Construction of axonometric projections

5.5.1. General provisions. Orthogonal projections of an object give a complete picture of its shape and size. However, the obvious disadvantage of such images is their low visibility - the figurative form is composed of several images made on different projection planes. Only as a result of experience does the ability to imagine the shape of an object develop—“read drawings.”

Difficulties in reading images in orthogonal projections led to the emergence of another method, which was supposed to combine the simplicity and accuracy of orthogonal projections with the clarity of the image - the method of axonometric projections.

Axonometric projection called the visual image obtained as a result parallel projection an object together with the axes of rectangular coordinates to which it is related in space to any plane.

The rules for performing axonometric projections are established by GOST 2.317-69.

Axonometry (from the Greek axon - axis, metreo - measure) is a construction process based on reproducing the dimensions of an object in the directions of its three axes - length, width, height. The result is a three-dimensional image that is perceived as a tangible thing (Fig. 56b), in contrast to several flat images that do not give a figurative form of the object (Fig. 56a).

Rice. 56. Visual representation of axonometry

IN practical work Axonometric images are used for various purposes, so different types have been created. What is common to all types of axonometry is that one or another arrangement of axes is taken as the basis for the image of any object. OX, OY, OZ, in the direction of which the dimensions of an object are determined - length, width, height.

Depending on the direction of the projecting rays in relation to the picture plane, axonometric projections are divided into:

A) rectangular– projecting rays are perpendicular to the picture plane (Fig. 57a);

b) oblique– the projecting rays are inclined to the picture plane (Fig. 57b).

Rice. 57. Rectangular and oblique axonometry

Depending on the position of the object and the coordinate axes relative to the projection planes, as well as depending on the direction of projection, units of measurement are generally projected with distortion. The sizes of projected objects are also distorted.

The ratio of the length of an axonometric unit to its true value is called coefficient distortion for a given axis.

Axonometric projections are called: isometric, if the distortion coefficients on all axes are equal ( x=y=z); dimetric, if the distortion coefficients are equal along two axes( x=z);trimetric, if the distortion coefficients are different.

For axonometric images of objects, five types of axonometric projections established by GOST 2.317 - 69 are used:

rectangular – isometric And dimetric;

oblique– frontal dimetric, frontalisometric, horizontal isometric.

Having orthogonal projections of any object, you can construct it axonometric image.

It is always necessary to choose from all types best view of this image is the one that provides good clarity and ease of constructing axonometry.

5.5.2. General order of construction. The general procedure for constructing any type of axonometry comes down to the following:

a) select coordinate axes on the orthogonal projection of the part;

b) construct these axes in an axonometric projection;

c) build an axonometry of the complete image of the object, and then its elements;

d) draw the contours of the section of the part and remove the image of the cut-off part;

d) circle the remaining part and put down the dimensions.

5.5.3. Rectangular isometric projection. This type of axonometric projection is widespread due to the good clarity of the images and the simplicity of construction. In rectangular isometry, axonometric axes OX, OY, OZ located at angles of 120 0 to one another. Axis OZ vertical. Axles OX And OY It is convenient to build by setting aside angles of 30 0 from the horizontal using a square. The position of the axes can also be determined by setting aside five arbitrary equal units from the origin in both directions. Through the fifth divisions, vertical lines are drawn down and 3 of the same units are laid on them. The actual distortion coefficients along the axes are 0.82. To simplify the construction, a reduced coefficient of 1 is used. In this case, when constructing axonometric images, measurements of objects parallel to the directions of the axonometric axes are laid aside without abbreviations. The location of the axonometric axes and the construction of a rectangular isometry of a cube, into the visible faces of which circles are inscribed, are shown in Fig. 58, a, b.

Rice. 58. Location of axes of rectangular isometry

Circles inscribed in the rectangular isometry of squares - the three visible faces of the cube - are ellipses. The major axis of the ellipse is 1.22 D, and small – 0.71 D, Where D– diameter of the depicted circle. The major axes of the ellipses are perpendicular to the corresponding axonometric axes, and the minor axes coincide with these axes and with the direction perpendicular to the plane of the cube face (thickened strokes in Fig. 58b).

When constructing a rectangular axonometry of circles lying in coordinate or parallel planes, they are guided by the rule: The major axis of the ellipse is perpendicular to the coordinate axis that is absent in the plane of the circle.

Knowing the dimensions of the ellipse axes and the projections of diameters parallel to the coordinate axes, you can construct an ellipse from all points, connecting them using a pattern.

The construction of an oval using four points - the ends of the conjugate diameters of the ellipse, located on the axonometric axes, is shown in Fig. 59.

Rice. 59. Constructing an oval

Through the point ABOUT the intersection of the conjugate diameters of the ellipse draw horizontal and vertical lines and from it describe a circle with a radius equal to half the conjugate diameters AB=SD. This circle will intersect the vertical line at points 1 And 2 (centers of two arcs). From points 1, 2 draw arcs of circles with radius R=2-A (2-D) or R=1-C (1-B). Radius OE make notches on the horizontal line and get two more centers of mating arcs 3 And 4 . Next, connect the centers 1 And 2 with centers 3 And 4 lines that intersect with arcs of radius R give junction points K, N, P, M. The extreme arcs are drawn from the centers 3 And 4 radius R 1 =3-M (4-N).

The construction of a rectangular isometry of a part, specified by its projections, is carried out in the following order (Fig. 60, 61).

1. Select coordinate axes X, Y, Z on orthogonal projections.

2. Construct axonometric axes in isometry.

3. Build the base of the part - a parallelepiped. To do this, from the origin along the axis X lay down the segments OA And OB, respectively equal to the segments O 1 A 1 And About 1 In 1, taken from the horizontal projection of the part, and get the points A And IN, through which straight lines parallel to the axes are drawn Y, and lay down segments equal to half the width of the parallelepiped.

Get points C, D, J, V, which are isometric projections of the vertices of the lower rectangle, and connect them with straight lines parallel to the axis X. From the origin ABOUT along the axis Z set aside a segment OO 1, equal to the height of the parallelepiped O 2 O 2´; through the point O 1 draw axes X 1, Y 1 and construct an isometry of the upper rectangle. The vertices of the rectangles are connected by straight lines parallel to the axis Z.

4. Construct an axonometry of the cylinder. Axis Z from O 1 set aside a segment O 1 O 2, equal to the segment О 2 ´О 2 ´´, i.e. height of the cylinder, and through the point O 2 draw axes X 2,Y2. The upper and lower bases of the cylinder are circles located in horizontal planes X 1 O 1 Y 1 And X 2 O 2 Y 2; construct their axonometric images - ellipses. The outlines of the cylinder are drawn tangentially to both ellipses (parallel to the axis Z). The construction of ellipses for a cylindrical hole is carried out similarly.

5. Construct an isometric image of the stiffener. From point O 1 along the axis X 1 set aside a segment O 1 E=O 1 E 1. Through the point E draw a straight line parallel to the axis Y, and lay on both sides segments equal to half the width of the edge E 1 K 1 And E 1 F 1. From the obtained points K, E, F parallel to the axis X 1 draw straight lines until they meet an ellipse (points P, N, M). Next, draw straight lines parallel to the axes Z(the lines of intersection of the rib planes with the surface of the cylinder), and segments are laid on them RT, MQ And N.S., equal to the segments R 2 T 2, M 2 Q 2, And N 2 S 2. Points Q, S, T connect and trace along the pattern, and the points K, T And F,Q connected by straight lines.

6. Construct a cutout of a part of a given part, for which two cutting planes are drawn: one through the axes Z And X, and the other – through the axes Z And Y.

The first cutting plane will cut the lower rectangle of the parallelepiped along the axis X(line segment OA), top – along the axis X 1, and the edge – along the lines EN And ES, cylinders - along the generatrices, the upper base of the cylinder - along the axis X 2.

Similarly, the second cutting plane will cut the upper and lower rectangles along the axes Y And Y 1, and the cylinders - along the generatrices, the upper base of the cylinder - along the axis Y2.

The flat figures obtained from the section are shaded. To determine the direction of hatching, it is necessary to plot equal segments on the axonometric axes from the origin of coordinates, and then connect their ends.

Rice. 60. Construction of three projections of a part

Rice. 61. Performing rectangular isometry of a part

Hatch lines for a section located in a plane XOZ, will be parallel to the segment 1-2 , and for a section lying in the plane ZOY, – parallel to the segment 2-3 . Remove all invisible lines and trace the contour lines. Isometric projection is used in cases where it is necessary to construct circles in two or three planes, parallel to the coordinate axes.

5.5.4. Rectangular dimetric projection. Axonometric images constructed with rectangular dimensions have the best clarity, but constructing images is more difficult than in isometry. The location of the axonometric axes in dimetry is as follows: axis OZ is directed vertically, and the axes OH And OY are made up with a horizontal line drawn through the origin of coordinates (point ABOUT), the angles are 7º10´ and 41º25´, respectively. The position of the axes can also be determined by laying eight equal segments from the origin in both directions; Through the eighth divisions, lines are drawn down and one segment is laid out on the left vertical, and seven segments are laid on the right vertical. By connecting the obtained points with the origin of coordinates, the direction of the axes is determined OH And OU(Fig. 62).

Rice. 62. Arrangement of axes in rectangular diameter

Axis distortion coefficients OH, OZ are equal to 0.94, and along the axis OY– 0.47. To simplify in practice, the following distortion coefficients are used: along the axes OX And OZ the coefficient is 1, along the axis OY– 0,5.

The construction of a rectangular cube with circles inscribed in its three visible faces is shown in Fig. 62b. Circles inscribed in faces are two types of ellipses. The axis of an ellipse located on a face that is parallel coordinate plane XOZ, are equal: major axis – 1.06 D; small – 0.94 D, Where D– the diameter of a circle inscribed in the face of a cube. In the other two ellipses the major axes are 1.06 D, and small ones - 0.35 D.

To simplify constructions, you can replace ellipses with ovals. In Fig. 63 provides techniques for constructing four center ovals that replace ellipses. An oval in the front face of a cube (rhombus) is constructed as follows. Perpendiculars are drawn from the middle of each side of the rhombus (Fig. 63a) until they intersect with the diagonals. Received points 1-2-3-4 will be the centers of the connecting arcs. The junction points of the arcs are located in the middle of the sides of the rhombus. The construction can be done in another way. From the midpoints of the vertical sides (points N And M) draw horizontal straight lines until they intersect with the diagonals of the rhombus. The intersection points will be the desired centers. From the centers 4 And 2 draw arcs with a radius R, and from the centers 3 And 1 – radius R 1.

Rice. 63. Constructing a circle in rectangular dimensions

An oval replacing the other two ellipses is made as follows (Fig. 63b). Direct LP And MN drawn through the midpoints of opposite sides of a parallelogram intersect at a point S. Through the point S draw horizontal and vertical lines. Direct LN, connecting the midpoints of adjacent sides of the parallelogram, is divided in half, and a perpendicular is drawn through its midpoint until it intersects the vertical line at the point 1 .

lay a segment on a vertical line S-2 = S-1.Direct 2-M And 1-N intersect a horizontal line at points 3 And 4 . Received points 1 , 2, 3 And 4 will be the centers of the oval. Direct 1-3 And 2-4 determine the junction points T And Q.

from centers 1 And 2 describe arcs of circles TLN And QPM, and from the centers 3 And 4 – arcs M.T. And NQ. The principle of constructing the rectangular dimetry of a part (Fig. 64) is similar to the principle of constructing the rectangular isometry shown in Fig. 61.

When choosing one or another type of rectangular axonometric projection, you should keep in mind that in rectangular isometry the rotation of the sides of the object is the same and therefore the image is sometimes not clear. In addition, often the diagonal edges of an object in the image merge into one line (Fig. 65b). These shortcomings are absent in images made in rectangular dimetry (Fig. 65c).

Rice. 64. Construction of a part in rectangular dimensions

Rice. 65. Comparison various types axonometry

5.5.5. Oblique frontal isometric projection.

The axonometric axes are located as follows. Axis OZ- vertical, axis OH– horizontal, axis OU relative to the horizontal line is located above an angle of 45 0 (30 0, 60 0) (Fig. 66a). On all axes, dimensions are plotted without abbreviations, in true size. In Fig. Figure 66b shows the frontal isometry of the cube.

Rice. 66. Construction of oblique frontal isometry

Circles located in planes parallel to the frontal plane are depicted in natural size. Circles located in planes parallel to the horizontal and profile planes are depicted as ellipses.

Rice. 67. Detail in oblique frontal isometry

The direction of the ellipse axes coincides with the diagonals of the cube faces. For planes XOY And ZОY the major axis is 1.3 D, and small – 0.54 D (D– diameter of the circle).

An example of frontal isometry of a part is shown in Fig. 67.

What is dimetria

Dimetry is one of the types of axonometric projection. Thanks to axonometry, with one three-dimensional image, you can view an object in three dimensions at once. Since the distortion coefficients of all sizes along the 2 axes are the same, this projection and was named dimetria.

Rectangular dimetry

When the Z" axis is positioned vertically, the X" and Y" axes form angles of 7 degrees 10 minutes and 41 degrees 25 minutes from the horizontal segment. In rectangular dimetry, the distortion coefficient along the Y axis will be 0.47, and along the X and Z axes twice as much, that is, 0.94.

To construct approximately axonometric axes of ordinary dimetry, it is necessary to assume that tg 7 degrees 10 minutes is equal to 1/8, and tg 41 degrees 25 minutes is equal to 7/8.

How to build dimetry

First you need to draw axes to depict the object in dimetry. In any rectangular diameter, the angles between the X and Z axes are 97 degrees 10 minutes, and between the Y and Z axes - 131 degrees 25 minutes and between Y and X - 127 degrees 50 minutes.

Now you need to plot the axes on the orthogonal projections of the depicted object, taking into account the selected position of the object for drawing in the dimetric projection. After you have completed transferring the overall dimensions of an object to a three-dimensional image, you can begin drawing minor elements on the surface of the object.

It is worth remembering that circles in each dimetric plane are represented by corresponding ellipses. In a dimetric projection without distortion along the X and Z axes, the major axis of our ellipse in all 3 projection planes will be 1.06 times the diameter of the drawn circle. And the minor axis of the ellipse in the XOZ plane is 0.95 diameters, and in the ZОY and ХОY planes it is 0.35 diameters. In a dimetric projection with distortion along the X and Z axes, the major axis of the ellipse is equal to the diameter of the circle in all planes. In the XOZ plane, the minor axis of the ellipse is 0.9 diameters, and in the ZOY and XOY planes it is 0.33 diameters.

To obtain a more detailed image, it is necessary to cut through the parts on the dimetry. When crossing out a cutout, shading should be applied parallel to the diagonal of the projection of the selected square onto the required plane.

What is isometry

Isometry is one of the types of axonometric projection, where the distances of unit segments on all 3 axes are the same. Isometric projection is widely used in mechanical engineering drawings to show appearance objects, as well as in a variety of computer games.

In mathematics, isometry is known as a transformation of metric space that preserves distance.

Rectangular isometry

In rectangular (orthogonal) isometry, the axonometric axes create angles between themselves that are equal to 120 degrees. The Z axis is in a vertical position.

How to draw isometry

Constructing an isometry of an object makes it possible to obtain the most expressive idea of ​​the spatial properties of the depicted object.

Before you start constructing a drawing in isometric projection, you need to choose such an arrangement of the depicted object so that its spatial properties are maximally visible.

Now you need to decide on the type of isometry you will draw. There are two types of it: rectangular and horizontal oblique.

Draw the axes with light, thin lines so that the image is centered on the sheet. As mentioned earlier, the angles in rectangular isometric projection should be 120 degrees.

Start drawing isometry from the top surface of the image of the object. From the corners of the resulting horizontal surface, you need to draw two vertical straight lines and mark the corresponding linear dimensions of the object on them. In an isometric projection, all linear dimensions along all three axes will remain multiples of one. Then you need to sequentially connect the created points on vertical lines. The result is the outer contour of the object.

It is worth considering that when depicting any object in an isometric projection, the visibility of curved details will necessarily be distorted. The circle should be depicted as an ellipse. The segment between the points of the circle (ellipse) along the axes of the isometric projection must be equal to the diameter of the circle, and the axes of the ellipse will not coincide with the axes of the isometric projection.

If the imaged object has hidden cavities? complex elements, try to do some shading. It can be simple or stepped, it all depends on the complexity of the elements.

Remember that all construction must be carried out strictly using drawing tools. Use multiple pencils with different types hardness

Construction of the third type based on two given

When constructing the view on the left, which is a symmetrical figure, the plane of symmetry is taken as the reference for the dimensions of the projected elements of the part, depicting it as an axial line.

The names of views in drawings made in projection connection are not indicated.

Construction of axonometric projections

For visual images of objects, products and their components unified system design documentation (GOST 2.317-69) recommends using five types of axonometric projections: rectangular - isometric and dimetric projections, oblique - frontal isometric, horizontal isometric and frontal dimetric projections.

Using orthogonal projections of any object, you can always construct its axonometric image. For axonometric constructions, they are used geometric properties flat figures, features spatial forms geometric bodies and their location relative to the projection planes.

The general procedure for constructing axonometric projections is as follows:

1. Select the coordinate axes of the orthogonal projection of the part;

2. Construct the axes of the axonometric projection;

3. Construct an axonometric image of the main shape of the part;

4. Construct an axonometric image of all elements that determine the actual shape of a given part;

5. Construct a cutout of a part of this part;

6. Put down the dimensions.

Rectangular geometric projection

The position of the axis in a rectangular isometric projection is shown in Fig. 17.12. The actual distortion coefficients along the axes are 0.82. In practice, the given coefficients are used, equal to 1. In this case, the images are enlarged by 1.22 times.

Methods for constructing isometric axes

The direction of axonometric axes in isometry can be obtained in several ways (see Fig. 11.13).

The first method is using a 30° square;

The second method is to divide a circle of arbitrary radius into 6 parts with a compass; straight line O1 is the x axis, straight line O2 is the oy axis.

The third way is to construct the ratio of parts 3/5; lay down five parts along a horizontal line (we get point M) and down three parts (we get point K). Connect the resulting point K to the center O. ROKOM is equal to 30°.

Methods for constructing flat figures in isometry

In order to correctly construct an isometric image of spatial figures, you must be able to construct the isometry of plane figures. To construct isometric images, you must perform the following steps.

1. Give the appropriate direction to the x and oy axes in isometry (30°).

2. On the ox and oy axes, plot the natural (in isometry) or abbreviated along the axes (in dimetry - along the oy axis) values ​​of the segments (coordinates of the vertices of the points.

Since the construction is carried out according to the given distortion coefficients, the image is obtained with magnification:

for isometry – 1.22 times;

the construction progress is shown in Fig. 11.14.

In Fig. 11.14a gives orthogonal projections of three flat figures - hexagon, triangle, pentagon. In Fig. 11.14b, isometric projections of these figures are constructed in different axonometric planes - xou, yoz.

Constructing a circle in rectangular isometry

In rectangular isometry, the ellipses representing a circle of diameter d in the planes xou, xoz, yoz are the same (Fig. 11.15). Moreover, the major axis of each ellipse is always perpendicular to the coordinate axis that is absent in the plane of the depicted circle. Major axis of the ellipse AB = 1.22d, minor axis CD = 0.71d.

When constructing ellipses, the directions of the major and minor axes are drawn through their centers, on which segments AB and CD are respectively laid, and straight lines parallel to the axonometric axes, on which segments MN are laid, equal to the diameter of the depicted circle. The resulting 8 points are connected according to the pattern.

In technical drawing, when constructing axonometric projections of circles, ellipses can be replaced by ovals. In Fig. Figure 11.15 shows the construction of an oval without defining the major and minor axes of the ellipse.

The construction of a rectangular isometric projection of a part defined by orthogonal projections is carried out in the following order.

1. On orthogonal projections, select coordinate axes, as shown in Fig. 11.17.

2. Construct the x, y, z coordinate axis in an isometric projection (Fig. 11.18)

3. Build a parallelepiped - the base of the part. To do this, from the origin of coordinates along the x axis, segments OA and OB are laid off, respectively equal to segments o 1 a 1 and o 1 b 1 on the horizontal projection of the part (Fig. 11.17) and points A and B are obtained.

Through points A and B, draw straight lines parallel to the y-axis, and lay off segments equal to half the width of the parallelepiped. We obtain points D, C, J, V, which are isometric projections of the vertices of the lower rectangle. Points C and V, D and J are connected by straight lines parallel to the x axis.

From the origin of coordinates O along the z axis, a segment OO 1 is laid off, equal to the height of the parallelepiped O 2 O 2 ¢, the x 1, y 1 axes are drawn through the point O 1 and an isometric projection of the upper rectangle is constructed. The vertices of the rectangle are connected by straight lines parallel to the z axis.

4. construct an axonometric image of a cylinder of diameter D. Along the z axis from O 1, a segment O 1 O 2 is laid out, equal to the segment O 2 O 2 2, i.e. height of the cylinder, obtaining point O 2 and drawing the x 2, y 2 axes. The upper and lower bases of the cylinder are circles located in the horizontal planes x 1 O 1 y 1 and x 2 O 2 y 2. An isometric projection is constructed similarly to the construction of an oval in the xOy plane (see Fig. 11.18). The outlines of the cylinder are drawn tangent to both ellipses (parallel to the z axis). The construction of ellipses for a cylindrical hole with diameter d is performed in a similar way.

5. Construct an isometric image of the stiffener. From point O 1 along the x 1 axis, a segment O 1 E equal to oe is plotted. Through point E, draw a straight line parallel to the y-axis and lay off a segment on both sides equal to half the width of the edge (ek and ef). Points K and F are obtained. From points K, E, F, straight lines are drawn parallel to the x 1 axis until they meet the ellipse (points P, N, M). Straight lines are drawn parallel to the z axis (the line of intersection of the rib planes with the surface of the cylinder), and the segments PT, MQ and NS, equal to the segments p 3 t 3, m 3 q 3, n 3 s 3, are laid on them. Points Q, S, T are connected and traced along the pattern, from points K, T and F, Q are connected with straight lines.

6. Construct a cutout of a part of a given part.

Two cutting planes are drawn: one through the z and x axes, and the other through the z and y axes. The first cutting plane will cut the lower rectangle of the parallelepiped along the x-axis (segment OA), the upper one along the x1 axis, the edge along the lines EN and ES, the cylinders with diameters D and d along the generators, the upper base of the cylinder along the x2 axis. Similarly, the second cutting plane will cut the upper and lower rectangle along the y and y axes 1, and the cylinders along the generatrices and the upper base of the cylinder along the y axis 2. The planes obtained from the section are shaded. In order to determine the direction of the hatching lines, it is necessary to plot equal segments O1, O2, O3 from the origin of coordinates on the axonometric axes drawn next to the image (Fig. 11.19), and connect the ends of these segments. Hatch lines for sections located in the xOz plane should be drawn parallel to segment I2, for a section lying in the zOy plane - parallel to segment 23.

Remove all invisible lines and construction lines and trace the contour lines.

7. Put down the dimensions.

To apply dimensions, extension and dimension lines are drawn parallel to the axonometric axes.

Rectangular dimetric projection

The construction of coordinate axes for a dimetric rectangular projection is shown in Fig. 11.20.

For a dimetric rectangular projection, the distortion coefficients along the x and z axes are 0.94, and along the y axis – 0.47. In practice, the reduced distortion coefficients are used: along the x and z axes the reduced distortion coefficient is 1, along the y axis - 0.5. In this case, the image is obtained 1.06 times.

Methods for constructing flat figures in dimetry

In order to correctly construct a dimetric image of a spatial figure, you must perform the following steps:

1. Give the appropriate direction to the axes x and oy, in dimetry (7°10¢; 41°25¢).

2. Plot the natural values ​​along the x, z axes, and the reduced values ​​of the segments (coordinates of the vertices of the points) along the y axis according to the distortion coefficients.

3. Connect the resulting points.

The construction progress is shown in Fig. 11.21. In Fig. 11.21a gives orthogonal projections of three plane figures. In Fig. 11.21b, the construction of dimetric projections of these figures in different axonometric planes is hou; уоz/

Constructing a circle of rectangular diameter

The axonometric projection of a circle is an ellipse. The direction of the major and minor axis of each ellipse is indicated in Fig. 11.22. For planes parallel to the horizontal (xy) and profile (yoz) planes, the magnitude of the major axis is 1.06d, the minor axis is 0.35d.

For planes parallel to the frontal plane xoz, the magnitude of the major axis is 1.06d, and the minor axis is 0.95d.

In technical drawing, when constructing a circle, ellipses can be replaced by ovals. In Fig. Figure 11.23 shows the construction of an oval without defining the major and minor axes of the ellipse.

The principle of constructing a dimetric rectangular projection of a part (Fig. 11.24) is similar to the principle of constructing an isometric rectangular projection shown in Fig. 11.22, taking into account the distortion coefficient along the y-axis.

1

As already discussed, the axes of the isometric projection are located at an angle of 120° to each other.

They can be built in several ways.

A. Using a compass. Initially, draw the axis and select the intersection point of the axes on it ABOUT. From the point ABOUT draw an arc of any radius intersecting the axis at a point 1. From it, with the same radius on the arc, serifs are made at points 3 , 4 , through which the axes are drawn (Fig. 2.48).

B. The construction of axes using a ruler and a square with angles of 30°, 60° and 90° is shown in Fig. 2.49. Axles hiu carried out at an angle of 30° to the horizontal line.

ISOMETRIC PROJECTIONS OF POLYGONS

The construction of an isometric projection of objects usually begins with the image of some of its faces, which are based on flat figures. Let's consider the construction of some polygons based on given rectangular projections.

For all constructions, the x and axes are initially drawn at on rectangular projections and the corresponding axes in isometric projection, i.e. They link rectangular and axonometric axes.

A. Construction of a triangle located in a horizontal plane (Fig. 2.50). From point ABOUT plot along the x-axis segments equal to half the side of the triangle, and along the x-axis y - its height AND. The resulting points are connected by straight segments.

Triangles located in the frontal and profile planes are constructed similarly (Fig. 2.51).

B. Construction of a square located in a horizontal plane (Fig. 2.52). A segment is laid along the x-axis A, equal to the side of the square, along the axis y - line segment b, from the obtained points, draw segments parallel to the x and axes u.

B. Construction of a hexagon located in a horizontal plane (Fig. 2.53).

Construction of hexagons in planes n 2 And n 3 shown in Fig. 2.53, b.

To construct a hexagon, it is advisable to choose the axes of the isometric projection so that they pass through the center of the hexagon. Along the x-axis to the right and left of the point ABOUT lay down the segments equal to side hexagon. Along the y-axis symmetrically to the point ABOUT lay off segments equal to half the distance h between opposite parties.

From points obtained on the axis y, Draw segments equal to half the side of the hexagon to the right and left parallel to the x-axis. The resulting points are connected by straight segments.

When constructing the contours of complex, asymmetrical figures (Fig. 2.54), their vertices are 7, 2, ..., 7 are found by measuring the markings x p x 2, x 3, x 4, x 5 on a rectangular projection, and transferring them to an axis or straight lines parallel to this axis of the isometric projection. Do the same with sizes. at R y 2, y y 4. At the intersection of the corresponding lines, the vertices of a given flat figure are found and connected to each other.

Questions and tasks

• 1. In what sequence is a triangle constructed in isometric projection? Any flat figure?
• 2. From the problem book, complete one of the variants of task No. 32. In it you need to construct isometric projections of “flat” figures in the frontal and profile projection planes.

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

The essence of the axonometric projection method is that a given object, together with the coordinate system to which it is assigned 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. The distortion coefficient is understood as the ratio of the image size in an axonometric projection to the image size in an orthogonal projection.

Depending on the ratio of 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 plane of projections, 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 plane of projections, then such a projection is called oblique. Oblique axonometric projections include frontal isometric, horizontal isometric and frontal dimetric projections.

In 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 ease of construction, the indicator is taken equal to 1. As a result, the axonometric image turns out to be enlarged by times.

The isometric axes are shown in Figure 57.

Figure 57

The construction of isometric axes can be done 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, which intersects the Z axis at point A. From point A, draw a second circle with the same radius to intersections with the first at points B and C. The resulting point B is connected to point O - the direction of the X axis is obtained. In the same way, point C is connected to point O - the direction of the Y axis is obtained.

Figure 58

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

Figure 59

Constructing a circle in a rectangular isometric projection

The most difficult flat figure to draw in axonometry is a circle. As is known, a circle in isometry is projected into an ellipse, but constructing an ellipse is quite difficult, therefore 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 ones.

The size of the major axis of the ellipse is 1.22d, the minor axis is 0.7d, where d is the diameter of the circle whose isometry is being constructed. Figure 60 shows a graphical method for determining the major and minor axes of an isometric ellipse. To determine the minor axis of the ellipse, connect points C and D. From points C and D, as from centers, arcs of radii equal to CD are drawn until they intersect each other. 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 points O 1, O 2, O 3, O 4 are marked, which are the centers oval arcs (Figure 61).

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

Figure 61

The direction of the ellipse or oval axes 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 at a point, and the minor axis coincides with the direction of this axis (Figure 62).

Figure 62

Hatching and isometric projection

Hatch lines of sections in an 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 rates along the two axes X and Z, and along the Y axis the distortion rate is half as much.

According to GOST 2.317-69, in rectangular diameter the Z axis is used, located vertically, the X axis inclined at an angle of 7°, and the Y axis at an angle of 41° to the horizon line. The distortion indicators for the X and Z axes are 0.94, and for the Y axis - 0.47. Usually the given coefficients are used: k x =k z =1, k y =0.5, i.e. along the X and Z axes or in directions parallel to them, the actual dimensions are plotted, and along the Y axis the dimensions are halved.

To construct dimetric axes, use the method indicated in Figure 63, which is as follows:

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

Figure 63

Constructing a dimetric projection of a hexagon

Let's consider the construction in dimetry of a regular hexagon located in the plane P1 (Figure 64).

Figure 64

On the X axis we plot a segment equal to the value b, to let him the middle was at point O, and along the Y axis there was a segment A, the size of which is halved. Through the obtained points 1 and 2 we draw straight lines parallel to the OX axis, on which we lay down segments equal to the side of the hexagon in full size with the middle at points 1 and 2. We connect the resulting vertices. Figure 65a shows a hexagon in dimetry, located parallel to the frontal plane, and in Figure 66b, parallel to the profile plane of projection.

Figure 65

Constructing a circle in dimetry

In rectangular dimetry, all circles are depicted as ellipses,

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

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

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

Figure 66

Then, from points O1 and O2 we draw arcs whose radius 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

We will consider the construction of an oval, replacing an ellipse, from a circle located in the P 2 plane in Figure 68. We draw the dimetric axes: 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 plot the radius of the circle from the beginning and get 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 the perpendicular to it, give points O 1, O 2, O 3, O 4 - the centers of the oval arcs (Figure 68).

From centers O 3 and O 4 they describe an arc of radius R 2 = O 3 M, and from centers O 1 and O 2 - arcs of radius R 1 = O 2 N

Figure 68

Hatching of rectangular diameter

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 located in isometry?
3. What shape does the 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 to construct a dimetric projection?
6. At what angles are the axes in dimetry located?
7. What figure will be the dimetric projection of the square?
8. How to construct a dimetric projection of a circle located in the frontal plane of the projections?
9. Basic rules for applying shading in axonometric projections.