Construction of sections parallel to the diagonal of the base. How to draw an inclined section

Section- an image of a figure obtained by mentally dissecting an object with one or more planes.
The section shows only what is obtained directly in the cutting plane.

Sections are usually used to reveal the transverse shape of an object. The cross-sectional figure in the drawing is highlighted by shading. Dashed lines are applied in accordance with the general rules.

The order of section formation:
1. A cutting plane is introduced at the part where it is necessary to more fully reveal its shape. 2. The part of the part located between the observer and the cutting plane is mentally discarded. 3. The section figure is mentally rotated to a position parallel to the main projection plane P. 4. The cross-section image is formed in accordance with the general projection rules.

Sections not included in the composition are divided into:

Taken out;
- superimposed.

Outlined sections are preferred and can be placed in the gap between parts of the same type.
The contour of the extended section, as well as the section included in the section, is depicted with solid main lines.

Superimposed called section, which is placed directly on the view of the object. The contour of the superimposed section is made with a solid thin line. The section figure is placed in the place of the main view where the cutting plane passes and is shaded.

Overlay of sections: a) symmetrical; b) asymmetrical

Axis of symmetry the superimposed or removed section is indicated by a thin dash-dotted line without letters and arrows, and the section line is not drawn.

Sections in the gap. Such sections are placed in the gap in the main image and are made as a solid main line.
For asymmetrical sections located in a gap or superimposed, the section line is drawn with arrows, but not marked with letters.

The section in the gap: a) symmetrical; b) asymmetrical

Outlined sections have:
- anywhere in the drawing field;
- in place of the main view;
- with a turn with the addition of a “turned” sign

If the secant plane passes through the axis of the surface of rotation, limiting the hole or recesses, then their contour in the section is shown in full, i.e. performed according to the cut rule.

If the section turns out to consist of two or more separate parts, then a cut should be applied, up to changing the direction of view.
The cutting planes are chosen so as to obtain normal cross sections.
For several identical sections related to one object, the section line is designated with one letter and one section is drawn.

Remote elements.
Detail element - a separate enlarged image of a part of an object to present details not indicated on the corresponding image; may differ from the main image in content. For example, the main image is a view, and the detail is a section.

In the main image, part of the object is distinguished by a circle of arbitrary diameter, made with a thin line; from it there is a leader line with a shelf, above which a capital letter of the Russian alphabet is placed, with a height greater than the height of the dimensional numbers. The same letter is written above the extension element and to the right of it in parentheses, without the letter M, the scale of the extension element is indicated.

Practical lesson: “Parallelepiped. Construction of sections of a parallelepiped."

1. Purpose of practical work : . To consolidate knowledge of theoretical material about polyhedra,skills in solving problems on constructing sections,ability to analyze a drawing.

2. Didactic equipment for practical work : Workstation, models and developments of polyhedra, measuring instruments, scissors, glue, thick paper.

Time:2 hours

Tasks for work:

Task 1

Construct a section of the parallelepiped ABCDA 1 B 1 C 1 D 1 plane passing through points M, N, P lying on lines, respectively, A 1 B 1, AD, DC

Sample and the sequence of solving the problem:

1.Points N and P lie in the section plane and in the plane of the lower base of the parallelepiped. Let's construct a straight line passing through these points. This straight line is the trace of the cutting plane onto the plane of the base of the parallelepiped.

2. Let us continue the straight line on which side AB of the parallelepiped lies. Lines AB and NP intersect at some point S. This point belongs to the section plane.

3. Since point M also belongs to the section plane and intersects line AA 1 at some point X.

4.Points X and N lie in the same plane of face AA 1 D 1 D, connect them and get straight line XN.

5. Since the planes of the faces of the parallelepiped are parallel, then through the point M we can draw a straight line to the face A 1 B 1 C 1 D 1 , parallel to the line NP. This line will intersect side B 1 WITH 1 at point Y.

6. Similarly, draw straight line YZ, parallel to straight line XN. We connect Z with P and get the desired section - MYZPNX.

Task 2

Option 1. Construct a section of the parallelepiped АВСDA1В1С1D1 by the plane defined by the following pointsM, NAndP

Level 1: All three points lie on the edges emerging from vertex A

Level 2.Mlies in the face AA1D1D,Nlies on the face AA1B1B,Plies in the face CC1D1D.

Level 3.Mlies on the diagonal B1D,Nlies on the diagonal AC1,Plies on the edge C1D1.

Option2.Construct a section of the parallelepiped ABCDA1B1C1D1 by a plane passing through the line DQ, where point Q lies on the edge CC1 and point P, defined as follows

Level 1: All three points lie on the edges emerging from vertex C

Level 2: M lies on the continuation of edge A1B1, and point A1 is located between points B1 and P.

Level 3: P lies on the diagonal B1D

Work order:

1.Study theoretical material on the following topics:

Parallelepiped.

Right parallelepiped.

Inclined parallelepiped.

Opposite faces of a parallelepiped.

Properties of parallelepiped diagonals.

Pthe concept of a cutting plane and the rules for its construction.

What types of polygons are obtained in the section of a cube and parallelepiped.

2. BuildparallelepipedABCDA 1 B 1 C 1 D 1

3. Analyze the solution to problem No. 1

4.Consistently build a sectionparallelepipedABCDA 1 B 1 C 1 D 1 plane passing through points P, Q, R of problem No. 1.

5.Construct three more parallelepipeds and select sections for problems of levels 1, 2, and 3 on them

Evaluation criteria :

Literature: Atanasyan L.S. Geometry: Textbook for 10-11 grades. general education institutions. L.S. Atanasyan, V.F. Butuzov, S.B. Kodomtsev et al. - M.: Education, 2010 Ziv B.G. Geometry problems: A manual for students of grades 7-11. general education institutions. / B.G. Ziv, V.M. Mailer, A.G. Bakhansky. - M.: Education, 2010. V. N. Litvinenko Tasks for the development of spatial concepts. Book for teachers. - M.: Education, 2010

Didactic material for the practical lesson assignment

To task No. 1:

Some possible sections:

Construct sections of a parallelepiped with a plane passing through these points

The task itself usually sounds like this: "build a natural view of a section figure". Of course, we decided not to leave this issue aside and try, if possible, to explain how the inclined section is constructed.

In order to explain how an inclined section is constructed, I will give several examples. I will, of course, start with the elementary ones, gradually increasing the complexity of the examples. I hope that after analyzing these examples of section drawings, you will understand how it is done and will be able to complete your study assignment yourself.

Let's consider a “brick” with dimensions 40x60x80 mm and an arbitrary inclined plane. The cutting plane cuts it at points 1-2-3-4. I think everything is clear here.

Let's move on to constructing a natural view of the section figure.
1. First of all, let's draw the section axis. The axis should be drawn parallel to the section plane - parallel to the line into which the plane is projected in the main view - usually it is in the main view that the task for construction of an inclined section(Further I will always mention the main view, keeping in mind that this almost always happens in educational drawings).
2. On the axis we plot the length of the section. In my drawing it is designated as L. The size L is determined in the main view and is equal to the distance from the point of entry of the section into the part to the point of exit from it.
3. From the resulting two points on the axis, perpendicular to it, we plot the width of the section at these points. The width of the section at the point of entry into the part and at the point of exit from the part can be determined in the top view. IN in this case both segments 1-4 and 2-3 are equal to 60 mm. As you can see from the picture above, the edges of the section are straight, so we simply connect our two resulting segments, obtaining a rectangle 1-2-3-4. This is the natural appearance of the cross section of our brick by an inclined plane.

Now let's complicate our part. Let's place a brick on a base 120x80x20 mm and add stiffening ribs to the figure. Let's draw a cutting plane so that it passes through all four elements of the figure (through the base, brick and two stiffeners). In the picture below you can see three views and a realistic image of this part.

Let's try to build a natural view of this inclined section. Let's start again with the section axis: draw it parallel to the section plane indicated in the main view. On it we plot the length of the section equal to A-E. Point A is the entry point of the section into the part, and in a particular case, the entry point of the section into the base. The exit point from the base is point B. Mark point B on the section axis. In a similar way, we mark the entry and exit points to the edge, to the “brick” and to the second edge. From points A and B, perpendicular to the axis, we will lay out segments equal to the width of the base (40 in each direction from the axis, 80 mm in total). Let's connect the extreme points - we get a rectangle, which is a natural cross-section of the base of the part.

Now it’s time to build a piece of the section, which is a section of the edge of the part. From points B and C we will put perpendiculars of 5 mm in each direction - we will get segments of 10 mm. Let's connect the extreme points and get a section of the rib.

From points C and D we lay out perpendicular segments equal to the width of the “brick” - completely similar to the first example of this lesson.

By setting aside perpendiculars from points D and E equal to the width of the second edge and connecting the extreme points, we obtain a natural view of its section.

All that remains is to erase the jumpers between the individual elements of the resulting section and apply shading. It should look something like this:

If we divide the figure along a given section, we will see the following view:

I hope that you are not intimidated by the tedious paragraphs describing the algorithm. If you have read all of the above and still do not fully understand, how to draw an inclined section, I strongly advise you to pick up a piece of paper and a pencil and try to repeat all the steps after me - this will almost 100% help you learn the material.

I once promised a continuation of this article. Finally, I am ready to present you with a step-by-step construction of an inclined section of a part, closer to the level of homework. Moreover, the inclined section is defined in the third view (the inclined section is defined in the left view)

or write down our phone number and tell your friends about us - someone is probably looking for a way to complete the drawings

or Create a note about our lessons on your page or blog - and someone else will be able to master drawing.

Yes, everything is fine, but I would like to see how to do the same thing on a more complex part, with chamfers and a cone-shaped hole, for example.

Thank you. Aren't the stiffening ribs hatched on the sections?
Exactly. They are the ones who do not hatch. Because these are the general rules for making cuts. However, they are usually shaded when making cuts in axonometric projections - isometry, dimetry, etc. When making inclined sections, the area related to the stiffener is also shaded.

Thank you, very accessible. Tell me, can an inclined section be done in the top view, or in the left view? If so, I would like to see a simple example. Please.

It is possible to make such sections. But unfortunately I don’t have an example at hand right now. And there is another interesting point: on the one hand, there is nothing new there, but on the other hand, in practice, such sections are actually more difficult to draw. For some reason, everything starts to get confused in the head and most students have difficulties. But don't give up!

Yes, everything is fine, but I would like to see how the same thing is done, but with holes (through and not through), otherwise they never turn into an ellipse in the head

help me with a complex problem

It's a pity that you wrote here. If you could write to us by email, maybe we could have time to discuss everything.

You explain well. What if one of the sides of the part is semicircular? There are also holes in the part.

Ilya, use the lesson from the section on descriptive geometry “Section of a cylinder by an inclined plane.” With its help you can figure out what to do with the holes (they are essentially cylinders too) and with the semicircular side.

I thank the author for the article! It’s brief and easy to understand. About 20 years ago I was gnawing on the granite of science, now I’m helping my son. I forgot a lot, but your article returned a fundamental understanding of the topic. I’ll go figure out the inclined section of the cylinder)

Add your comment.

Do you know what is called the section of polyhedra by a plane? If you still doubt the correctness of your answer to this question, you can check yourself quite simply. We suggest you take a short test below.

Question. What is the number of the figure that shows the section of a parallelepiped by a plane?

So, the correct answer is in Figure 3.

If you answer correctly, it confirms that you understand what you are dealing with. But, unfortunately, even the correct answer to a test question does not guarantee you the highest grades in lessons on the topic “Sections of polyhedra.” After all, the most difficult thing is not recognizing sections in finished drawings, although this is also very important, but their construction.

To begin with, let us formulate the definition of a section of a polyhedron. So, a section of a polyhedron is a polygon whose vertices lie on the edges of the polyhedron, and whose sides lie on its faces.

Now let’s practice quickly and accurately constructing intersection points a given straight line with a given plane. To do this, let's solve the following problem.

Construct the intersection points of straight line MN with the planes of the lower and upper bases of the triangular prism ABCA 1 B 1 C 1, provided that point M belongs to the side edge CC 1, and point N belongs to edge BB 1.

Let's start by extending straight line MN in both directions in the drawing (Fig. 1). Then, in order to obtain the intersection points required by the problem, we extend the lines lying in the upper and lower bases. And now comes the most difficult moment in solving the problem: which lines in both bases need to be extended, since each of them has three lines.

In order to correctly complete the final step of construction, it is necessary to determine which of the direct bases are in the same plane as the straight line MN of interest to us. In our case, this is straight CB in the lower and C 1 B 1 in the upper bases. And it is precisely these that we extend until they intersect with the straight line NM (Fig. 2).

The resulting points P and P 1 are the points of intersection of the straight line MN with the planes of the upper and lower bases of the triangular prism ABCA 1 B 1 C 1 .

After analyzing the presented problem, you can proceed directly to constructing sections of polyhedra. The key point here will be reasoning that will help you arrive at the desired result. As a result, we will eventually try to create a template that will reflect the sequence of actions when solving problems of this type.

So, let's consider the following problem. Construct a section of a triangular prism ABCA 1 B 1 C 1 with a plane passing through points X, Y, Z belonging to edges AA 1, AC and BB 1, respectively.

Solution: Let's draw a drawing and determine which pairs of points lie in the same plane.

Pairs of points X and Y, X and Z can be connected, because they lie in the same plane.

Let's construct an additional point that will lie on the same face as point Z. To do this, we will extend the lines XY and CC 1, because they lie in the plane of the face AA 1 C 1 C. Let's call the resulting point P.

Points P and Z lie in the same plane - in the plane of the face CC 1 B 1 B. Therefore, we can connect them. The straight line PZ intersects the edge CB at a certain point, let's call it T. Points Y and T lie in the lower plane of the prism, connect them. Thus, the quadrilateral YXZT was formed, and this is the desired section.

Let's summarize. To construct a section of a polyhedron with a plane, you must:

1) draw straight lines through pairs of points lying in the same plane.

2) find the lines along which the section planes and faces of the polyhedron intersect. To do this, you need to find the intersection points of a straight line belonging to the section plane with a straight line lying in one of the faces.

The process of constructing sections of polyhedra is complicated because it is different in each specific case. And no theory describes it from beginning to end. In fact, there is only one sure way to learn how to quickly and accurately construct sections of any polyhedra - this is constant practice. The more sections you build, the easier it will be for you to do this in the future.

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Today we’ll look again at how construct a section of a tetrahedron with a plane.
Let's consider the simplest case (mandatory level), when 2 points of the section plane belong to one face, and the third point belongs to another face.

Let us remind you algorithm for constructing sections of this type (case: 2 points belong to the same face).

1. We are looking for a face that contains 2 points of the section plane. Draw a straight line through two points lying on the same face. We find the points of its intersection with the edges of the tetrahedron. The part of the straight line that ends up in the face is the side of the section.

2. If the polygon can be closed, the section has been constructed. If it is impossible to close, then we find the intersection point of the constructed line and the plane containing the third point.

1. We see that points E and F lie on the same face (BCD), draw a straight line EF in the plane (BCD).
2. Find the point of intersection of the straight line EF with the edge of the tetrahedron BD, this is point H.
3. Now you need to find the point of intersection of the straight line EF and the plane containing the third point G, i.e. plane (ADC).
The straight line CD lies in the planes (ADC) and (BDC), which means it intersects the straight line EF, and point K is the point of intersection of the straight line EF and the plane (ADC).
4. Next, we find two more points lying in the same plane. These are points G and K, both lie in the plane of the left side face. We draw a line GK and mark the points at which this line intersects the edges of the tetrahedron. These are points M and L.
4. It remains to “close” the section, i.e. connect the points lying on the same face. These are points M and H, and also L and F. Both of these segments are invisible, we draw them with a dotted line.

The cross-section turned out to be a quadrangle MHFL. All its vertices lie on the edges of the tetrahedron. Let's select the resulting section.

Now let's formulate "properties" of a correctly constructed section:

1. All vertices of a polygon, which is a section, lie on the edges of a tetrahedron (parallelepiped, polygon).

2. All sides of the section lie on the faces of the polyhedron.
3. Each face of a polygon can contain no more than one (one or none!) side of the section