bounded by surfaces of revolution (ball, ellipsoid of revolution, ring) or a surface of revolution and one or more planes (cone of revolution, cylinder of revolution, etc.). Images on projection planes parallel to the axis of rotation are limited by outline lines. These sketch lines are the boundary of the visible and invisible parts of geometric bodies. Therefore, when constructing projections of lines belonging to surfaces of revolution, it is necessary to construct points located on the outlines.

8.3.1. Rotation cylinder.

P 1, then the cylinder will be projected on this plane in the form of a circle, and on the other two projection planes in the form of rectangles, the width of which is equal to the diameter of this circle. Such a cylinder is projecting to P 1 .

If the axis of rotation is perpendicular P 2, then on P 2 it will be projected as a circle, and on P 1 and P 3 in the form of rectangles.

Similar reasoning for the position of the axis of rotation perpendicular to P 3(fig.8.3).

Cylinder F intersects with planes R, S , L and G(fig.8.3).

2 GPZ, 1 algorithm (Module №3)

F ^P 3

R, S, L, G ^ P 2

F Ç R = a(6 5 and )

F ^P 3 Þ Ф 3 \u003d a 3 (6 3 \u003d 5 3 and \u003d)

a 2 and a 1 built on belonging to the surface F .

F Ç S = b (5 4 3 )

F Ç S = s (2 3 ) The reasoning is similar to the previous one.

F G \u003d d (12 and

The tasks in Figures 8.4, 8.5, 8.6 are solved similarly to the problem in Figure 8.3, since the cylinder

everywhere profile-projecting, and holes - surfaces projecting relatively

P 1- 2GPZ, 1 algorithm (Module No. 3).

If both cylinders have the same diameters (Fig. 8.7), then their intersection lines will be two ellipses (Monge's theorem, module No. 3). If the axes of rotation of these cylinders lie in a plane parallel to one of the projection planes, then the ellipses will be projected onto this plane in the form of intersecting line segments.

8.3.2 Cone of rotation

The tasks in Figures 8.8, 8.9, 8.10, 8.11, 8.12 -2 GPZ (module No. 3) are solved according to the 2nd algorithm, since the surface of the cone cannot be projective, and the secant planes are frontally projecting everywhere.

Figure 8.13 shows a cone of revolution (body) intersected by two front-projecting planes G and L. Intersection lines are built according to the 2nd algorithm.

In Figure 8.14, the surface of the cone of revolution intersects with the surface of the profile-projecting cylinder.

2 GPZ, 2 solution algorithm (module No. 3), that is, the profile projection of the intersection line is on the drawing, it coincides with the profile projection of the cylinder. Two other projections of the intersection line are built according to belonging to the cone of revolution.

Fig.8.14

8.3.3. Sphere.

The surface of a sphere intersects with a plane and with all surfaces of revolution with it, in circles. If these circles are parallel to the projection planes, then they are projected onto them into a circle of natural size, and if not parallel, then in the form of an ellipse.

If the axes of rotation of the surfaces intersect and are parallel to one of the projection planes, then all intersection lines - circles - are projected onto this plane in the form of straight line segments.

On fig. 8.15 - sphere, G- plane, L- cylinder, F- frustum.

S З Г = a- circle;

S Ç L=b- circle;

S Ç F \u003d s- circle.

Since the axes of rotation of all intersecting surfaces are parallel P 2, then all lines of intersection are circles on P 2 are projected into line segments.

On the P 1: circle "a" is projected to the true value since it is parallel to it; circle "b" is projected into a straight line segment, since it is parallel P 3; circle "with" is projected in the form of an ellipse, which is built according to the belonging to the sphere.

Points are built first. 1, 7 and 4, which define the minor and major axes of the ellipse. Then builds a point 5 , as lying on the sphere's equator.

For the remaining points (arbitrary), circles (parallels) are drawn on the surface of the sphere and horizontal projections of the points lying on them are determined by belonging to them.

9. Examples of tasks.

Task 4. Build three types of parts with the necessary cuts and apply dimensions.

Task 5. Build three views of the part and make the necessary cuts.

10. Axonometry

10.1. Brief theoretical information about axonometric projections

A complex drawing composed of two or three projections, having the properties of reversibility, simplicity, etc., at the same time has a significant drawback: it lacks visibility. Therefore, in order to give a more visual representation of the subject, along with a complex drawing, an axonometric drawing is given, which is widely used in describing product designs, in operating manuals, in assembly diagrams, for explaining drawings of machines, mechanisms and their parts.

Compare two images - an orthogonal drawing and an axonometric drawing of the same model. Which image makes it easier to read the form? Of course on the axonometric image. (fig.10.1)

The essence of axonometric projection is that a geometric figure, together with the axes of rectangular coordinates to which it is referred in space, is projected in parallel onto a certain projection plane, called the axonometric projection plane, or picture plane.

If we postpone on the coordinate axes x,y and z line segment l (lx,ly,lz) and project onto a plane P ¢ , then we get axonometric axes and segments on them l "x, l" y, l "z(fig.10.2)

lx, ly, lz- natural scale.

l=lx=ly=lz

l "x, l" y, l "z- axonometric scales.

The resulting set of projections on П¢ is called an axonometry.

The ratio of the length of axonometric scale segments to the length of natural scale segments is called the indicator or distortion coefficient along the axes, which are denoted Kx, Ky, Kz.

Types of axonometric images depend on:

1. From the direction of the projecting rays (they can be perpendicular P"- then the axonometry will be called orthogonal (rectangular) or located at an angle not equal to 90 ° - oblique axonometry).

2. From the position of the coordinate axes to the axonometric plane.

Three cases are possible here: when all three coordinate axes make some acute angles (equal and unequal) with the axonometric projection plane, and when one or two axes are parallel to it.

In the first case, only rectangular projection is applied, (s ^ P") in the second and third - only oblique projection (s П") .

If the coordinate axes OH, OY, OZ not parallel to the axonometric projection plane P", then will they be projected onto it in full size? Of course not. The image of lines in the general case is always less than natural size.

Consider an orthogonal drawing of a point BUT and its axonometric image.

The position of a point is determined by three coordinates - X A, Y A, Z A, obtained by measuring the links of a natural broken line OA X - A X A 1 - A 1 A(fig.10.3).

A"- main axonometric projection of a point BUT ;

BUT- secondary point projection BUT(projection of the projection of the point).

Axial distortion coefficients X", Y" and Z" will be:

k x = ; k y = ; k y =

In orthogonal axonometry, these indicators are equal to the cosines of the angles of inclination of the coordinate axes to the axonometric plane, and therefore they are always less than one.

They are linked by the formula

k 2 x + k 2 y + k 2 z= 2 (I)

In oblique axonometry, the distortion indicators are related by the formula

k x + k y + k z = 2+ctg a (III)

those. any of them can be less than, equal to or greater than one (here a is the angle of inclination of the projecting rays to the axonometric plane). Both formulas are a derivation from Polke's theorem.

Polke's theorem: the axonometric axes on the plane of the drawing (П¢) and the scales on them can be chosen quite arbitrarily.

(Hence, the axonometric system ( O"X"Y"Z") is generally determined by five independent parameters: three axonometric scales and two angles between the axonometric axes).

The angles of inclination of the natural coordinate axes to the axonometric projection plane and the direction of projection can be chosen arbitrarily, therefore, many types of orthogonal and oblique axonometries are possible.

They are divided into three groups:

1. All three distortion indicators are equal (k x = k y = k z). This type of perspective is called isometry. 3k 2 =2; k= » 0.82 - theoretical distortion factor. According to GOST 2.317-70, you can use K=1 - reduced distortion factor.

2. Any two indicators are equal (for example, kx=ky kz). This type of perspective is called dimetria. k x = k z ; k y = 1/2k x 2 ; k x 2 +k z 2 + k y 2 /4 = 2; k = » 0.94; kx = 0.94; ky = 0.47; kz = 0.94 - theoretical distortion coefficients. According to GOST 2.317-70, the distortion coefficients can be given - k x =1; k y =0.5; kz=1.

3. 3. All three indicators are different (k x ¹ k y ¹ k z). This type of perspective is called trimetry .

In practice, several types of both rectangular and oblique axonometry are used with the simplest ratios between distortion indicators.

From GOST2.317-70 and various kinds axonometric projections, consider orthogonal isometry and dimetry, as well as oblique dimetry, as the most commonly used.

10.2.1. Rectangular isometry

In isometry, all axes are inclined to the axonometric plane at the same angle, therefore the angle between the axes (120°) and the distortion coefficient will be the same. Select scale 1: 0.82=1.22; M 1.22: 1.

For the convenience of construction, the given coefficients are used, and then natural dimensions are plotted on all axes and lines parallel to them. Images thus become larger, but this does not affect visibility.

The choice of the type of axonometry depends on the shape of the depicted part. The easiest way to build a rectangular isometry, so such images are more common. However, when depicting details that include quadrangular prisms and pyramids, their clarity decreases. In these cases, it is better to perform rectangular dimetry.

Oblique dimetry should be chosen for parts that have a large length with a small height and width (such as a shaft) or when one of the sides of the part contains the largest number of important features.

In axonometric projections, all the properties of parallel projections are preserved.

Consider the construction of a flat figure ABCDE .

First of all, let's build the axes in axonometry. Figure 10.4 shows two ways to build axonometric axes in isometry. In Figure 10.4 a the construction of axes using a compass is shown, and in Fig. 10.4 b- construction using equal segments.

Fig.10.5

Figure ABCDE lies in the horizontal plane of projections, which is limited by the axes OH and OY(Fig. 10.5a). We build this figure in axonometry (Fig. 10.5b).

Each point lying in the projection plane, how many coordinates does it have? Two.

A point lying in a horizontal plane - coordinates X and Y .

Consider the construction v.A. At what coordinate do we start building? From coordinates X A .

To do this, we measure the value on the orthogonal drawing OA X and set aside on the axis X", we get a point A X" . A X A 1 what axis is it parallel to? axes Y. So from t. A X" draw a line parallel to the axis Y"and put a coordinate on it Y A. Received point BUT" and will be an axonometric projection v.A .

All other points are constructed similarly. Dot With lies on the axis OY, so it has one coordinate.

In figure 10.6, a five-sided pyramid is given, in which the base is the same pentagon ABCDE. What needs to be completed to make a pyramid? Gotta make a point S, which is its vertex.

Dot S is a point in space, so it has three coordinates X S , Y S and Z S. First, a secondary projection is built S(S1), and then all three dimensions are transferred from the orthogonal drawing. By connecting S" c A", B", C", D" and E", we get an axonometric image of a three-dimensional figure - a pyramid.

10.2.2. Circle isometry

Circles are projected onto the projection plane at full size when they are parallel to that plane. And since all planes are inclined to the axonometric plane, the circles lying on them will be projected onto this plane in the form of ellipses. In all types of axonometries, ellipses are replaced by ovals.

When depicting ovals, it is necessary, first of all, to pay attention to the construction of the major and minor axes. You need to start by determining the position of the minor axis, and the major axis is always perpendicular to it.

There is a rule: the minor axis coincides with the perpendicular to this plane, and the major axis is perpendicular to it, or the direction of the minor axis coincides with the axis that does not exist in this plane, and the major axis is perpendicular to it (Fig. 10.7)

The major axis of the ellipse is perpendicular to the coordinate axis, which is absent in the plane of the circle.

The major axis of the ellipse is 1.22 ´ d env; the minor axis of the ellipse is 0.71 ´ d env.

In Figure 10.8, there is no axis in the plane of the circle Z Z ".

In Figure 10.9, there is no axis in the plane of the circle X, so the major axis is perpendicular to the axis X ".

Now consider how an oval is drawn in one of the planes, for example, in the horizontal plane XY. There are many ways to construct an oval, let's get acquainted with one of them.

The sequence for constructing an oval is as follows (Fig. 10.10):

1. The position of the minor and major axis is determined.

2. Through the intersection point of the minor and major axes, we draw lines parallel to the axes X" and Y" .

3. On these lines, as well as on the minor axis, from the center with a radius equal to the radius of a given circle, set aside points 1 and 2, 3 and 4, 5 and 6 .

4. Connect the dots 3 and 5, 4 and 6 and mark the points of their intersection with the major axis of the ellipse ( 01 and 02 ). From a point 5 , radius 5-3 , and from the point 6 , radius 6-4 , draw arcs between points 3 and 2 and dots 4 and 1 .

5. Radius 01-3 draw an arc connecting the points 3 and 1 and radius 02-4 - points 2 and 4 . Similarly, ovals are built in other planes (Fig. 10.11).

For ease of constructing a visual image of the surface, the axis Z may coincide with the height of the surface, and the axes X and Y with axes of horizontal projection.

To build a point BUT belonging to the surface it is necessary to construct its three coordinates X A , Y A and Z A. A point on the surface of a cylinder and other surfaces is constructed in a similar way (Fig. 10.13).

The major axis of the oval is perpendicular to the axis Y ".

When constructing an axonometry of a part limited by several surfaces, the following sequence should be followed:

Option 1.

1. The detail is mentally divided into elementary geometric shapes.

2. The axonometry of each surface is drawn, the construction lines are saved.

3. A 1/4 cutout of the part is built to show the internal configuration of the part.

4. Hatching is applied in accordance with GOST 2.317-70.

Consider an example of constructing an axonometry of a part, the outer contour of which consists of several prisms, and inside the part there are cylindrical holes of different diameters.

Option 2. (Fig. 10.5)

1. A secondary projection of the part is built on the projection plane P.

2. The heights of all points are plotted.

3. A cutout of 1/4 of the part is being built.

4. Hatching is applied.

For this part, option 1 will be more convenient for construction.

10.3. Stages of making a visual representation of a part.

1. The part fits into the surface of a quadrangular prism, the dimensions of which are equal to the overall dimensions of the part. This surface is called wrapping.

An isometric image of this surface is performed. The wrapping surface is built according to the overall dimensions (Fig. 10.15 a).

Rice. 10.15 a

2. From this surface, protrusions are cut, located on the top of the part along the axis X and a 34 mm high prism is built, one of the bases of which will be the upper plane of the wrapping surface (Fig. 10.15 b).

Rice. 10.15 b

3. From the remaining prism, a lower prism is cut out with bases 45 ´35 and a height of 11mm (Fig. 10.15 in).

Rice. 10.15 in

4. Two cylindrical holes are built, the axes of which lie on the axis Z. The upper base of the large cylinder lies on the upper base of the part, the second one is 26 mm lower. The lower base of the large cylinder and the upper base of the small one lie in the same plane. The lower base of the small cylinder is built on the lower base of the part (Fig. 10.15 G).

Rice. 10.15 G

5. A cut is made in 1/4 of the part to open its inner contour. The incision is made by two mutually perpendicular planes, that is, along the axes X and Y(fig.10.15 d).

Fig.10.15 d

6. The sections and the rest of the part are outlined, and the cut out part is removed. Hidden lines are erased and sections are shaded. The hatching density should be the same as in the orthogonal drawing. The direction of the dashed lines is shown in Figure 10.15 e in accordance with GOST 2.317-69.

The hatching lines will be lines parallel to the diagonals of squares lying in each coordinate plane, the sides of which are parallel to the axonometric axes.

Fig.10.15 e

7. There is a peculiarity of hatching of the stiffener in axonometry. According to the rules

GOST 2.305-68 in a longitudinal section, the stiffener in the orthogonal drawing is not

shaded, and shaded in axonometry. Figure 10.16 shows an example

hatching of the stiffener.

10.4 Rectangular dimetry.

A rectangular dimetric projection can be obtained by rotating and tilting the coordinate axes about P ¢ so that the distortion indicators along the axes X" and Z" took an equal value, and along the axis Y"- half as much. Distortion indicators " k x" and " kz" will be equal to 0.94, and " k y "- 0,47.

In practice, they use the given indicators, i.e. along the axes X" and Z" set aside natural dimensions, and along the axis Y"- 2 times less than natural ones.

Axis Z" usually placed vertically X"- at an angle of 7°10¢ to the horizontal line, and the axis Y"- at an angle of 41°25¢ to the same line (Fig. 12.17).

1. A secondary projection of a truncated pyramid is built.

2. Point heights are built 1,2,3 and 4.

The easiest way to build an axis X ¢ , setting aside 8 equal parts on a horizontal line and down the vertical line 1 the same part.

To build an axis Y" at an angle of 41 ° 25¢, it is necessary to set aside 8 parts on the horizontal line, and 7 of the same parts on the vertical line (Fig. 10.17).

Figure 10.18 shows a truncated quadrangular pyramid. To make it easier to build it in axonometry, the axis Z must match the height, then the vertices of the base ABCD will lie on the axles X and Y (A and C О X ,AT and D Î y). How many coordinates do points 1 and have? Two. Which? X and Z .

These coordinates are plotted in actual size. The resulting points 1¢ and 3¢ are connected to the points A¢ and C¢.

Points 2 and 4 have two Z coordinates and Y. Since they have the same height, the coordinate Z deposited on the axis Z". through the given point 0 ¢ draw a line parallel to the axis Y, on which the distance is plotted on both sides of the point 0 1 4 1 reduced by half.

Received points 2 ¢ and 4 ¢ connect with dots AT ¢ and D" .

10.4.1. Construction of circles in rectangular dimetry.

Circles lying on coordinate planes in rectangular dimetry, as well as in isometry, will be displayed as ellipses. Ellipses located on the planes between the axes X" and Y",Y" and Z" in the reduced dimetry will have a large axis equal to 1.06d, and a small one - 0.35d, and in the plane between the axes X" and Z"- the major axis is also 1.06d, and the minor one is 0.95d (Fig. 10.19).

Ellipses are replaced by four-cent ovals, as in isometrics.

10.5. Oblique dimetric projection (frontal)

If we arrange the coordinate axes X and Y parallel to the plane П¢, then the distortion indicators along these axes will become equal to unity (k = t=1). Axis Distortion Index Y usually taken equal to 0.5. Axonometric axes X" and Z" form a right angle, axis Y" usually drawn as the bisector of this angle. Axis X can be directed both to the right of the axis Z", and to the left.

It is preferable to use the right system, as it is more convenient to depict objects in a dissected form. In this type of axonometry, it is good to draw details that have the shape of a cylinder or cone.

For the convenience of the image of this part, the axis Y must be aligned with the axis of rotation of the surfaces of the cylinders. Then all circles will be depicted in full size, and the length of each surface will be halved (Fig. 10.21).

11. Inclined sections.

When making drawings of machine parts, it is often necessary to use inclined sections.

When solving such problems, it is necessary first of all to understand: how the cutting plane should be located and which surfaces are involved in the section in order for the part to be read better. Consider examples.

Given a tetrahedral pyramid, which is dissected by an inclined frontally projecting plane A-A(fig.11.1). The section will be a quadrilateral.

First, we construct its projections on P 1 and on P 2. The frontal projection coincides with the projection of the plane, and we build the horizontal projection of the quadrilateral by belonging to the pyramid.

Then we build the natural size of the section. For this, an additional projection plane is introduced P 4, parallel to the given cutting plane A-A, project a quadrilateral onto it, and then combine it with the drawing plane.

This is the fourth main task of complex drawing transformation (module #4, page 15 or task #117 from the Descriptive Geometry Workbook).

Constructions are performed in the following sequence (Fig. 11.2):

1. 1. In the free space of the drawing, we draw an axial line parallel to the plane A-A .

2. 2. From the points of intersection of the edges of the pyramid with the plane, we draw projecting rays perpendicular to the cutting plane. points 1 and 3 will lie on a line perpendicular to the axis.

3. 3. Distance between points 2 and 4 transferred from a horizontal projection.

4. Similarly, the true value of the cross section of the surface of revolution is constructed - an ellipse.

Distance between points 1 and 5 the major axis of the ellipse. The minor axis of the ellipse must be built by dividing the major axis in half ( 3-3 ).

Distance between points 2-2, 3-3, 4-4 transferred from a horizontal projection.

Consider more complex example, including polyhedral surfaces and surfaces of revolution (Fig. 11.3)

Given a four-sided prism. There are two holes in it: a prismatic one located horizontally and a cylindrical one, the axis of which coincides with the height of the prism.

The cutting plane is frontally projecting, therefore the frontal projection of the section coincides with the projection of this plane.

A quadrangular prism projecting to the horizontal plane of projections, and hence the horizontal projection of the section is also in the drawing, it coincides with the horizontal projection of the prism.

The natural size of the section into which both prisms and the cylinder fall, we build on a plane parallel to the secant plane A-A(fig.11.3).

Execution sequence oblique section:

1. The axis of the section is drawn, parallel to the cutting plane, in the free field of the drawing.

2. A section of the outer prism is built: its length is transferred from the frontal projection, and the distance between the points from the horizontal.

3. The section of the cylinder is built - part of the ellipse. First, characteristic points are constructed that determine the length of the minor and major axes ( 5 4 , 2 4 -2 4 ) and points bounding the ellipse (1 4 -1 4 ) , then additional points (4 4 -4 4 and 3 4 -3 4).

4. A section of a prismatic hole is built.

5. Hatching is applied at an angle of 45° to the main inscription, if it does not coincide with the contour lines, and if it does, then the hatching angle can be 30° or 60°. The hatching density in the section is the same as in the orthogonal drawing.

The slanted section can be rotated. In this case, the designation is accompanied by the sign . It is also allowed to show a half figure of an oblique section if it is symmetrical. A similar arrangement of an inclined section is shown in Fig. 13.4. Designations of points when constructing an inclined section can be omitted.

Figure 11.5 shows a visual representation of a given figure with a section by a plane A-A .

test questions

1. What is called a view?

2. How is an image of an object obtained on a plane?

3. What names are assigned to the views on the main projection planes?

4. What is called the main view?

5. What is called an additional view?

6. What is called a local species?

7. What is called a cut?

8. What designations and inscriptions are established for cuts?

9. What is the difference simple incisions from complex?

10. What convention is observed when making broken cuts?

11. What cut is called local?

12. Under what conditions is it allowed to combine half of the view and half of the section?

13. What is called a section?

14. How are sections arranged in the drawings?

15. What is called a remote element?

16. How is it simplified to show repeating elements in the drawing?

17. How is the image of objects of great length conditionally reduced in the drawing?

18. How do axonometric projections differ from orthogonal ones?

19. What is the principle of formation of axonometric projections?

20. What types of axonometric projections are established?

21. What are the features of isometry?

22. What are the features of dimetria?

Bibliographic list

1. Suvorov, S.G. Engineering drawing in questions and answers: (reference book) / S.G. Suvorov, N.S. Suvorova.-2nd ed. revised and additional - M.: Mashinostroenie, 1992.-366s.

2. Fedorenko V.A. Handbook of engineering drawing / V.A. Fedorenko, A.I. Shoshin, - Ed.16-ster.; m Perepech. from the 14th edition of 1981 - M .: Alliance, 2007.-416s.

3.Bogolyubov, S.K. Engineering Graphics: Textbook for Wednesdays. specialist. textbook establishments on special tech. profile / S.K. Bogolyubov.-3rd ed., corrected. and add.-M .: Mashinostroenie, 2000.-351s.

4. Vyshnepolsky, I.S. Technical drawing e. Proc. for the beginning prof. education / I.S. Vyshnepolsky.-4th ed., revised. and add.; Vulture MO.- M.: Higher. school: Academy, 2000.-219p.

5. Levitsky, V.S. Engineering drawing and automation of drawings: textbook. for higher education institutions / V.S. Levitsky. - 6th ed., revised. and add.; Vulture MO.-M.: Higher. school, 2004.-435s.

6. Pavlova, A.A. Descriptive geometry: textbook. for universities / A.A. Pavlova-2nd ed., revised. and add.; Vulture MO.- M.: Vlados, 2005.-301s.

7. GOST 2.305-68*. Images: views, sections, sections / Unified system for design documentation. - M.: Publishing house of standards, 1968.

8. GOST 2.307-68. Application of dimensions and limit deviations / Unified system

design documentation. - M.: Publishing house of standards, 1968.