Projection drawing of a group of geometric bodies, Fig. 83. Axonometric projections of geometric bodies. Examples of solving problems on constructing projections of figures

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 determine their relative positions; develop logical thinking and the ability to express your thoughts in graphic language.

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 work; 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. Checking 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 revolution).

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 the 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. Demonstration of the social and practical significance of 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 as an isosceles triangle, and on the projection plane H - as 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 shows 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 their activities.

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

To develop spatial imagination, it is useful to make complex drawings of a group of geometric bodies and simple models from life.

Figure 147

A visual representation of a group of geometric bodies is shown in Figure 147, a. 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. Using vertical communication lines, a frontal projection of the figures is constructed. A profile projection is constructed using vertical and horizontal communication lines (Figure 147, b) drawn from the vertices and points of the base line.

8 Technical drawing

Technical drawing is a visual image that has the basic properties of axonometric projections or a perspective drawing, made without the use of drawing tools, on a visual scale, in compliance with proportions and possible shading of the form.

Engineers, designers, and architects, when designing new models of equipment, products, and structures, use technical drawings as a means of fixing the first, intermediate and final solutions to a technical concept. In addition, technical drawings serve to verify the correct reading of a complex shape shown in a drawing.

A technical drawing can be performed using the central projection method, and thereby obtain a perspective image of the object, or the parallel projection method (axonometric projections), constructing a visual image without perspective distortions.

Technical drawing can be performed without revealing chiaroscuro, with shading of volume, as well as with the transfer of color and material of the depicted object.

In technical drawings, it is allowed to reveal the volume of objects using the techniques of shading (parallel strokes), scribbling (strokes applied in the form of a grid) and dot shading.

8.1 Shading methods

Chiaroscuro is applied to a linear drawing by shading, scribbling, shading with dots and other methods.

8.1.1 General concepts

To give the drawing greater clarity and expressiveness in technical drawing, conventional means of conveying volume using shading - chiaroscuro - are used. Chiaroscuro called the distribution of light on the surfaces of an object. The illumination of an object depends on the angle of inclination of the light rays. In technical drawing, it is conventionally accepted that the light source is located at the top left and behind the painter. Light rays make an angle of inclination to the horizon approximately equal to 45 ° . The convexity of the drawing of an object is achieved by gradation of light and shadow: the most illuminated surfaces are shaded lighter than surfaces further away from the light.

Chiaroscuro consists of the following elements: own shadow, falling shadow, reflex, halftone, light and highlight.

Your own shadow called a shadow located on the unlit part of an object.

Falling shadow called the shadow cast by an object on any surface. Since the technical drawing is mainly of a conventional, applied nature, falling shadows are not shown on it.

Reflex called reflected light on the surface of an object in its unlit part. With the help of a reflex, a convex, stereoscopic pattern is created.

Dimly lit areas on the surfaces of an object are called halftones. Halftones make a gradual, smooth transition from shadow to light so that the drawing does not turn out to be too contrasting. Halftone reveals the volumetric shape of an object.

Light- the most illuminated part of the surface of an object.

Blik- the lightest spot on an object. In technical drawing, highlights are shown mainly on surfaces of revolution.

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. 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. 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).

Rice. 79. Pyramid: projection: b drawing in a system of rectangular projections; in isometric projection 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. 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).

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. 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). 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 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. 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 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.

Prism A prism is a polyhedron whose side faces are rectangles or parallelograms, and the bases are two equal polygons. If the base of a prism is regular polygons, and the height is perpendicular to the base, then the prism is regular and straight. Depending on the number of sides of the base, prisms can be triangular, quadrangular, etc.

Pyramid A pyramid is a polyhedron whose side faces are triangles with a common vertex. At the base of the pyramid is a polygon. Depending on the number of sides of the base, the pyramid is called three-, four-, pentagonal, etc. If the base of the pyramid is a regular polygon and the height is perpendicular to the base, then the pyramid is regular and straight

Right circular cone A right circular cone is a body of revolution bounded by a conical surface and a plane perpendicular to the axis of rotation. For a right circular cone, the conical surface is formed by the rotation of a straight line (generator) intersecting the axis of rotation at a point (vertex) around this axis of rotation. A cone whose axis is perpendicular to the horizontal plane of projection is called a straight cone.

Construction of projections of a straight regular hexagonal pyramid d=50 mm h=60 mm s S S x y"y" y z

Determination of the missing projections of point “a” located on the surface of the pyramid according to a given frontal projection s 1 2(6) 3(5) 4 S 56 S 6(5) 1(4) 2(3) a´ n´ n a a

Determination of the missing projections of points “a” and “b” located on the surface of the cylinder, according to the given frontal projections Z y Yx a´ a a" b´ c c"

Hello, dear readers! The topic of our lesson today is creating projections of a group of geometric bodies. I am creating this lesson at the request of a reader.

As you remember, we had a lesson on creating . And for the development of spatial imagination, it is also proposed to perform a complex drawing groups of geometric bodies.

So, let's get down to business. Let's take the task from Bogolyubov's collection, page 81, option 10.

Creating models of a group of geometric bodies

We will create sketches for three models in the zx (horizontal) plane, xyz isometry. And the last sketch, a hexagon, will be made on the xy plane.

Our procedure is as follows: a) we create four sketches independent of each other, b) we create models of bodies using form-building operations of extrusion and sections, c) we create three projections of a group of geometric bodies.

1 Create the first sketch of a hexagon, which is the base of the pyramid, zx plane.

2 Create an auxiliary plane parallel to zx at a distance of 60 mm. We create a point in this plane - the top of the pyramid.

Using the “Operation by Sections” command we create a pyramid.

3 Create a sketch of the base of the truncated cone.

4 Similar to the pyramid, we create an auxiliary plane at a distance of 60 mm. On this plane we make a sketch of the upper base of the truncated cone - a circle with a diameter of 14 mm.

Using the operation of sections we create a cone model.