Axonometric projections are used to visually depict various objects. The subject here is depicted as it is seen (from a certain angle of view). This image shows all three spatial dimensions, so reading an axonometric drawing usually does not cause difficulties.
An axonometric drawing can be obtained using either a rectangular projection or an oblique projection. The object is positioned so that the three main directions of its measurements (height, width, length) coincide with the coordinate axes and, together with them, are projected onto the plane. The direction of projection should not coincide with the direction of the coordinate axes, i.e. none of the axes will be projected to the point. Only in this case will you get a clear image of all three axes.
To obtain rectangular axonometric projections, the coordinate axes are tilted relative to the projection plane R A so that their direction does not coincide with the direction of the projecting rays. With oblique projection, you can vary both the direction of projection and the inclination of the coordinate axes relative to the projection plane. In this case, the coordinate axes, depending on their angle of inclination to the axonometric plane of projections and the direction of projection, will be projected with different distortion coefficients. Depending on this, different axonometric projections will be obtained, differing in the location of the coordinate axes. GOST 2.317-69 (ST SEV 1979-79) provides for the following axonometric projections: rectangular isometric projection; rectangular dimetric projection; oblique frontal isometric projection; oblique horizontal isometric projection; oblique frontal dimetric projection.
§ 26. RECTANGULAR AXONOMETRIC PROJECTIONS
Isometric projection is highly visual and is widely used in practice. When obtaining an isometric projection, the coordinate axes are tilted relative to the axonometric plane of the projections so that they have the same angle of inclination (Fig. 236). In this case, they are projected with the same distortion factor (0.82) and at the same angle to each other (120°).
In practice, the distortion coefficient along the axes is usually taken equal to one, i.e., they set aside the actual size of the size. The image is enlarged by 1.22 times, but this does not lead to distortion of the shape and does not affect clarity, but simplifies the construction.
Axonometric axes in isometry are carried out by first constructing angles between the axes x, y And z(120°) or axle tilt angles X And at to the horizontal line (30°). Constructing axes in isometry with 
using a compass is shown in Fig. 237, where is the radius R taken arbitrarily. In Fig. 238 shows a method for constructing axes X And at using a tangent of 30°. From point ABOUT- the intersection points of the axonometric axes lay five identical segments of arbitrary length to the left or right along a horizontal line and, having drawn a vertical line through the last division, lay three identical segments up and down on it. The constructed points are connected to the point ABOUT and get axes Oh And Oh.
You can plot (construct) dimensions and make measurements in axonometry only along axes Ooh, oh And Oz or on straight lines parallel to these axes.
In Fig. 239 shows the construction of a point A in isometry according to an orthogonal drawing (Fig. 239, a). Dot A located in the plane V. To construct it, it is enough to construct a secondary projection A" dots A(Fig. 239, b) on the plane xOz by coordinates X A And Z A . Point image A coincides with its secondary projection. Secondary projections of a point are the images of its orthogonal projections in axonometry.
In Fig. 240 shows the construction of point B in isometry. First, construct a secondary projection of point B on the plane xOy. To do this, from the origin along the axis Oh set aside the coordinate X in(Fig. 240, b), obtain a secondary projection of the point b x. From this point parallel to the axis Oh draw a straight line and mark the coordinate on it Y B .
Constructed point b on the axonometric plane will be the secondary projection of the point IN. Swiping from a point b a straight line parallel to the Oz axis, plot the coordinate Z B and get point B, i.e. an axonometric image of point B. The axonometry of point B can also be constructed from secondary projections on the plane zOh or zОу.
Rectangular dimetric projection. The coordinate axes are positioned so that the two axes Oh And Oz had the same tilt angle and were projected with the same distortion factor (0.94), and the third axis Oh would be tilted so that the projection distortion coefficient would be half as large (0.47). Typically the axial distortion factor is Oh And Oz is taken equal to unity, and along the axis Oh- 0.5. The image turns out to be enlarged by 1.06 times, but this, just like in isometry, does not affect the clarity of the image, but simplifies the construction. The location of the axes in rectangular diameter is shown in Fig. 241. They are constructed by laying off angles of 7° 10" and 41° 25" from the horizontal line along a protractor, or by laying off identical segments of arbitrary length, as shown in Fig. 241. Connect the resulting points with a point ABOUT. When constructing rectangular dimetry, it must be remembered that actual dimensions are plotted only on the axes Oh And Oz or on lines parallel to them. Axial dimensions Oh and parallel to it they are laid off with a distortion factor of 0.5.
§ 27. OBLIQUE AXONOMETRIC PROJECTIONS
Frontal isometric view. The location of the axonometric axes is shown in Fig. 242. Axle tilt angle Oh to the horizontal is usually 45°, but can be 30 or 60°.
Horizontal isometric projection. The location of the axonometric axes is shown in Fig. 243. Axle tilt angle Oh to the horizontal is usually 30°, but can be 45 or 60°. In this case, the angle is 90° between the axes Oh And Oh must be preserved.
Frontal and horizontal oblique isometric projections are constructed without distortion along the axes Ooh, oh And Oz.
Frontal dimetric projection. The location of the axes is shown in Fig. 244. Fig. 245 illustrates the projection of coordinate axes onto the axonometric projection plane. Plane xOz parallel to the plane R. Allowed axis Oh carried out at an angle of 30 or 60° to the horizontal, axial distortion coefficient Oh And Oz taken equal to 1, and along the axis Oh- 0,5.
CONSTRUCTION OF FLAT GEOMETRIC FIGURES IN AXONOMETRY
The basis of a number of geometric bodies is a flat geometric figure: a polygon or a circle. To construct a geometric body in axonometry, you must be able to construct, first of all, its base, i.e., a flat geometric figure. For example, consider the construction of flat figures in rectangular isometric and dimetric projection. The construction of polygons in axonometry can be done using the coordinate method, when each vertex of the polygon is constructed in axonometry as a separate point (the construction of a point by the coordinate method is discussed in § 26), then the constructed points are connected by straight line segments and a broken closed line is obtained in the form of a polygon. This problem can be solved differently. In a regular polygon, construction begins with the axis of symmetry, and in an irregular polygon, an additional line is drawn, which is called the base, parallel to one of the coordinate axes in the orthogonal drawing.
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 is called dimetry.
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 mark the axes on orthogonal projections of the depicted object, taking into account the selected position of the object for drawing in a 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 previously stated, the angles in a rectangular isometric view 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 depicted object has hidden cavities or complex elements, try to shade it. 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
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 relevant 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. Dimensional lines
They are carried out parallel to the measured segment.
Construction examplesisometry and dimetry look
You can display various geometric objects using drawings and computer graphics using the principles of isometry and axonometry. What are the specifics of each of them?
What is axonometry?
Under axonometry or axonometric projection refers to a method of graphically displaying certain geometric objects through parallel projections.
Axonometry
In this case, a geometric object is most often drawn using a specific coordinate system - so that the plane on which it is projected does not correspond to the position of the plane of other coordinates of the corresponding system. It turns out that the object is displayed in space through 2 projections and looks three-dimensional.
Moreover, for the reason that the display plane of the object is not located strictly parallel to any of the axes of the coordinate system, individual elements of the corresponding display may be distorted - according to one of the following 3 principles.
Firstly, distortion of object display elements can be observed along all 3 axes used in the system, to an equal extent. In this case, the isometric projection of the object, or isometry, is fixed.
Secondly, distortion of elements can only be observed along 2 axes of equal magnitude. In this case, a dimetric projection is observed.
Third, element distortion can be recorded as varying along all 3 axes. In this case, a trimetric projection is observed.
Let us therefore consider the specifics of the first type of distortions formed within the framework of axonometry.
What is isometry?
So, isometry- this is a type of axonometry that is observed when drawing an object if the distortion of its elements along all 3 coordinate axes is the same.
Isometric The type of axonometric projection under consideration is actively used in industrial design. It allows you to clearly view certain details within the drawing. The use of isometrics is also widespread in the development of computer games: with the help of the appropriate type of projection, it becomes possible to effectively display three-dimensional images.
It can be noted that in the field of modern industrial developments, isometry generally means a rectangular projection. But sometimes it can be presented in an oblique variety.
Comparison
The main difference between isometry and axonometry is that the first term corresponds to a projection, which is only one of the varieties of the one denoted by the second term. Isometric projection, thus, differs significantly from other types of axonometry - dimetry and trimetry.
Let us display more clearly the difference between isometry and axonometry in a small table.