Image of circles in isometric projection

Let's look at how circles are depicted in an 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 depicted in isometry as three identical ellipses.

Rice. 3.16.

To simplify the work, ellipses are replaced by ovals outlined by circular arcs; they are constructed as follows (Fig. 3.17). Draw a rhombus into which an oval should fit, depicting given circle in isometric projection. To do this, the axes are plotted from the point ABOUT 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 to form a rhombus. Its sides are equal to the diameter of the depicted circle.

Rice. 3.17.

From the vertices of obtuse angles (points A And IN) describe between points A And b, and With And d arc radius R, equal to length direct Va or Bb(Fig. 3.17, b).

Points WITH and D lying at the intersection of the diagonal of the rhombus with straight lines Va And Bb, are the centers of small arcs conjugating large ones.

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

Construction of isometric projections of parts

Let's consider the construction of an isometric projection of a part, two views of which are given in Fig. 3.18, A.

The construction is carried out in the following order. First, draw out the original shape of the part - a square. Then ovals are built to represent an arc (Fig. 3.18, b) and circles (Fig. 3.18, c).

Rice. 3.18.

To do this, find a point on a vertical plane ABOUT, through which the isometric axes are drawn X And z. This construction produces a rhombus into which half of the oval is inscribed (Fig. 3.18, b). Ovals on parallel planes are constructed by moving the centers of the arcs to a segment equal to the distance between these planes. Double circles in Fig. Figure 3.18 shows the centers of these arcs.

On the same axes X And z build a rhombus with side equal to the diameter circle d. An oval is inscribed in the rhombus (Fig. 3.18, c).

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

The concept of dimetric rectangular projection

The location of the dimetric projection axes and the method of their construction are shown in Fig. 3.19. Axis z carried 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 construct axes using a ruler and compass. To do this from the point ABOUT laid horizontally to the right and left in eights equal divisions(Fig. 3.19, b). Perpendiculars are drawn from the extreme points. Their height is equal to: for 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 axial dimensions at is reduced by 2 times, and along the axes X And z postponed without cuts.

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

Rice. 3.20.

Technical drawing

Technical drawing – This is a visual image made according to the rules 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 when designing, inventing and rationalizing, as well as when learning to read drawings, when using a technical drawing you need to explain the shape of a part presented in the drawing.

When 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 construction sequence are observed.

In some cases, it is more convenient to begin constructing axonometric projections by constructing a base figure. Therefore, let us consider how flat geometric figures located horizontally are depicted in axonometry.

1. square shown in Fig. 1, a and b.

Along the axis X lay down the side of the square a, along the axis at- half a side a/2 for frontal dimetric projection and side A for isometric projection. The ends of the segments are connected by straight lines.

Rice. 1. Axonometric projections of a square:

2. Construction of an axonometric projection triangle shown in Fig. 2, a and b.

Symmetrical to a point ABOUT(origin of coordinate axes) along the axis X lay aside half the side of the triangle A/ 2, and along the axis at- its height h(for frontal dimetric projection half height h/2). The resulting points are connected by straight segments.

Rice. 2. Axonometric projections of a triangle:

a - frontal dimetric; b - isometric

3. Construction of an axonometric projection regular hexagon shown in Fig. 3.

Axis X to the right and left of the point ABOUT lay down the segments equal to side hexagon. Axis at symmetrical to the point ABOUT lay down the segments s/2, equal to half distances between opposite sides hexagon (for frontal dimetric projection, these segments are halved). From points m And n, obtained on the axis at, swipe right and left parallel to the axis X segments equal to half the side of the hexagon. The resulting points are connected by straight segments.

Rice. 3. Axonometric projections of a regular hexagon:

a - frontal dimetric; b - isometric

4. Construction of an axonometric projection circle .

Frontal dimetric projection convenient for depicting objects with curvilinear outlines, similar to those shown in Fig. 4.

Fig.4. Frontal dimetric projections of parts

In Fig. 5. given frontal dimetric projection of a cube with circles inscribed in its faces. Circles located on planes perpendicular to the x and z axes are represented by ellipses. The front face of the cube, perpendicular to the y-axis, is projected without distortion, and the circle located on it is depicted without distortion, i.e., described by a compass.

Fig.5. Frontal dimetric projections of circles inscribed in the faces of a cube

Construction of a frontal dimetric projection of a flat part with a cylindrical hole .

The frontal dimetric projection of a flat part with a cylindrical hole is performed as follows.

1. Construct the outline of the front face of the part using a compass (Fig. 6, a).

2. Straight lines are drawn through the centers of the circle and arcs parallel to the y-axis, on which half the thickness of the part is laid. The centers of the circle and arcs located on the rear surface of the part are obtained (Fig. 6, b). From these centers a circle and arcs are drawn, the radii of which must be equal to the radii of the circle and arcs of the front face.

3. Draw tangents to the arcs. Remove excess lines and outline the visible contour (Fig. 6, c).

Rice. 6. Construction of a frontal dimetric projection of a part with cylindrical elements

Isometric projections of circles .

A square in isometric projection is projected into a rhombus. Circles inscribed in squares, for example, located on the faces of a cube (Fig. 7), are depicted as ellipses in an isometric projection. In practice, ellipses are replaced by ovals, which are drawn with four arcs of circles.

Rice. 7. Isometric projections of circles inscribed in the faces of a cube

Construction of an oval inscribed in a rhombus.

1. Construct a rhombus with a side equal to the diameter of the depicted circle (Fig. 8, a). To do this, through the point ABOUT draw isometric axes X And y, and on them from the point ABOUT lay down segments equal to the radius of the depicted circle. Through dots a, b, WithAnd d carry out direct parallel to the axes; get a rhombus. The major axis of the oval is located on the major diagonal of the rhombus.

2. Fit an oval into a rhombus. To do this, from the vertices of obtuse angles (points A And IN) describe arcs with a radius R, equal to the distance from the top obtuse angle(points A And IN) to points a, b or s, d respectively. From point IN to the points A And b draw straight lines (Fig. 8, b); the intersection of these lines with the larger diagonal of the rhombus gives the points WITH And D, which will be the centers of small arcs; radius R 1 minor arcs is equal to Sa (Db). Arcs of this radius conjugate the large arcs of the oval.

Rice. 8. Construction of an oval in a plane, perpendicular to the axis z.

This is how an oval is built, lying in a plane perpendicular to the axis z(oval 1 in Fig. 7). Ovals located in planes perpendicular to the axes X(oval 3) and at(oval 2), build in the same way as oval 1, only oval 3 is built on the axes at And z(Fig. 9, a), and oval 2 (see Fig. 7) - on the axes X And z(Fig. 9, b).

Rice. 9. Construction of an oval in planes perpendicular to the axes X And at

Constructing an isometric projection of a part with a cylindrical hole.

If on an isometric projection of a part you need to depict a through cylindrical hole drilled perpendicular to the front face, shown in the figure. 10, a.

The construction is carried out as follows.

1. Find the position of the center of the hole on the front face of the part. Isometric axes are drawn through the found center. (To determine their direction, it is convenient to use the image of the cube in Fig. 7.) On the axes from the center, segments equal to the radius of the depicted circle are laid (Fig. 10, a).

2. Construct a rhombus, the side of which is equal to the diameter of the depicted circle; carry out large diagonal rhombus (Fig. 10, b).

3. Describe large oval arcs; find centers for small arcs (Fig. 10, c).

4. Small arcs are carried out (Fig. 10, d).

5. Construct the same oval on the back face of the part and draw tangents to both ovals (Fig. 10, e).

Rice. 10. Construction of an isometric projection of a part with a cylindrical hole

Look at Figure 59. How many objects are shown on it? various shapes?

You see one object depicted in different ways. Can you answer what the names of images a, b, c are?

Pay attention to images 6 and c. They're called. as you already know, with visual images. It is easier to imagine the shape of an object using them than from Figure 59, a. Figure 60 shows how one of these visual images is produced. The front and rear faces of the cube are located parallel to the projection plane P (Fig. 60, a).

Rice. 59. Various images

By projecting the cube together with the coordinate axes X0, Y0, Z0 onto the plane P with parallel rays directed to it at an angle less than 90°, an oblique frontal dimetric projection is obtained (Fig. 60, c). In what follows we will briefly call it the frontal dimetric projection. You saw an object depicted in such a projection in Figure 59, b.

Rice. 60. Formation of axonometric projections: a, c - frontal dimetric: b, d - isometric

If the faces of a cube are tilted to the plane P under equal angles(Fig. 60, b) and project the cube along with the coordinate axes onto the plane with rays perpendicular to it, you will get another visual image, which is called a rectangular isometric projection (Fig. 60.). In what follows we will briefly call it an isometric projection.

You saw the image of an object in isometric projection in Figure 59, c.

Now compare images c and d (Fig. 60). What is the name of image c and what is the name of image d?

Frontal dimetric (Fig. 60, c) and isometric (Fig. 60.d) projections are combined into one common name- axonometric projections. The word "axonometry" is Greek. Translated, it means “measurement along the axes.”

Hence the name “dimetry”, which in Greek means “double dimension”. Hence the name “isometry”. which in Greek means " equal measurements»

The x, y and z axes on the plane of axonometric projections are called axonometric. When such projections are constructed, dimensions are plotted along the x, y and z axes.

Axonometric projections are classified as visual images.

1. What axonometric projections are given in Figure 59?
2. How are the projecting rays directed relative to the projection planes to obtain the images given in Figure 59, b and c?

§ 7. Construction of axonometric projections

7.1. Axes position. The construction begins by drawing the axonometric axes x, y and z. The axis of the frontal dimetric projection is positioned as shown in Figure 61, a: the X axis is horizontal, the z axis is vertical, the y axis is at an angle of 45° to horizontal line.

An angle of 45° can be constructed using a drawing square with angles of 45, 45 and 90°, as shown in Figure 61, c. The y-axis is tilted to the left or right.

In the frontal dimetric projection, natural dimensions are plotted along the x and z axes (and parallel to them), halved along the y axis (and parallel to it).

The position of the isometric projection axes is shown in Figure 61, b. The x and y axes are positioned at an angle of 30° to the horizontal line (120° between the axes). They are also convenient to carry out using a square. But in this case, the square is taken with angles of 30, 60 and 90° (Fig. 61, d).

When constructing an isometric projection along the x, y, z axes and parallel to them, the natural dimensions of the object are plotted.

Figure 61. e and f shows the construction of axes on paper. lined in a checkered pattern. It is used when performing technical drawings. To obtain an angle of 15°, the axis is drawn along the diagonals of the cells (Fig. 61, e). The ratio of segments of 3 and 5 cells gives an axis tilt of approximately 30° (Fig. 61, e).

What dimensions are laid down when making a drawing along the axonometric axes in isometric and frontal dimetric projections?

Rice. 61. Image of the axes of axonometric projections: a, 6 - position of the axes; c, d techniques for constructing axes; d, f - construction of axes when performing technical drawings

7.2. Axonometric projections flat figures . Let's consider the construction of axonometric projections of flat geometric shapes, located horizontally (Table 1). Such constructions will be needed later when performing axonometric projections geometric bodies. The construction begins by drawing the axonometric x and y axes.

Table 1. Method for constructing axonometric projections of flat figures

7.3. Axonometric projections of flat-sided objects.

Let's consider general method constructing axonometric projections of flat-edged objects (Table 2) using the example of a part, two views of which are given in Figure 62.

Figure 62. Part drawing

Table 2. Method for constructing axonometric projections of flat-sided objects

From the example discussed in the table it is clear that the rules for constructing isometric and frontal dimetric projections are generally the same. The only difference is in the location of the axes and in the length of the segments laid along the y-axis.

Rice. 63. Exercise task

Please note that when drawing dimensions on the axonometric projection of an object, extension lines are drawn parallel to the axonometric axes, dimension lines are drawn parallel to the measured segment.

1. How are the axes of the frontal dimetric projection located? isometric projection?
2. What dimensions are laid along the axes of the frontal dimetric and isometric projections and parallel to them?
3. List general stages construction of axonometric projections.

1. Construct a frontal dimetric projection equilateral triangle with a side of 40 mm.

Construct an isometric projection of a regular hexagon with sides also 40 mm. Place them parallel to the frontal plane of projections.

1. Construct frontal dimetric and isometric projections of the part shown in Figure 63.

§ 8. Axonometric projections of objects with round surfaces

8.1. Frontal dimetric projections of circles. If they want some elements in the axonometric image. for example, circles (Fig. 64) are kept undistorted, then a frontal dimetric projection is used. The construction of a frontal dimetric projection of a part with a cylindrical hole, two views of which are given in Figure 64, a, is performed as follows:

1. Using the x, y, z axes, draw outlines with thin lines external form details (Fig. 64, b).
2. Find the center of the hole on the front face. The axis of the hole is drawn through it parallel to the y-axis and half the thickness of the part is laid on it. The center of the hole located on the back face is obtained.
3. From the obtained points, as from centers, circles are drawn, the diameter of which is equal to the diameter of the hole (Fig. 64, c).
4. Remove excess lines and trace the visible outline of the part (Fig. 64, d).

Rice. 64. Construction of a frontal dimetric projection

Build in workbook frontal dimetric projection of the part shown in Figure 64, a. Point the y-axis in the other direction. Enlarge the image size approximately twice.

8.2. Isometric projections of circles. The isometric projection of a circle (Fig. 65) is a curve called an ellipse. Ellipses are difficult to construct. In drawing practice, ovals are often built instead. An oval is a closed curve outlined by arcs of circles. It is convenient to construct an oval by fitting it into a rhombus, which is an isometric projection of a square.

Rice. 65. Image in isometric projection of circles inscribed in a cube

The construction of an oval inscribed in a rhombus is performed in the following sequence.

First, a rhombus is built with a side equal to the diameter of the depicted circle (Fig. 66, a). To do this, the isometric x and y axes are drawn through point O. On them, from point O, segments equal to the radius of the depicted circle are laid. Through points a, b, c and d, draw straight lines parallel to the axes; get a rhombus.

Rice. 66. Constructing an oval

The major axis of the oval is located on the major diagonal of the rhombus.

After this, an oval is inscribed in the rhombus. To do this, arcs are drawn from the vertices of obtuse angles (points A and B). Their radius R is equal to the distance from the vertex of the obtuse angle (points A and B) to points c, d or a, b, respectively (Fig. 66, b).

Straight lines are drawn through points B and a, B and b. At the intersection of straight lines Ba and Bb with the larger diagonal of the rhombus, points C and D are found (Fig. 66, a). These points will be the centers of the small arcs. Their radius R1 is equal to Ca (or Db). Arcs of this radius smoothly connect the large arcs of the oval.

We examined the construction of an oval lying in a plane perpendicular to the z axis (oval 1 in Figure 65). Ovals located in planes perpendicular to the y-axis (oval 2) and the x-axis (oval 3) are also constructed. Only for oval 2 the construction is carried out on the x and z axes (Fig. 67, a), and for oval 3 - on the y and z axes (Fig. 67, b). Let's consider how the studied constructs are applied in practice.

Rice. 67. Construction of ovals: a lying in a plane perpendicular to the y-axis; b - lying in a plane perpendicular to the x axis

Rice. 68. Construction of an isometric projection of a part with a cylindrical hole

8.3. A method for constructing axonometric projections of objects with round surfaces. Figure 68a shows an isometric projection of the bar. It is necessary to depict a cylindrical hole drilled perpendicular to the front edge. The construction is done like this:

1. Find the center of the hole on the front face. Determine the direction of the isometric axes to construct a rhombus (see Fig. 65). Axes are drawn from the found center (Fig. 68, a) and segments equal to the radius of the circle are laid on them.
2. They are building a rhombus. Draw it along a large diagonal (Fig. 68, b).
3. Describe large arcs. Find the centers for small arcs (Fig. 68.c).
4. Small arcs are drawn from the found centers.

The same oval is built on the back face, but only its visible part is outlined (Fig. 68, d).

1. In Figure 69, a the axes are drawn to construct three rhombuses. Indicate on which face of the cube - top, side right, side left (see Fig. 65) - each rhombus will be located. Which axis will the plane of each of these rhombuses be perpendicular to? And to which axis is the plane of each oval perpendicular (Fig. 69, b)?

Rice. 69. Exercise task

1. The sides of the rhombuses in Figure 65 are 30 mm. What are the diameters of the circles whose projections are represented by ovals inscribed in these rhombuses?
2. Construct ovals corresponding to the projections of circles inscribed in the faces of a cube given in an isometric projection (following the example in Figure 65). The side of the cube is 80 mm.

§ 9. Technical drawing

To simplify the work of making visual images, technical drawings are often used.

Technical drawing- this is an image made by hand, according to the rules of axonometry, observing proportions by eye. In this case, the same rules are followed as when constructing axonometric projections: the axes are placed at the same angles, the dimensions are laid along the axes or parallel to them.

It is convenient to make technical drawings on checkered paper. Figure 70, a shows the construction using the cells of a circle. First on center lines four strokes are applied from the center at a distance equal to the radius of the circle. Then four more strokes are applied between them. Finally, draw a circle (Fig. 70, b).

It is easier to draw an oval by inscribing it in a rhombus (Fig. 70, d). To do this, as in the previous case, first strokes are applied inside the rhombus, outlining the shape of an oval (Fig. 70, c).

Rice. 70. Constructions that facilitate the execution of technical drawings

To better display the volume of an object, shading is applied to technical drawings (Fig. 71). In this case, it is assumed that the light falls on the object from the top left. Illuminated surfaces are left light, and shaded ones are covered with shading, which is more frequent the darker the surface of the object.

Rice. 71. Technical drawing of a part with shading

Look at Fig. 92. It shows a frontal dimetric projection of a cube with circles inscribed in its faces.

Circles located on planes perpendicular to the x and z axes are represented by ellipses. The front face of the cube, perpendicular to the y-axis, is projected without distortion, and the circle located on it is depicted without distortion, i.e., described by a compass. Therefore, the frontal dimetric projection is convenient for depicting objects with curvilinear outlines, such as those shown in Fig. 93.

Construction of a frontal dimetric projection of a flat part with a cylindrical hole. The frontal dimetric projection of a flat part with a cylindrical hole is performed as follows.

1. Construct the outline of the front face of the part using a compass (Fig. 94, a).

2. Straight lines are drawn through the centers of the circle and arcs parallel to the y-axis, on which half the thickness of the part is laid. The centers of the circle and arcs located on the rear surface of the part are obtained (Fig. 94, b). From these centers a circle and arcs are drawn, the radii of which must be equal to the radii of the circle and arcs of the front face.

3. Draw tangents to the arcs. Remove excess lines and outline the visible contour (Fig. 94, c).

Isometric projections of circles. A square in isometric projection is projected into a rhombus. Circles inscribed in squares, for example, located on the faces of a cube (Fig. 95), are depicted as ellipses in an isometric projection. In practice, ellipses are replaced by ovals, which are drawn with four arcs of circles.

Construction of an oval inscribed in a rhombus.

1. Construct a rhombus with a side equal to the diameter of the depicted circle (Fig. 96, a). To do this, the isometric axes x and y are drawn through point O and segments equal to the radius of the depicted circle are laid on them from point O. Through points a, w, c and d, draw straight lines parallel to the axes; get a rhombus. The major axis of the oval is located on the major diagonal of the rhombus.

2. Fit an oval into the rhombus. To do this, arcs of radius R are drawn from the vertices of obtuse angles (points A and B), equal to the distance from the vertex of the obtuse angle (points A and B) to points a, b or c, d, respectively. Straight lines are drawn through points B and a, B and b (Fig. 96, b); the intersection of these lines with the larger diagonal of the rhombus gives points C and D, which will be the centers of the minor arcs; the radius R 1 of small arcs is equal to Ca (Db). Arcs of this radius conjugate the large arcs of the oval. This is how an oval is built, lying in a plane perpendicular to the z axis (oval 1 in Fig. 95). Ovals located in planes perpendicular to the x (oval 3) and y (oval 2) axes are constructed in the same way as oval 1, only the construction of oval 3 is carried out on the y and z axes (Fig. 97, a), and ovals 2 (see Fig. 95) - on the x and z axes (Fig. 97, b).

Constructing an isometric projection of a part with a cylindrical hole.

How to apply the discussed constructions in practice?

An isometric projection of the part is given (Fig. 98, a). It is necessary to draw a through cylindrical hole drilled perpendicular to the front edge.

The construction is carried out as follows.

1. Find the position of the center of the hole on the front face of the part. Isometric axes are drawn through the found center. (To determine their direction, it is convenient to use the image of a cube in Fig. 95.) On the axes from the center, segments equal to the radius of the depicted circle are laid (Fig. 98, a).

2. Construct a rhombus, the side of which is equal to the diameter of the depicted circle; draw a large diagonal of the rhombus (Fig. 98, b).

3. Describe large oval arcs; find centers for small arcs (Fig. 98, c).

4. Draw small arcs (Fig. 98, d).

5. Construct the same oval on the back face of the part and draw tangents to both ovals (Fig. 98, e).

Answer the questions

1. What figures are depicted in the frontal dimetric projection of circles located on planes perpendicular to the x and y axes?

2. Is a circle distorted in a frontal dimetric projection if its plane is perpendicular to the y-axis?

3. When depicting what parts is it convenient to use frontal dimetric projection?

4. What figures are used to represent circles in an isometric projection located on planes perpendicular to the x, y, z axes?

5. What figures in practice replace ellipses depicting circles in isometric projection?

6. What elements does the oval consist of?

7. What are the diameters of the circles depicted as ovals inscribed in rhombuses in Fig. 95 if the sides of these rhombuses are 40 mm?

Tasks for § 13 and 14

Exercise 42

In Fig. 99 axes are drawn to construct three rhombuses representing squares in an isometric projection. Look at Fig. 95 and write down on which face of the cube - the top, right side or left side will be located each rhombus, built on the axes given in Fig. 99. Which axis (x, y or z) will the plane of each rhombus be perpendicular to?