The distortion coefficient in a rectangular isometric projection is equal to. Isometric projection

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 the length of the part 45 along the x-axis, and the width of the part 30 along the y-axis. 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. Hatching 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.

6.1. General provisions

Complex (technical) drawings are constructed using the method of rectangular projection on projection planes, and the number of images of the object in these drawings should be the smallest, but fully revealing its shape and dimensions. Such drawings are reversible, easily measurable, but not sufficiently visual, since the spatial image of an object in the mind very often has to be reproduced from several of its images. Therefore, there was a need for drawings that would be visual, but at the same time reversible and give general idea about the relative size and shape of an object.

An axonometric projection is a visual image of an object obtained by parallel projection of it onto one axonometric plane of projections P along with axes of the spatial coordinate system Oxyz, to which it is assigned (an object is referred to a coordinate system if its projection onto one of the coordinate planes is known.). Projection of the object

and onto plane P called axonometric (axonometry);

projections of coordinate axes - corresponding axonometric axes(they are simply denoted x, y, z instead of x, y, z); the ratio of the length of the axonometric projection of a segment parallel to the coordinate axis to the natural length of the segment - distortion indicator along the corresponding axonometric axis. If the direction of projection is perpendicular to the plane P, then the axonometry is called rectangular, and if not, then oblique.

To construct visual technical images, GOST 2.317-69* recommends standard axonometry with good clarity.

6.2. Rectangular isometric projection(isometric)

This type of axonometry is obtained with the same inclination of all coordinate planes associated with the object to the axonometric plane of projections. Therefore, in isometry, the distortion coefficients along the x, y and z axes are the same (they are equal to 0.82), and the axonometric axes form angles of 120° with each other (Fig. 6.1). They can be constructed using compasses or squares with

		angles of 30° and 60°, positioning
		z axis is vertical. In Fig.
		6.1 x and y axes are drawn with
		slope 4:7 to horizontal
		no line of the drawing.
		To simplify isomet-
		riya is built using the
		data distortion indicators
		along the axes equal to 1. In this
		case image of the object
		in isometry	performed in
		enlarged scale 1.22:1.
		Rectangular isomet-
		ria is most convenient for
		items	curvilinear
		shapes, length, width and
	the heights of which differ from each other not very significantly.
	6.3. Rectangular dimetric projection
		(dimetria)

Dimetry is obtained with the same inclination to the axonometric plane of the coordinate planes xOy and yOz, therefore the distortion indicators along the x and z axes are the same and equal to 0.94, and along the y axis - 0.47. Using in practice the given distortion indicators (1 each for the x and z axes and 0.5 for the y axis), dimetry is performed on an enlarged scale.

ratio 1.06:1.

When constructing axonometric axes (Fig. 6.2), the axis

z is carried out vertically, and for

drawing x and y axes
it is not the angles of their inclination to the horizontal
zontal straight line
(respectively 7 10 and
and their deviations towards this
(1:8 and 7:8 respectively).
Rectangular diameter
it is advisable to use
prismatic and

pyramidal shapes, as well as for elongated objects in which the length significantly exceeds the width and height, directing the length parallel to the x or z axis. In this case, the length is not subject to severe distortion and the idea of ​​the shape of the object and the ratio of its main dimensions is not lost.

6.4. Drawing circles in axonometry

A circle lying in the coordinate plane or a plane parallel to it is projected in rectangular axonometry into an ellipse, the major axis of which is perpendicular to the “free” axonometric axis, and the minor axis is parallel to it. Free axonometric axis - projection of the coordinate axis, perpendicular to the plane circle (for example, for a circle whose plane is parallel to the yOz plane, the “free” axis is the x axis).

The construction, based on the given distortion indicators, of ellipses into which circles are projected, the planes of which are parallel to the coordinate ones, is shown for standard isometry and dimetry in Fig. 6.1 and 6.2 respectively.

The major axes of these ellipses in isometry are equal to 1.22d, and the minor axes are 0.71d (d is the diameter of the circle). Ellipses in isometry (Fig. 6.1) are constructed along the major and minor axes (4 points) and points on diameters parallel to the coordinate axes (4 more points).

In dimetry, the major axes of ellipses are equal to 1.06d, and the minor axes are equal to 0.35d for circles lying in the xOy and yOz planes and parallel to them, and 0.94d for circles located in the xOz plane and planes parallel to it. To construct ellipses in dimetry, 8 points are used, similar to the points along which an ellipse is drawn in isometry (Fig. 6.2). To more accurately construct ellipses into which circles parallel to the xOy and yOz planes are projected, additional points are used, obtained due to the symmetry of the ellipse points relative to the major and minor axes.

In Fig. 6.1 and 6.2 near the axes of the ellipses and their diameters indicate the reduced distortion indicators in these directions.

Axonometric projections of circles (arcs) of large radius, circles not lying in planes parallel to coordinate planes, and curved lines are constructed from axonometric projections of their points.

6.5. Examples of axonometric projections of various objects

The axonometry of an object is usually constructed according to its technical drawing, on which the projections of the axes of the Oxyz spatial coordinate system to which the object is assigned can be indicated.

The construction of axonometry begins with drawing axonometric axes.

Axonometric projections of figures are constructed using axonometric projections of their characteristic points. Axonometric projections of points are constructed using the coordinates of these points, taking into account distortion indicators along the axonometric axes.

Axonometric projections of segments are constructed based on the axonometric projections of two of their points. Axonometric projections of parallel lines are parallel. In this case, axonometric projections of straight lines parallel to the coordinate axes are parallel to the corresponding axonometric axes and have the same distortion indicators.

In Fig. 6.3a, 6.4a and 6.5a show technical drawings of a parallelepiped, hemisphere and cone of rotation, respectively, in Fig. 6.3b and 6.4b show isometries of the first two figures, and in Fig. 6.5b - dimetry of the third.

A 1 E 1

a) z 2

a) z 2

b) z

x

The outline of a sphere in rectangular projection is always a circle with a radius equal to the radius of the sphere R. When using the above distortion indicators, the radius of the outline of the sphere in isometry is increased to 1.22R, and in dimetry - to 1.06R.

When constructing the axonometry of an object, they strive, if possible, to combine the coordinate plane xOy with the plane of the base of the object, and the coordinate axes with its edges or axes of symmetry.

In Fig. 6.6a and 6.7a show complex drawings of objects, and in Fig. 6.6c and 6.7b, respectively, are isometric projections of these objects with a one-quarter cutout.

Cutouts in axonometric images are necessary in the same way as cuts in technical drawings to reveal the hidden internal shapes of an object.

Axonometric sections can be constructed in two ways. The first method is to build a complete image

the object in thin lines, followed by drawing the contours of the sections formed by each cutting plane of the cutout, and removing the image of the cut-off part of the object (Fig. 6.6b).

According to the second method, the contours of sections of the object are first constructed using secant planes (shown in Fig. 6.6b by the main lines), and then an image of the rest of the object is drawn.

In axonometry, as a rule, do not use full cuts, in which at least one of the three main dimensions of the object disappears(length, width, height). Otherwise, axonometry would be deprived of its main advantage - clarity.

To determine the direction of hatching in sections, an arbitrary segment b is laid out on the axonometric axes, and half of this segment in dimetry on the y axis. The straight lines connecting the ends of the segments specify the direction of hatching for the corresponding planes (Fig. 6.1 and 6.2).

If the secant plane passes through stiffeners, solid protrusions or thin walls, then the sections of these parts elements are always shaded. In axonometry, holes located on round flanges or disks are not rotated into the cut plane (Fig. 6.6).

IN Axonometry may not show small structural elements of an object (chamfers, roundings, etc.). Lines of smooth transition from one surface to another are shown with conditionally thin lines (Fig. 6.7b).

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 off 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 lines 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 extreme points Draw vertical edges of the required size from this segment. 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.

Rectangular isometric projection.

The location of the axonometric axes is shown in the figure. All three axes form among themselves equal angles V

120 0 . Axis OZ located vertically.

Distortion factor equal on all three axes 0,82 . In practice, rectangular isometric projection

Usually built without reducing dimensions along the axes - all sizes, parallel to the axes, are taken with a coefficient

Distortion equal unit.

The result is an image similar to an exact projection, but magnified 1.22 times. The picture shows

Directions of the axes of ellipses depicting circles located in planes parallel to coordinates

Planes.

Large AB axis is perpendicular to the corresponding axonometric axes. Small CD axis

Perpendicular to AB and parallel corresponding axonometric axes. All three ellipses are equal.

Ellipse axes dimensions in relation to diameter d circle :

When building accurate projection with coefficient distortion 0.82 AB = d; CD = 0.58d.

When constructed without reducing dimensions along all axes AB = 1.22d; CD = 0.71d.

Construction examplesisometry and dimetry look

The isometry of the ball is shown in the figure. The outer contour of the ball is a circle. When constructing an exact

Projections R = d/2. When plotted with the distortion coefficient reduced to unity,R = 1.22d/2.

d- diameter of the ball.

Construction examplesisometry and dimetry look

Hatching of sections in axonometry.

The hatching lines of the sections are drawn parallel to one of the diagonals of the squares (conventionally depicted) lying

In the corresponding coordinate planes. The sides of a conventional square are parallel to the axonometric axes.

Different sections of the same part are hatched with an inclination in different directions.

Extension lines in axonometric drawings are drawn parallel to the axonometric axes. Dimension lines

They are carried out parallel to the measured segment.

Construction examplesisometry and dimetry look