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An axonometric projection is called an isometric if. Axonometry

Axonometric (Axonometry translated from Greek("Ahop" - axis; "metreo" - measuring) means an axial image.) projections are images obtained by projecting with parallel rays of a figure (object) together with the coordinate axes onto an arbitrarily located plane, which is called "Axonometric"(or picture). Usually 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 understanding of the size of the depicted object, therefore they are used as an illustration to a drawing to facilitate understanding of the constructive form of an object. On (Fig. 270) shows the receipt of an axonometric projection of the part.

On axonometric projections, the following designations are adopted: axonometric plane is denoted by P "; axonometric coordinate axes - x", y ", z"; axonometric projections of points A, B, etc. are designated A ", B", etc. The origin is indicated by O ".
2. Types of axonometric projections.
Depending on the direction of the projecting rays, the axonometric projections are divided into: rectangular or orthogonal (the projecting rays are perpendicular to the axonometric plane P ") and oblique (the 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 dimensions of the axonometric projections of the segments having the direction of the coordinate axes (It is known that a straight line segment inclined to a plane is projected onto it reduced; the greater the angle of inclination, the smaller the projection of the segment will be.), - all axonometric projections are divided into three main types:
1) isometric, i.e. the same dimension (the z ", x" and y "axes are inclined in the same way; therefore, the reduction in dimensions in 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 dimensions along these two axes will be the same, and along the third axis - different);
3) trimetric, i.e. triple dimension (all axes have a different inclination; therefore, the reduction in dimensions in the direction of all three axes is different).
In mechanical engineering, from rectangular axonometric projections, isometric and dimetric projections are most often used, and from oblique - dimetric, which is otherwise called frontal dimetric projection.
In isometric projection, the angles between the axonometric axes x ", y" and z "are the same (120 ° each); the z-axis is vertical; 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 indices 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 having 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 the dimetric projection, the angle between the axonometric axes z "and x" is 97 ° 10 ", and the angles between the axonometric axes x" and y ", as well as z" and y "are the same, that is, 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 an inclination of the axonometric axes, the distortion index 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 z "and x" axes is 90 °, and the angles between the axonometric x "and y" axes, as well as between the axonometric z "and y" axes, are the same, that is, 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).
Distortion indices along the axonometric axes x "and z" are equal to 1.0 and along the y-axis - 0.5.
Such a frontal dimetric projection is called cabinet; it is recommended to use it when you want to show without changing the outline of figures located in planes parallel to the frontal plane of projections.
For comparison of images made in axonometric projections, (Fig. 272) shows different axonometric projections of the same cube.

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

What is dimetry

Dimetry is one of the types of axonometric projection. Thanks to axonometry, with one volumetric image, an object can be viewed in three dimensions at once. Since the distortion coefficients of all sizes along the 2 axes are the same, this projection is called dimetry.

Rectangular dimetry

When the Z-axis is "vertical, while 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 carry out the construction of approximately axonometric axes of ordinary dimetry, it is necessary to assume that tg 7 degrees 10 minutes is 1/8, and tg 41 degrees 25 minutes is 7/8.

How to build dimetry

First, you need to draw the axes to depict the object in dimetry. In any rectangular dimetry, 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 the Y and X axes - 127 degrees 50 minutes.

Now it is required 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 complete the transfer to the volumetric image of the overall dimensions of the object, you can start drawing minor elements on the surface of the object.

It is worth remembering that circles in each dimetric plane are depicted 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 of the diameter of the drawn circle. And the minor axis of the ellipse in the XOZ plane is 0.95 of the diameter, and in the ZOY and XOY plane - 0.35 of the diameter. 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 the ZOY and XOY planes are 0.33 diameters.

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

What is isometry

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

In mathematics, isometry is known as a metric space transformation 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 upright.

How to draw isometric

Isometric construction of an object makes it possible to get the most expressive idea of ​​the spatial properties of the depicted object.

Before you start building a drawing in isometric projection, you must 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 isometric that you will draw. There are two types of it: rectangular and horizontal oblique.

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

Start drawing isometry from the top surface of the subject image. Two vertical lines should be drawn from the corners of the resulting horizontal surface and the corresponding linear dimensions of the object should be laid on them. In 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 should be borne in mind that when depicting any object in an isometric projection, the visibility of curved details will necessarily be distorted. The circle should be drawn as an ellipse. The segment between the points of a 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 depicted object has hidden cavities or complex elements, try to perform shading. It can be simple or stepwise, 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 kinds hardness.

GOST 2.317-68 * establishes rectangular and oblique axonometric projections.

The construction of axonometric projections is that geometric shape together with the axes of rectangular coordinates, to which this figure is referred in space, in parallel (rectangular or oblique) ways are projected onto the selected projection plane. Thus, an axonometric projection is a projection onto one plane. In this case, the direction of projection is chosen so that it does not coincide with any of the coordinate axes.

When constructing axonometric projections, the depicted object is rigidly associated with the natural coordinate system Oxyz. In general, an axonometric drawing is obtained consisting of a parallel projection of the object, supplemented by an image of the coordinate axes with natural scale segments along these axes. The name "axonometry" comes from the words - axon - axis and metreo - I measure.

Types of axonometric projections

Axonometric projections, depending on the direction of projection, are divided into:

  • obliquewhen the direction of projection is not perpendicular to the plane of axonometric projections;
  • rectangularwhen the direction of projection is perpendicular to the plane of the axonometric projections.

Depending on the comparative value of the distortion coefficients along the axes, three types of axonometry are distinguished:

  • isometry - all three distortion factors are equal to each other;
  • dimetry - two distortion factors are equal to each other and differ from the third;
  • trimetry - all three distortion factors are not equal to each other.

Rectangular isometry

In a rectangular isometry, the angles between the axes are equal to 120 °. When constructing an isometric projection along the x, y, z axes and parallel to them, the natural dimensions of the object are laid. Hence the name "isometry", which in Greek means "equal measurements"


Creation of isometric projections of flat geometric shapes

Consider the construction of a triangle on a horizontal plane in isometric projection. When constructing, it is initially necessary to determine the location of the figure relative to the origin. For this, the distance m is plotted along the x-axis, which is equal to the displacement of the triangle axis relative to the y-axis. From the found point, draw a straight line parallel to the y-axis, and lay on it a segment equal to k - the displacement of the base of the triangle from the x-axis, we got point 1. Symmetrically to point 1 along a straight line parallel to the x-axis, segments equal to half of the base of the triangle are laid on both sides - points 3, 4 are found. From point 1 along a straight line parallel to the y-axis, a segment equal to the height of the triangle is laid - point 2 is determined. The obtained points are connected. The frontal and profile projection of the figure is similarly built.

Instructions

Construct using a ruler and protractor or compass and ruler for a rectangular (horizontal) isometric view. In this type of axonometric projection, all three axes - OX, OY, OZ - are angles of 120 ° between each other, while the OZ axis has a vertical direction.

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 isometric (with the adopted 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 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 of the length of the part along one axis (for example, along the Y axis). Draw through the found point a segment of the required size (part width) parallel to the other axis (OX).

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

If this face has a circular hole, draw it. In isometric view, a circle is drawn as an ellipse because we are looking 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, 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 face equal to the height of the part. Connect the ribs at their bottom points.

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

In isometric projection, all coefficients are equal to each other:

k = t = n;

3 k 2 = 2,

k = yj 2UZ - 0.82.

Therefore, when constructing an isometric projection, the dimensions of the object, laid down along the axonometric axes, are multiplied by 0.82. This recalculation of sizes is inconvenient. Therefore, for simplicity, an isometric projection is usually performed without reducing the dimensions (distortion) along the axes x, y, i, those. take the reduced distortion factor equal to one. The resulting image of an object in isometric projection has a slightly larger size than in reality. The increase in this case is 22% (expressed by the number 1.22 = 1: 0.82).

Each line segment directed along the axes x, y, z or parallel to them, retains its value.

The location of the axes of the isometric projection is shown in Fig. 6.4. In fig. 6.5 and 6.6 show orthogonal (but) and isometric (b) point projection BUT and segment Л IN.

Hexagonal prism in isometric view. The construction of a hexagonal prism according to this drawing in the system of orthogonal projections (on the left in Fig. 6.7) is shown in Fig. 6.7. On the isometric axis I lay off height H, draw lines parallel to the axes hiu. Mark on a line parallel to the axis NS, position of points / and 4.

To plot a point 2 determine the coordinates of this point in the drawing - x 2 and at 2 and, putting these coordinates on the axonometric image, build a point 2. The points are built in the same way. 3, 5 and 6.

The constructed points of the upper base are connected to each other, an edge is drawn from the point / to the intersection with the x-axis, then -

dotted edges 2 , 3, 6. The ribs of the lower base are drawn parallel to the ribs of the upper one. Plotting a point L, located on the side face, by coordinates x A(or at A) and 1 A evident from

Circle isometry. Circles in isometry are depicted as ellipses (Fig. 6.8) indicating the values ​​of the axes of the ellipses for the given distortion coefficients equal to one.

The major axis of the ellipses is located at an angle of 90 ° for ellipses lying IN THE PLANE xC> 1 to the OSI y, IN THE PLANE u01 To the X axis, in the plane hoy To the OSI ?.


When constructing an isometric image by hand (like a picture), an ellipse is performed at eight points. For example, trays 1, 2, 3, 4, 5, 6, 7 and 8 (see fig. 6.8). Points 1, 2, 3 and 4 are found on the corresponding axonometric axes, and the points 5, 6, 7 and 8 plotted according to the values ​​of the corresponding major and minor axes of the ellipse. When drawing ellipses in isometric projection, you can replace ovals and build them as follows 1. The construction is shown in Fig. 6.8 on the example of an ellipse lying in a plane xOz. From point / as from the center, make a radius serif R = D on the continuation of the minor axis of the ellipse at point O, (a point symmetrical to it is also constructed in a similar way, which is not shown in the drawing). From point O, as from the center, an arc is drawn CGC radius D, which is one of the arcs that make up the outline of the ellipse. From point O, as from the center, an arc of radius is drawn O ^ G before the intersection with the major axis of the ellipse in points OU Drawing through the points O p 0 3 straight line, found at the intersection with an arc CGC point TO, which defines 0 3 C- the value of the radius of the closing arc of the oval. Points TO are also the conjugation points of the arcs that make up the oval.

Cylinder isometry. An isometric view of a cylinder is defined by isometric images of the circles of its base. Isometric Creation of a Cylinder with Height H according to the orthogonal drawing (Fig. 6.9, left) and point C on its lateral surface is shown in Fig. 6.9, right.


Proposed by Yu.B. Ivanov.

An example of constructing in an isometric projection of a round flange with four cylindrical holes and one triangular one is shown in Fig. 6.10. When constructing the axes of cylindrical holes, as well as the edges of a triangular hole, their coordinates are used, for example, the coordinates x 0 and y 0.