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

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

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

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

For rectangular axonometric projections, the projection is carried out perpendicular to the plane of projections, and the object is located 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 coefficient of distortion in all three axes is the same. 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 set aside the value of the length of the part 45 along the x-axis, and the value of the width of the part 30 along the y-axis. From each point of the quadrangle we will raise the top of the vertical segments by the height of the base of the part 7 (Fig. 2). On axonometric images, when applying dimensions, extension lines are drawn parallel to the axonometric axes, dimension lines - 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 the circles in all planes will be projected into ellipses on the axonometric image. Therefore, first we will learn how to build approximately ellipses.

If a circle is inscribed in a square, then 8 characteristic points can be marked in it: 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). Fig. 4c and Fig. 4b show the exact way of constructing the points of intersection of the diagonal of a square with a circle. Figure 4e 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 build a projection of a quadrilateral into which a square is projected. Next, we build an ellipse Fig.6.

Next, we rise to a height of 16mm and transfer the ellipse there (Fig. 7). We remove extra lines. We turn to the construction of holes. To do this, we build an ellipse at the top, into which a hole with a diameter of 14 is projected (Fig. 8). Further, in order to show a hole with a diameter of 6 mm, it is necessary to mentally cut out a quarter of the part. To do this, we will build 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 upper part of the part we draw already 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 cut a quarter of the part and remove invisible lines (Fig. 11).

Let's proceed to the construction of the stiffener. To do this, on the upper plane of the base, we measure 3 mm from the edge of the part and draw a segment half the thickness of the rib (1.5 mm) long (Fig. 12), we also mark the rib on the far side of the part. An angle of 40 degrees does not suit us when constructing axonometry, so we calculate the second leg (it will be equal to 10.35mm) and build the second point of the angle along the plane of symmetry using it. To build the border of the rib, we build a straight line at a distance of 1.5mm from the axis on the upper plane of the part, then we draw the lines parallel to the x-axis until they intersect with the outer ellipse and lower the vertical straight line. Draw a straight line through the lower point of the rib boundary 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, we 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 the distance at which the upper point of the rib boundary is (5.24mm) and set aside the same distance on a vertical straight line from 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. The hatching lines of sections in axonometric projections are applied parallel to one of the diagonals of the projections of the squares lying in the corresponding coordinate planes, whose sides 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). It remains to depict the side holes. To do this, we mark the centers of the axes of rotation of the holes, and build ellipses, as indicated above. Similarly, we build rounding radii (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 the oblique frontal dimetric projection. It is good because circles parallel to this plane will be projected onto the plane defined by the X and Z axes in the true value (the angle between the X and Z axes is 90 degrees, the Y axis is tilted at an angle of 45 degrees to the horizon). "Dimetric" projection means that the coefficients of distortion along the two axes X and Z are the same, along the Y axis the coefficient of distortion is two times less.

When choosing an axonometric projection, it is necessary to strive for the largest number of elements to be 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 projection plane.

The layout of the axes and the axonometric image of the part "Rack" in an oblique frontal dimetric projection is shown in Fig.18.

6.1. General provisions

Complex (technical) drawings are built according to the method of rectangular projection on the projection plane, while 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, measurable, but not visual enough, 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 a general idea of ​​the relative size and shape of the object.

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

that on the plane P called axonometric (axonometry);

projections of coordinate axes - corresponding axonometric axes(they are simply referred to as 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 build visual technical images, GOST 2.317-69 * recommends standard axonometries that have good clarity.

6.2. Rectangular Isometric View(isometry)

This type of axonometry is obtained with the same inclination of all coordinate planes associated with the object to the axonometric projection plane. 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° between themselves (Fig. 6.1). They can be built using a compass or squares with

		angles 30O and 60O, placing
		z-axis is vertical. On fig.
		6.1 the x and y axes are drawn with
		slope 4:7 to the horizontal
		line of the drawing.
		To simplify isomet-
		Riya is built using the
		given indicators of distortion
		along the axes equal to 1. In this
		case image of the subject
		in isometry	performed in
		scaled up 1.22:1.
		Rectangular isomet-
		ria is most convenient for
		items	curvilinear
		shape, length, width and
	the height 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), the 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

plotting the x and y axes
it is not the angles of their inclination to the horizontal
umbrella line
(respectively 7 10 and
and their biases towards this
(respectively 1:8 and 7:8).
Rectangular dimetria
appropriate to apply
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 subjected to strong 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 a rectangular axonometry into an ellipse, the major axis of which is perpendicular to the “free” axonometric axis, and the minor one 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 of the given indicators of distortion 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 1.22d, and the minor axes are 0.71d (d is the diameter of the circle). Ellipses in isometry (Fig. 6.1) are built along major and minor axes (4 points) and points on diameters parallel to the coordinate axes (4 more points).

In dimetry, the major axes of the 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 build ellipses into which circles are projected parallel to the xOy and yOz planes, additional points are used, obtained due to the symmetry of the points of the ellipses relative to the major and minor axes.

On fig. 6.1 and 6.2, near the axes of the ellipses and their diameters, the reduced distortion indicators in these directions are indicated.

Axonometric projections of circles (arcs) of large radius, circles that do not lie in planes parallel to the coordinate ones, and curved lines are built according to the axonometric projections of their points.

6.5. Examples of axonometric projections of various objects

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

The construction of axonometry begins with the axonometric axes.

Axonometric projections of figures are built on axonometric projections of their characteristic points. Axonometric projections of points are built according to the coordinates of these points, taking into account the distortion indicators along the axonometric axes.

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

On fig. 6.3a, 6.4a and 6.5a are technical drawings of the parallelepiped, hemisphere and cone of revolution, 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

a) z

x

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

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

On fig. 6.6a and 6.7a are complex drawings of objects, and in fig. 6.6c and 6.7b, respectively, are isometric projections of these objects with a cutout of one quarter.

Cut-outs in axonometric images are necessary in the same way as cuts in technical drawings, in order to reveal the hidden internal forms of the object.

Sections in axonometry can be built in two ways. The first way is to build a complete image

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 the sections of the object are first built by cutting planes (shown in Fig. 6.6b by the main lines), and then the image of the rest of the object is performed.

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 - visibility.

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

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

AT axonometry, it is allowed not to show small structural elements of the object (chamfers, fillets, etc.). The lines of a smooth transition from one surface to another are shown by conditionally thin lines (Fig. 6.7b).

Instruction

Construct with a ruler and protractor or a compass and ruler for a rectangular (orogonal) isometric projection. In this type of axonometric projection, all three axes - OX, OY, OZ - are angles of 120 ° to each other, 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 factor to one. By the way, “isometric” itself means “equal size”. In fact, when displaying a three-dimensional object on a plane, the ratio of the length of any projected segment parallel to the coordinate axis to the actual length of this segment is 0.82 for all three axes. Therefore, the linear dimensions of the 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 face. Measure along the OZ axis from the center of intersection of the coordinate axes the height of the part. Draw thin lines for the X and Y axes through this point. From the same point, set aside 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 desired size (part length) parallel to the first axis (OY). The two drawn line segments must intersect. Complete the rest of the top face.

If this face has a round hole, draw it. In isometry, a circle is shown as an ellipse because we are looking at it from 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, the b-axis is always 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 from the three corners of the top face vertical edges equal to the height of the part. Connect the edges through their bottom points.

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

Rectangular isometric view.

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

120 0 . Axis oz situated vertically.

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

Usually built without reducing the 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 figure shows

The directions of the axes of ellipses depicting circles located in planes parallel to the coordinate

Planes.

Large axes AB are perpendicular to the corresponding axonometric axes. Small axes CD

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

Dimensions of the axes of the ellipse in relation to diameter d circles :

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

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

Construction examplesisometric and dimetric see

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 constructed with the distortion factor reduced to unity,R = 1.22d/2.

d- ball diameter.

Construction examplesisometric and dimetric see

Hatching of cuts in axonometry.

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

in the respective coordinate planes. The sides of the conditional 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

Conducted parallel to the measured segment.

Construction examplesisometric and dimetric see