Axonometry rectangular isometry. Construction of an axonometric image of a part

You can display various geometric objects using drawings and computer graphics using the principles of isometry and axonometry. What are the specifics of each of them?

What is axonometry?

Under axonometry or axonometric projection refers to a method of graphically displaying certain geometric objects through parallel projections.

Axonometry

In this case, a geometric object is most often drawn using a specific coordinate system - so that the plane on which it is projected does not correspond to the position of the plane of other coordinates of the corresponding system. It turns out that the object is displayed in space through 2 projections and looks three-dimensional.

Moreover, for the reason that the display plane of the object is not located strictly parallel to any of the axes of the coordinate system, individual elements of the corresponding display may be distorted - according to one of the following 3 principles.

Firstly, distortion of object display elements can be observed along all 3 axes used in the system, to an equal extent. In this case, the isometric projection of the object, or isometry, is fixed.

Secondly, distortion of elements can only be observed along 2 axes in equal amounts. In this case, a dimetric projection is observed.

Third, the distortion of elements can be recorded as varying along all 3 axes. In this case, a trimetric projection is observed.

Let us therefore consider the specifics of the first type of distortions formed within the framework of axonometry.

What is isometry?

So, isometry- this is a type of axonometry that is observed when drawing an object if the distortion of its elements along all 3 coordinate axes is the same.

The type of axonometric projection under consideration is actively used in industrial design. It allows you to clearly view certain details within the drawing. The use of isometrics is also widespread in the development of computer games: with the help of the appropriate type of projection, it becomes possible to effectively display three-dimensional images.

It can be noted that in the field of modern industrial developments, isometry is generally understood as a rectangular projection. But sometimes it can be presented in an oblique variety.

Comparison

The main difference between isometry and axonometry is that the first term corresponds to a projection, which is only one of the varieties of the one denoted by the second term. Isometric projection, therefore, differs significantly from other types of axonometry - dimetry and trimetry.

Let us display more clearly the difference between isometry and axonometry in a small table.

Instructions

Construct using a ruler and protractor or compass and ruler for a rectangular (otrogonal) isometric projection. In this type of axonometric projection, all three axes - OX, OY, OZ - have angles of 120° between themselves, while the OZ axis has a vertical orientation.

For simplicity, draw an isometric projection without distortion along the axes, since it is customary to equate the isometric distortion coefficient to unity. By the way, “isometric” itself means “equal size”. In fact, when mapping a three-dimensional object onto a plane, the ratio of the length of any projected segment parallel to the coordinate axis to the actual length of this segment is equal to 0.82 for all three axes. Therefore, the linear dimensions of an object in isometry (with the accepted distortion coefficient) increase by 1.22 times. In this case, the image remains correct.

Start projecting the object onto the axonometric plane from its top edge. Measure the height of the part along the OZ axis from the center of intersection of the coordinate axes. Draw thin lines on the X and Y axes through this point. From the same point, lay down half the length of the part along one axis (for example, along the Y axis). Draw a segment of the required size (part width) through the found point parallel to the other axis (OX).

Now along the other axis (OX) set aside half the width. Through this point, draw a segment of the required size (part length) parallel to the first axis (OY). The two drawn segments must intersect. Complete the rest of the top edge.

If there is a round hole in this face, draw it. In isometry, a circle is depicted as an ellipse because we look at it at an angle. Calculate the dimensions of the axes of this ellipse based on the diameter of the circle. They are equal: a = 1.22D and b = 0.71D. If the circle is located on a horizontal plane, the a-axis of the ellipse is always horizontal, and the b-axis is vertical. In this case, the distance between the points of the ellipse on the X or Y axis is always equal to the diameter of the circle D.

Draw vertical edges from the three corners of the top edge equal to the height of the part. Connect the edges through their lowest points.

If the shape has a rectangular hole, draw it. Place a vertical (parallel to the Z axis) segment of the required length from the center of the edge of the top face. Through the resulting point, draw a segment of the required size parallel to the top edge, and therefore the X axis. From the extreme points of this segment, draw vertical edges of the required size. Connect their lower points. From the bottom right point of the drawn diamond, draw the inner edge of the hole, which should be parallel to the Y axis.

Axonometric (Axonometry translated from Greek (“achop” - axis; “metreo” - measure) means an octagonal image.) projections are images obtained by projecting parallel rays of a figure (object) together with coordinate axes onto an arbitrarily located plane, which is called "axonometric"(or picture). Typically, the plane (or object) is positioned so that three sides are visible on the axonometric projection of the object: top (or bottom), front and left (or right).
The main advantage of axonometric projections is clarity and an idea of ​​the size of the depicted object, therefore they are used as an illustration to the drawing to facilitate understanding of the structural form of the object. Figure 270 shows an axonometric projection of the part.

The following notations are used on axonometric projections: the axonometric plane is designated P"; the axonometric coordinate axes are x", y", z"; axonometric projections of points A, B, etc. are designated A", B", etc. The origin of coordinates is designated O".
2. Types of axonometric projections.
Depending on the direction of the projecting rays, axonometric projections are divided into: rectangular or orthogonal (projecting rays are perpendicular to the axonometric plane P") and oblique (projecting rays are inclined to the axonometric plane).
Depending on the inclination of the coordinate axes to the axonometric plane, and, consequently, on the degree of reduction in the size of the axonometric projections of segments having the direction of the coordinate axes (It is known that a straight segment inclined to a plane is projected onto it reduced; the greater the angle of inclination, the smaller the projection of the segment.), - all axonometric projections are divided into three main types:
1) isometric, i.e. of the same dimension (the z, x, and y axes are inclined equally; therefore, the reduction in size along the direction of all three axes is the same);
2) dimetric, i.e. double dimension (two coordinate axes have the same slope, and the third - another; therefore, the reduction in size along these two axes will be the same, and along the third axis - another);
3) trimetric, i.e. triple dimension (all axes have different slopes; therefore, the reduction in dimensions in the direction of all three axes is different).
In mechanical engineering drawing, from rectangular axonometric projections, isometric and dimetric ones are most often used, and from oblique angles - dimetric, which is otherwise called frontal dimetric projection.
In an isometric projection, the angles between the axonometric axes x", y" and z" are the same (120° each); the z" axis is located vertically; therefore, the x" and y" axes are inclined to the horizontal line at an angle of 30° (Fig. 271, a).

With this position of the axes, the distortion indicators for all axes are the same and equal to 0.82.
The distortion indicator is the ratio of the size of the axonometric projection of a segment in the direction of any coordinate axis to its actual size. For example, with an actual size of 100 mm and a distortion index of 0.82, the size of the axonometric projection is 100 × 0.82 = 82 mm.
In a dimetric projection, the angle between the axonometric axes z" and x" is equal to 97°10", and the angles between the axonometric axes x" and y", as well as z" and y" are the same, i.e. 131°25". The axonometric z-axis has a vertical position, therefore, the x-axis is inclined to the horizontal line at an angle of 7°10" and the y-axis at an angle of 41°25" (Fig. 271, b).
With such a tilt of the axonometric axes, the distortion indicator for the z" and x" axes is 0.94, and for the y" axis - 0.47.
In the frontal dimetric projection, the angle between the axonometric axes z" and x" is equal to 90°, and the angles between the axonometric axes x" and y", as well as between the axonometric axes z" and y" are the same, i.e. 135°. The z" axis has a vertical position, therefore, the x" axis will have a horizontal position, and the y" axis is inclined to the horizontal line at an angle of 45° (Fig. 271, c).
The distortion indicators along the axonometric axes x" and z" are equal to 1.0 and along the y" axis - 0.5.
This frontal dimetric projection is called cabinet; it is recommended to be used when they want to show without changing the outlines of figures located in planes parallel to the frontal plane of projections.
To compare images made in axonometric projections, (Fig. 272) shows different axonometric projections of the same cube.

To simplify the calculation of distortion indicators, GOST 3453-59 recommends constructing an isometric projection without reduction along the axonometric axes x", y" and z", and a dimetric projection without reduction along the axonometric axes x" and y", and with a reduction of 0.5 along the axonometric axis y" . In this case, the image turns out to be slightly enlarged, but its clarity does not deteriorate.

Lecture 6. Axonometric projections

1. General information about axonometric projections.

2. Classification of axonometric projections.

3. Examples of constructing axonometric images.

1 General information about axonometric projections

When drawing up technical drawings, sometimes it becomes necessary, along with images of objects in the system of orthogonal projections, to have more visual images. For such images, the method is used axonometric projection(axonometry is a Greek word, literally translated it means measurement along axes; axon - axis, metreo - measure).

The essence of the axonometric projection method: an object, together with the axes of rectangular coordinates to which it is related in space, is projected onto a certain plane so that none of its coordinate axes is projected onto it to a point, which means the object itself is projected onto this projection plane in three dimensions.

Fuck it. 88 the coordinate system x, y, z located in space is projected onto a certain projection plane P. Projections x р, y р,

z p coordinate axes on the plane P are called axonometric axes.

Figure 88

Equal segments e are plotted on the coordinate axes in space. As can be seen from the drawing, their projections e x , e y , e z onto the plane P in general

case are not equal to the segment e and are not equal to each other. This means that the dimensions of an object in axonometric projections along all three axes are distorted. The change in linear dimensions along the axes is characterized by distortion indicators (coefficients) along the axes.

Distortion index is the ratio of the length of a segment on the axonometric axis to the length of the same segment on the corresponding axis of a rectangular coordinate system in space.

The distortion indicator along the x-axis is denoted by the letter k, along the y-axis

– letter m, along the z axis – letter n, then: k = e x /e; m = e y /e; n = e z /e.

The magnitude of the distortion indicators and the relationship between them depend on the location of the projection plane and the direction of projection.

In the practice of constructing axonometric projections, they usually use not the distortion coefficients themselves, but some values ​​proportional to the values ​​of the distortion coefficients: K:M:N = k:m:n. These quantities are called given distortion coefficients.

2 Classification of axonometric projections

The entire set of axonometric projections is divided into two groups:

1 Rectangular projections – obtained with a projection direction perpendicular to the axonometric plane.

2 Oblique projections – obtained with a projection direction chosen at an acute angle to the axonometric plane.

In addition, each of these groups is also divided according to the ratio of axonometric scales or distortion indicators (coefficients). Based on this feature, axonometric projections can be divided into the following types:

a) Isometric - the distortion indicators on all three axes are the same (isos - the same).

b) Dimetric - distortion indicators along two axes are equal to each other, but the third is not equal (di - double).

c) Trimetric - distortion indicators on all three axes are not equal

us among ourselves. This is axonometry (does not have much practical application).

2.1 Rectangular axonometric projections

Rectangular isometric projection

IN rectangular isometry, all coefficients are equal between

k = m = n, k2 + m2 + n2 =2,

then this equality can be written in the form 3k 2 =2, whence k =.

Thus, in isometry, the distortion index is ~0.82. This means that in a rectangular

isometry, all dimensions of the depicted object are reduced by 0.82 times. For

Rectangular dimetric projection

In rectangular dimetry, the distortion indicators along two axes are the same, i.e. k = p. Third

We choose the distortion indicator to be half as large as the other two, i.e. m =1/2k. Then the equality k 2 +m 2 +n 2 = 2 will take the following form: 2k 2 +1/4k 2 =2; from where k= 0.94;

Figure 91

Figure 92

are equal: k = n=1.

2.2 Oblique projections

Frontal isometric view

In Fig. 91 shows the position of the axonometric axes for frontal isometry.

According to GOST 2.317-69, it is allowed to use frontal isometric projections with a y-axis inclination angle of 30° and 60°. The distortion factors are exact and equal to:

k = m = n=1.

Horizontal isometric projection

In Fig. 92 shows the position of the axonometric axes for frontal isometry. According to GOST 2.317-69, it is allowed to use horizontal isometric projections with a y-axis tilt angle of 45° and 60° while maintaining the angle between the x and y axes at 90°. The distortion coefficients are exact and equal to: k=m= n= 1.

Frontal dimetric projection

The position of the axes is the same as for frontal isometry (Fig. 91). It is also possible to use frontal dimetry with a y-axis inclination angle of 30° and 60°.

Distortion factors are accurate and m=0.5

All three types of standard oblique projections are obtained by placing one of the coordinate planes (horizontal or frontal) parallel to the axonometric plane. Therefore, all figures located in these planes or parallel to them are projected onto the drawing plane without distortion.

3 Examples of constructing axonometric images

Both in rectangular (orthogonal projections) and in axonometric projections, one projection of a point does not determine its position in space. In addition to the axonometric projection of a point, it is necessary to have another projection, called secondary. Secondary point projection- this is an axonometry of one of its rectangular projections (usually horizontal).

Techniques for constructing axonometric images do not depend on the type of axonometric projections. For all projections, the construction techniques are the same. An axonometric image is usually built on the basis of rectangular projections of an object.

3.1 Axonometry of a point

We begin constructing the axonometry of a point based on its given orthogonal projections (Fig. 93, a) by determining its secondary projection (Fig. 93, b). To do this, on the axonometric x axis from the origin of coordinates we plot the value of the X coordinates of point A - X A; along the y axis – segment Y A (for diameter Y A ×0.5, since the distortion indicator along this axis is m=0.5).

At the intersection of communication lines drawn parallel to the axes from the ends of the measured segments, point A 1 is obtained - a secondary projection of point A.

The axonometry of point A will be at a distance Z A from the secondary projection of point A.

Figure 93

3.2 Axonometry of a straight segment (Fig. 94)

We find secondary projections of points A, B. To do this, plot the corresponding coordinates of points A and B along the x and y axes. Then mark on straight lines drawn from secondary projections parallel to the z axis, the heights of points A and B (Z A and Z B). We connect the resulting points - we get the axonometry of the segment.

Figure 94

3.3 Axonometry of a flat figure

In Fig. Figure 95 shows the construction of an isometric projection of triangle ABC. We find secondary projections of points A, B, C. To do this, plot along the x and y axes the corresponding coordinates of points A, B and C. Then we mark on straight lines drawn from secondary projections parallel to the z axis, the heights of points A, B and C. We connect the resulting points with lines - we get the axonometry of the segment.

Figure 95

If a flat figure lies in the projection plane, then the axonometry of such a figure coincides with its projection.

3.4 Axonometry of circles located in projection planes

Circles in axonometry are depicted as ellipses. To simplify constructions, the construction of ellipses is replaced by the construction of ovals outlined by circular arcs.

Rectangular circle isometry

a certain value of the diameter of the depicted circle, and find the points n of intersection of this circle with the axonometric axes x and y.

From the points O 1, O 2 of the intersection of the auxiliary circle with the z axis, as

from centers with radius R = O 1 n = O 2 n, draw two arcs nDn and pСn circles belonging to the oval.

The construction of ovals lying in planes parallel to the planes π 2 and π 3 begins by drawing the horizontal AB and vertical CD axes of the oval:

AB x axis for an oval lying in a plane parallel to the planes π 3;

AB y-axis for an oval lying in a plane parallel to

planes π 2; Further construction of ovals is similar to the construction of an oval,

lying in a plane parallel to π1.

Figure 98

Rectangular dimetry of a circle (Fig. 99)

In Fig. 99 in rectangular isometry shows a cube with an edge α, in the faces of which circles are inscribed. Two faces of the cube will be depicted as equal parallelograms with sides equal to 0.94d and 0.47d, the third face - as a rhombus with sides equal to 0.94d. Two circles inscribed in the faces of a cube are projected as identical ellipses, the third ellipse is close in shape to a circle.

equal to 1.06 d, the minor axes of two ellipses are equal to 0.35 d, the minor axis of the third ellipse is equal to 0.94 d.

dimetric projection of a circle located parallel to the π 2 plane (Figure 100, a).

Through point O we draw axes parallel to the x and z axes. From the center O with a radius equal to the radius of the given circle, we draw an auxiliary circle that intersects with the axes at points 1, 2, 3, 4. From points 1 and 3 (in the direction of the arrows) we draw horizontal lines until they intersect with the axes AB and CD of the oval and get points O 1, O 2, O 3, O 4. Taking points O 1, O 4 as centers, we draw arcs 1 2 and 3 4 with radius R. Taking points O 2 and O 3 as centers, we draw arcs of radius R 1 closing the oval.

Let us analyze the simplified construction of a dimetric projection of a circle lying in the π 1 plane (Figure 100, c).

Through the intended point O we draw straight lines parallel to the x and y axes, as well as the major axis of the oval AB perpendicular to the minor axis CD. From the center O with a radius equal to the radius of the given circle, we draw an auxiliary circle and obtain points n and n 1.

On a straight line parallel to the z axis, to the right and left of the center O

we set aside segments equal to the diameter of the auxiliary circle and get points O 1 and O 2. Taking these points as centers, we draw oval arcs with radius R = O 1 n 1. Connecting points O 2 with straight lines to the ends of the arc n 1 n 2, on the line of the major axis AB of the oval we obtain points O 4 and O 3. Taking them as centers, we draw arcs of radius R 1 closing the oval.

Figure 100

3.5 Axonometry of a geometric body

Axonometry of a hexagonal prism (Fig. 101)

The base of a straight prism is a regular hexagon

Construction of an axonometric image of the part, the drawing of which is shown in Fig.a.

All axonometric projections must be carried out in accordance with GOST 2.317-68.

Axonometric projections are obtained by projecting an object and its associated coordinate system onto one projection plane. Axonometry is divided into rectangular and oblique.

For rectangular axonometric projections, the projection is carried out perpendicular to the projection plane, and the object is positioned so that all three planes of the object are visible. This is possible, for example, when the axes are located as on a rectangular isometric projection, for which all projection axes are located at an angle of 120 degrees (see Fig. 1). The word "isometric" projection means that the distortion coefficient is the same on all three axes. According to the standard, the distortion coefficient along the axes can be taken equal to 1. The distortion coefficient is the ratio of the size of the projection segment to the true size of the segment on the part, measured along the axis.

Let's build an axonometry of the part. First, let's set the axes as for a rectangular isometric projection. Let's start from the foundation. Let us plot along the x-axis the value of the length of the part 45, and along the y-axis the value of the width of the part 30. From each point of the quadrilateral we will raise vertical segments to the top by the height of the base of the part 7 (Fig. 2). On axonometric images, when drawing dimensions, extension lines are drawn parallel to the axonometric axes, dimension lines are drawn parallel to the measured segment.

Next, we draw the diagonals of the upper base and find the point through which the axis of rotation of the cylinder and the hole will pass. We erase the invisible lines of the lower base so that they do not interfere with our further construction (Fig. 3)

.

The disadvantage of a rectangular isometric projection is that circles in all planes will be projected into ellipses in the axonometric image. Therefore, first we will learn how to construct approximately ellipses.

If you inscribe a circle into a square, then you can mark 8 characteristic points: 4 points of contact between the circle and the middle of the side of the square and 4 points of intersection of the diagonals of the square with the circle (Fig. 4, a). Figure 4, c and Figure 4, b show the exact method of constructing the points of intersection of the diagonal of a square with a circle. Figure 4d shows an approximate method. When constructing axonometric projections, half of the diagonal of the quadrilateral into which the square is projected will be divided in the same ratio.

We transfer these properties to our axonometry (Fig. 5). We construct a projection of a quadrilateral into which a square is projected. Next, we build the ellipse Fig. 6.

Next, we rise to a height of 16mm and transfer the ellipse there (Fig. 7). We remove unnecessary lines. Let's move on to creating holes. To do this, we build an ellipse on the top into which a hole with a diameter of 14 will be projected (Fig. 8). Next, to show a hole with a diameter of 6mm, you need to mentally cut out a quarter of the part. To do this, we will construct the middle of each side, as in Fig. 9. Next, we build an ellipse corresponding to a circle with a diameter of 6 on the lower base, and then at a distance of 14 mm from the top of the part we draw two ellipses (one corresponding to a circle with a diameter of 6, and the other corresponding to a circle with a diameter of 14) Fig. 10. Next, we make a quarter section of the part and remove the invisible lines (Fig. 11).

Let's move on to constructing the stiffener. To do this, on the upper plane of the base, measure 3 mm from the edge of the part and draw a segment half the thickness of the rib (1.5 mm) (Fig. 12), and also mark the rib on the far side of the part. An angle of 40 degrees is not suitable for us when constructing axonometry, so we calculate the second leg (it will be equal to 10.35 mm) and use it to construct the second point of the angle along the plane of symmetry. To construct the edge boundary, we draw a straight line at a distance of 1.5 mm from the axis on the upper plane of the part, then draw lines parallel to the x axis until they intersect with the outer ellipse and lower the vertical line. Through the lower point of the rib boundary, draw a straight line parallel to the rib along the cut plane (Fig. 13) until it intersects with the vertical line. Next, we connect the intersection point with a point in the cut plane. To construct the far edge, draw a straight line parallel to the X axis at a distance of 1.5 mm to the intersection with the outer ellipse. Next, we find at what distance the upper point of the rib border is located (5.24mm) and put the same distance on a vertical straight line on the far side of the part (see Fig. 14) and connect it to the far lower point of the rib.

We remove the extra lines and hatch the section planes. Hatch lines of sections in axonometric projections are drawn parallel to one of the diagonals of the projections of squares lying in the corresponding coordinate planes, the sides of which are parallel to the axonometric axes (Fig. 15).

For a rectangular isometric projection, the hatch lines will be parallel to the hatch lines shown in the diagram in the upper right corner (Fig. 16). All that remains is to draw the side holes. To do this, mark the centers of the axes of rotation of the holes, and build ellipses, as indicated above. We similarly construct the radii of roundings (Fig. 17). The final axonometry is shown in Fig. 18.

For oblique projections, projection is carried out at an angle to the projection plane other than 90 and 0 degrees. An example of an oblique projection is an oblique frontal dimetric projection. It is good because on the plane defined by the X and Z axes, circles parallel to this plane will be projected to their true size (the angle between the X and Z axes is 90 degrees, the Y axis is inclined at an angle of 45 degrees to the horizontal). “Dimetric” projection means that the distortion coefficients along the two axes X and Z are the same, and along the Y axis the distortion coefficient is half as much.

When choosing an axonometric projection, you must strive to ensure that the greatest number of elements are projected without distortion. Therefore, when choosing the position of a part in an oblique frontal dimetric projection, it must be positioned so that the axes of the cylinder and holes are perpendicular to the frontal plane of the projections.

The layout of the axes and the axonometric image of the “Stand” part in an oblique frontal dimetric projection are shown in Fig. 18.