Projections of geometric bodies prism. Summary of the drawing lesson "projections of a group of geometric bodies"

Projection of regular triangular and hexagonal prisms. The bases of the prisms, parallel to the horizontal projection plane, are depicted on it in full size, and on the frontal and profile planes - as straight segments. The side faces are depicted without distortion on those projection planes to which they are parallel, and in the form of straight segments on those to which they are perpendicular (Fig. 78). Edges. inclined to the projection planes are depicted distorted on them.

Fig 78. Prisms: a. g - projection; b, d - drawings in a system of rectangular projections: c, c - isometric projections The dimensions of the prisms are determined by their height and the dimensions of the base figure. The dash-dot lines in the drawing indicate the axes of symmetry.

Rice. 80. Cylinder and cone: a, d - projection; b, d drawings in a system of rectangular projections; V. e - isometric projections The frontal and profile projections of the cylinder in this case are rectangles, and the cones are isosceles triangles.

Please note that on all projections the axes of symmetry should be drawn, with which the drawings of the cylinder and cone begin. The frontal and profile projections of the cylinder are the same. The same can be said about cone projections. Therefore, in this case, profile projections in the drawing are unnecessary. In addition, thanks to the “diameter” icon, you can imagine the shape of a cylinder from one projection (Fig. 81). It follows that in such cases there is no need for three projections. Rice. 81. Image of a cylinder in one view The dimensions of the cylinder and cone are determined by their height h and base diameter d. Construction methods

isometric projection cylinder and cone are the same. To do this, draw the x and y axes, on which a rhombus is built. Its sides are equal to the diameter of the base of the cylinder or cone. An oval is inscribed in the rhombus (see Fig. 66). Projections of a group of geometric bodies. Figure 83 shows the projections of a group of geometric bodies. Can you tell how many geometric bodies are included in this group? What kind of bodies are these? Rice. 83. Drawing of a group of geometric bodies Having examined the images, it can be established that it contains a cone, a cylinder and 1. There are checkers on the table, as shown in Figure 84, a. Based on the drawing, count how many checkers are in the first columns closest to you. How many checkers are there on the table? If you find it difficult to count them according to the drawing, try first stacking the checkers in columns using the drawing. Now try to answer the questions correctly.

Rice. 84. Exercises 2. Checkers are arranged in four columns on the table. In the drawing they are shown in two projections (Fig. 84, b). How many checkers are on the table if there are equal numbers of black and white? To solve this problem, you need not only to know the rules of projection, but also to be able to reason logically.

GRAPHIC WORK

Subject:COMPLEX DRAWING OF A GROUP OF GEOMETRIC BODIES

Goals: acquire practical skills in performing a complex drawing of a group of geometric bodies, learn how to competently and accurately carry out drawings, and develop spatial concepts.

EXERCISE: build on A3 format in three projections a group of geometric bodies, the relative positions of which are presented on a horizontal projection and an isometric projection (according to options).

Guidelines

Each subject, from the point of view spatial form, is either a geometric body or a combination of various geometric bodies bounded by curved or flat surfaces. To correctly draw a drawing of an object, you must be able to draw drawings of individual geometric bodies.

To develop spatial imagination, it is useful to make complex drawings of a group of geometric bodies and simple models from life. A visual representation of a group of geometric bodies is shown in Fig. 1.

The construction of a complex drawing of this group of geometric bodies should begin with a horizontal projection, since the bases of the cylinder, cone and hexagonal pyramid are projected onto the horizontal projection plane without distortion. By using vertical lines connections we build a frontal projection. We build a profile projection using vertical and horizontal communication lines.

Rice. 1

Sequence of graphic work

We begin the construction of geometric bodies from a top view, the relative position of which is presented in a horizontal projection and an isometric projection (in the version shown in the drawing above). Then, using vertical communication lines, we obtain a frontal projection, and construct a profile projection using vertical and horizontal communication lines. Next, in the remaining space we build an axonometry of these geometric bodies.

Projection of cylinders. The simplest is to construct orthogonal projections straight circular cylinder with a vertical axis.

Side surface The cylinder is formed by the movement of generatrix AB around its axis along the guide circle of its base. Figure 1a shows a visual representation of this cylinder. Figure 2b shows the sequence of constructing its three projections - horizontal, frontal, profile. To simplify the construction of the base of the cylinder, it is assumed to be located on the horizontal projection plane - H.

a) b)

Rice. 2

The construction begins with an image of the base of the cylinder, i.e. two projections of the circle (Fig.2b ). Since the circle is located on the H plane, its horizontal projection will be identical to the circle itself; the frontal and profile projection of this circle is a segment of a horizontal straight line of length. Equal to the diameter of the base circle. After constructing the base, we will draw two contour (outline) generatrices on the front and profile and plot the height of the cylinder on them. Next, we will draw a segment of the horizontal straight line, which is the frontal projection and the profile projection of the upper base of the cylinder. The horizontal projections of the upper and lower bases of the cylinder coincide (merge).

Projection of cones. A visual representation of a straight circular cone is shown in Fig. 3a. The lateral surface of this cone is formed by the movement of the generatrixS.B.near the axis of the cone along the guide - the circumference of the base.

a) b)

Rice. 3

The construction begins with an image of the base of the cone (Fig. 3b). Since the circle is located on the H plane, its horizontal projection will be identical to the circle itself; the frontal and profile projection of this circle is a segment of a horizontal straight line of length. Equal to the diameter of the base circle. After constructing the base on the frontal projection and the profile from the middle, we set aside the height of the cone (Fig. 3b). We connect the resulting vertex of the cone with straight lines to the ends of the frontal projection of the base and the profile projection of the base.

Projection of the pyramids. The construction of three projections of a hexagonal pyramid (Fig. 4a) resembles the construction of the previous figures.

a) b)

Rice. 4

We begin the construction from the base of the pyramid - regular hexagonal (Fig. 4b). It can be constructed using a compass by dividing the circle into six equal parts. Then, using vertical communication lines, we obtain the frontal and profile projections of the base and from their middle we restore the perpendicular and plot the height of the pyramid on it. We get the top. We connect the vertex by straight lines, which are the frontal projections of the ribs, with the vertices of the corners of the hexagon (the profile projections of the three rear ribs coincide).

Projection of a straight pentagonal prism. The construction of three projections of a straight pentagonal prism (Fig. 5a) also resembles the construction of the previous figures.

a) b)

Rice. 5

We begin the construction from the base of the prism - a regular pentagon (Fig. 5b). It can be constructed using a compass by dividing the circle into five equal parts. Then, using vertical communication lines, we obtain a frontal projection, where we depict five edges, two of which are invisible, and a profile projection, where three vertical edges are depicted. We get the top. As with the projections of a cylinder, the horizontal projections of the upper and lower bases coincide.

Options for tasks.

Summarizing, Homework

Literature:

Brodsky A.M. Engineering graphics(metalworking): textbook for secondary vocational education - M. "Academy", 2008

Brodsky A.M. Workshop on engineering graphics: tutorial for secondary vocational education - M. "Academy", 2008

Kuprikov M.Yu. Engineering graphics: Textbook for secondary vocational education – M. “Drofa”, 2010

Bogolyubov S.N. Assignments for the drawing course. – M., Higher. school, 2008

– State Public Scientific and Technical Library of Russia.

Topic "Projections of a group of geometric bodies."

Target: Teaching students graphic literacy, developing spatial thinking, identifying the level of development of intellectual qualities in students.

Tasks:

I. Educational: Create conditions for the development of visual memory, spatial imagination and imaginative thinking; teach how to identify projections of the simplest geometric bodies on a drawing and how to define them relative position; develop logical thinking and the ability to express one's thoughts graphically.

II. Developmental: : develop spatial representation and spatial thinking, rationality taking into account individual abilities. Continue to develop students’ general educational competencies.

III. Educational: To cultivate accuracy and precision when performing graphic works; to cultivate the principles of aesthetic perception of the surrounding objective environment.

Equipment: models of geometric bodies, slide “Drawing of a group of geometric bodies,” repetition tests, task cards, textbook, ruler, pencil, format, compass.

Lesson type: combined

Forms and methods of teaching: individual; differentiated, visual, practical; method of independent activity.
During the classes:

I. Organizational stage. Greetings. Checking readiness for the lesson. Organization of attention. Revealing the lesson plan.

II. Examination homework : establish the correctness, completeness and awareness of completing homework. What line will be obtained at the intersection of a cylinder with an inclined plane intersecting all its generators? (If a cylinder is cut by an inclined plane so that all its generatrices intersect, then the line of intersection of the side surface with this plane will be an ellipse, the size and shape of which depend on the angle of inclination of the cutting plane to the planes of the bases of the cylinder).

III. Repetition of covered topics(test).

Question 1: What geometric bodies did we study? (polyhedra and bodies of rotation).

Question 2: Name the polyhedra...
Question 3: Name the bodies of revolution...
Question 4: Why are bodies of revolution called that?

1. Because at the base of these bodies lies a circle

2. Because these bodies are formed by rotating a flat figure around an axis

3. These bodies can be rotated

Question 5: by rotating which figure did we get a cylinder?

1. Trapezoid

2. Rectangle

3. Triangle

Question 6: A geometric body has 2 bases, the side faces are trapezoids, name it:

1. Truncated cone

2. Truncated pyramid

Question 7: What quantities determine the size of a hexagonal prism?

1. Height and width

2. Height and side of the hexagon

3. The height and diameter of a circle circumscribed around the base

Question 8: What quantities determine the size of a triangular pyramid?

1. The height of the pyramid and the side of the triangle

2. The height of the pyramid and the dimensions of the base

3. The apothem of the pyramid and the dimensions of the base

Question 9: List geometric shapes that have such a frontal projection

IV. Updating the subjective experience of students:

A) Work from drawings to identify geometric bodies. Drawings of geometric bodies are offered in A3 format one by one. If students correctly name a geometric body based on projections, then by turning the format over, we are convinced of the correctness; a visual image of the geometric body is pasted there.

B) Creation of a problematic situation. A drawing of a group of geometric bodies is proposed. A critical point is created: we can do it or we can’t.

C) Reporting the topic of the lesson. Formation of goals together with students. Show social and practical significance the material being studied. Formulation of the problem. Actualization of subjective experience.

V. Stage of learning new material. Ensuring students’ perception, comprehension and primary memorization of new material.

Let's look at the images of the drawing of a group of geometric bodies shown in Fig. 120. The group consists of three geometric bodies. The first geometric body (see from left to right) on the projection planes V is depicted isosceles triangle, and on the projection plane H - a circle. Only a cone has such projections. The cone axis is perpendicular to the horizontal projection plane.

The second geometric body was displayed on two projection planes (H, by two rectangles, and on the frontal one - by a circle. Such projections are inherent in a cylinder, the axis of which is perpendicular to the frontal projection plane. The third geometric body was displayed on all projection planes by rectangles. This means that this is a rectangular parallelepiped, the faces of which parallel to the projection planes. Thus, we can come to the conclusion that the drawing represents a group of geometric bodies composed of a cone, a cylinder and a parallelepiped.

On the frontal projection of a group of geometric bodies, the projection of the cylinder covers part of the projection of the cone. This suggests that the cylinder is in front of the cone. The assumption is confirmed by other projections. The front face of a rectangular parallelepiped lies in the same plane with one of the bases of the cylinder - this conclusion can be made by considering the horizontal projection of a group of geometric bodies.

Based on image analysis, we come to the conclusion that the parallelepiped and cylinder are closer to us, and the cone is located behind them (Fig. 120). This is how drawings of a group of geometric bodies are read.
VI. The stage of initial testing of new knowledge. To establish the correctness and awareness of the studied material by students. Identify gaps in initial understanding. Correct the identified gaps.

1.What geometric bodies are shown in the drawing" (Fig. 121)? Which body is located closer to us? Which bodies touch each other? Find all the projections of each geometric body one by one.

Consider the “Drawing of a group of geometric bodies” and answer the questions:
- How many bodies does a group of geometric bodies consist of?
- Which geometric body is depicted as a rectangle on plane P, and as a circle on plane P3?
- how is the base of the pyramid located on the P2 plane?
- what body is displayed on the plane P3 as a square, and on the plane P1 as a rectangle and P2 as rectangles?
- how is the cylinder axis located in relation to the planes P1, P2, P3?
- what body was reflected on three planes in different forms?
Conclusion. The drawing shows a group of geometric bodies: a prism, a cylinder and a pyramid.
. Analyze the drawing and answer the question: in what order are the geometric bodies arranged in the group? Conclusion. Closer to us are a prism and a cylinder and a pyramid are located behind them.

V. Consolidating new material: ensure that students retain the knowledge and methods of action that they need to work . Checking the completeness and awareness of students' assimilation of new knowledge. Identifying gaps in initial understanding. Eliminating ambiguity in understanding.

Draw a drawing of a group of geometric bodies in a notebook, swapping the places of the bodies indicated in the drawing by numbers 1 and 2.

VI. Homework: textbook paragraph 3.6, prepare A3 format, prepare drawing tools for work.

VII. Lesson summary stage: evaluate the work of the class and individual students.

Reflection. Initiate students about their emotional state of its activities.

Mobilizing students for reflection. Did you like the lesson? Questions about a new topic?

In order to imagine the shape of a part when making drawings, it is convenient to mentally dissect the part into geometric bodies. Geometric bodies bounded by flat figures - polygons - are called polyhedra (Fig. 13). Their flat figures are called faces, and the lines of their intersection are called edges. An angle formed by faces converging at one point - a vertex - will be a polyhedral angle. For example. Prism and pyramid are polyhedra. Bodies of revolution are limited by surfaces that are obtained as a result of rotation about the axis of some line AB, called the generatrix.

Rice. 13. Polyhedral bodies and bodies of revolution

Prism projections

Constructing a projection of a regular straight hexagonal prism (Fig. 14) begins with making its horizontal projection - a regular hexagon. From the vertices of this hexagon, vertical communication lines are drawn and a frontal projection of the lower base of the prism is constructed. This projection is represented by a horizontal line segment. From this straight line upward, the height of the prism is plotted and a frontal projection of the upper base is constructed. Then the frontal projections of the ribs are drawn - segments of vertical straight lines equal to the height of the prism. The frontal projections of the anterior and posterior ribs coincide. Horizontal projections of the side faces are depicted as straight segments.

Rice. 14. Projection of a hexagonal prism

Pyramid projections

The construction of projections of a trihedral pyramid begins with the construction of a base, the horizontal projection of which represents the actual appearance of the triangle (Fig. 15). The frontal projection of the base is depicted as a horizontal straight line segment. From the horizontal projection s of the vertex of the pyramid, the frontal projection s’ of the vertex is obtained. By connecting point s’ with points 1’, 2’ and 3’, frontal projections of the edges of the pyramid are obtained.

Horizontal projections of the ribs are obtained by connecting the horizontal projection s of the top of the pyramid with the horizontal projections of the 1st, 2nd and 3rd vertices of the base.

Rice. 15. Pyramid projections

Cylinder projections

The lateral surface of a right circular cylinder is formed by the movement of a segment AB around a vertical axis along a guide circle. In Fig. 16, and a visual image of the cylinder is given. The construction of horizontal and frontal projections of the cylinder is shown in Fig. 16, b and 16, c.

Rice. 16. Projections of a cylinder

The construction begins with an image of the base of the cylinder, i.e. two projections of a circle. Because If the circle is located on the H plane, then it is projected onto this plane without distortion. The frontal projection of a circle is a segment of a horizontal straight line equal to the diameter of the base circle.

After constructing the base, two outline (extreme) generators are drawn on the frontal projection and the height of the cylinder is plotted on them. Draw a segment of a horizontal straight line, which is the frontal projection of the upper base of the cylinder.

Projections of cones.

A visual representation of a right circular cone is shown in Fig. 17, a. The lateral surface of the cone is formed by rotation of the generatrix BS about the axis of the cone along a guide - the circle of the base. The sequence of constructing two projections of a cone is shown in Fig. 17, b and c. Two projections of the base are preliminarily constructed. The horizontal projection of the base is a circle. If we assume that the base of the cone lies on the H plane, then the frontal projection will be a straight segment equal to the diameter of this circle. On the frontal projection, a perpendicular is restored from the middle of the base and the height of the cone is plotted on it. The resulting frontal projection of the top of the cone is connected by straight lines to the ends of the frontal projection of the base and a frontal projection of the cone is obtained.

Rice. 17. Projections of a cone

So, you already know that the shape of most objects is a combination of various geometric bodies or their parts. Therefore, to read and complete drawings you need to know how geometric bodies are depicted.

11.1. Projection of a cube and a rectangular parallelepiped. The cube is positioned so that its edges are parallel to the projection planes. Then they will be depicted on life-size projection planes parallel to them as squares, and on perpendicular planes as straight segments (Fig. 76).

The projections of a cube are three equal squares.
In the drawing of a cube and a parallelepiped, three dimensions are indicated: length, height and width.

In Figure 77, the part is formed by two rectangular parallelepipeds, each having two square faces. Pay attention to how the dimensions are shown on the drawing. Flat surfaces are marked with thin intersecting lines.
Thanks to conventional sign□ The shape of the part is clear and from one view.

11.2. Projection of regular triangular and hexagonal prisms. The bases of the prisms, parallel to the horizontal projection plane, are depicted on it in full size, and on the frontal and profile planes - as straight segments. The side faces are depicted without distortion on those projection planes to which they are parallel, and in the form of straight segments on those to which they are perpendicular (Fig. 78). Faces inclined to projection planes appear distorted on them.

The dimensions of the prisms are determined by their height and the size of the base figure. The dash-dot lines in the drawing indicate the axes of symmetry.

The construction of isometric projections of the prism begins from the base. Then perpendiculars are drawn from each vertex of the base, on which segments equal to the height are laid, and straight lines parallel to the edges of the base are drawn through the resulting points.

A drawing in a system of rectangular projections also begins with a horizontal projection.

11.3. Projection of a regular quadrangular pyramid. The square base of the pyramid is projected onto the horizontal plane H in full size. On it, diagonals depict the lateral ribs running from the tops of the base to the top of the pyramid (Fig. 79).

The frontal and profile projections of the pyramid are isosceles triangles.

The dimensions of the pyramid are determined by the length b of the two sides of its base and the height h.

The isometric projection of the pyramid begins to be built from the base. A perpendicular is drawn from the center of the resulting figure, the height of the pyramid is plotted on it and the resulting point is connected to the vertices of the base.

11.4. Projection of a cylinder and a cone. If the circles lying at the bases of the cylinder and cone are located parallel to the horizontal plane H, their projections onto this plane will also be circles (Fig. 80, b and d).

The frontal and profile projections of the cylinder in this case are rectangles, and the cones are isosceles triangles.
Please note that on all projections the axes of symmetry should be drawn, with which the drawings of the cylinder and cone begin.

The frontal and profile projections of the cylinder are the same. The same can be said about cone projections. Therefore, in this case, profile projections in the drawing are unnecessary. In addition, thanks to the 0 sign, it is possible to represent the shape of a cylinder in one projection (Fig. 81). It follows that in such cases there is no need for three projections. The dimensions of the cylinder and cone are determined by their height h and base diameter d.

The methods for constructing an isometric projection of a cylinder and a cone are the same. To do this, draw the x and y axes, on which a rhombus is built. Its sides are equal to the diameter of the base of the cylinder or cone. An oval is inscribed in the rhombus (see Fig. 66).

11.5. Projections of the ball. All projections of the ball are circles, the diameter of which is equal to the diameter of the ball (Fig. 82). Center lines are drawn on each projection.
Thanks to the diameter sign, the ball can be depicted in one projection. But if it is difficult to distinguish the sphere from other surfaces from the drawing, add the word “sphere”, for example: “Sphere dmameter 45”.

11.6. Projections of a group of geometric bodies. Figure 83 shows the projections of a group of geometric bodies. Can you tell how many geometric bodies are included in this group? What kind of bodies are these?

Having examined the images, we can establish that it contains a cone, a cylinder and a rectangular parallelepiped. They are located differently relative to the projection planes and each other. How exactly?

The axis of the cone is perpendicular to the horizontal plane of projections, and the axis of the cylinder is perpendicular to the profile plane of projections. Two faces of the parallelepiped are parallel to the horizontal projection plane. On a profile projection, the image of a cylinder is to the right of the image of a parallelepiped, and on a horizontal projection it is below. This means that the cylinder is located in front of the parallelepiped, therefore part of the parallelepiped in the front projection is shown by a dashed line. From horizontal and profile projections it can be established that the cylinder touches the parallelepiped.

The frontal projection of the cone touches the projection of the parallelepiped. However, judging by the horizontal projection, the parallelepiped does not touch the cone. The cone is located to the left of the cylinder and parallelepiped. In profile projection, it partially covers them. Therefore, invisible sections of the cylinder and parallelepiped are shown with dashed lines.

20. How will the profile projection in Figure 83 change if a cone is removed from the group of geometric bodies?

Entertaining tasks

1. There are checkers on the table, as shown in Figure 84, a. Based on the drawing, count how many checkers are in the first columns closest to you. How many checkers are there on the table? If you find it difficult to count them according to the drawing, try first taking and stacking the checkers into columns using the drawing. Now try to complete the tasks correctly.

2. There are checkers on the table in four columns (Fig. 84, b). In the drawing they are shown in two projections. How many checkers are on the table if there are equal numbers of black and white? To solve this problem, you need not only to know the rules of projection, but also to be able to reason logically.