Construction of complex drawings of geometric bodies. Graphic work plan. Complex drawing of a group of geometric bodies. Finding projections of points

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

11.1. Projecting a cube and rectangular parallelepiped. The cube is positioned so that its faces are parallel to the projection planes. Then they will be depicted on the projection planes parallel to them in full size - by squares, and on perpendicular planes - by line segments (Fig. 76).

Three equal squares are projections of a cube.
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 edges. Pay attention to how the dimensions are plotted in the drawing. Flat surfaces are marked with thin intersecting lines.
Thanks to the conventional sign □ the shape of the part is clear and one by one.

11.2. Projecting 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 - by straight line segments. The side faces are depicted without distortion on those projection planes to which they are parallel, and in the form of line segments on those to which they are perpendicular (Fig. 78). The faces inclined to the projection planes are shown distorted on them.

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

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

A drawing in a rectangular projection system also begins to be performed with a horizontal projection.

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

Frontal and profile projections of the pyramid - isosceles triangles.

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

An isometric view of the pyramid begins to build from the base. A perpendicular is drawn from the center of the resulting figure, the height of the pyramid is laid on it and the resulting point is connected to the tops of the base.

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

In this case, the frontal and profile projections of the cylinder are rectangles, and the cones are isosceles triangles.
Note that on all projections, the axes of symmetry should be applied, from which the drawing of the cylinder and cone begins.

The frontal and profile projections of the cylinder are the same. The same can be said about cone projections. Therefore, in this case, the profile projections in the drawing are superfluous. In addition, thanks to the sign 0, it is possible to represent the shape of the 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. For this, the x and y axes are drawn, 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. Ball projections.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 the drawing is difficult to distinguish the sphere from other surfaces, add the word "sphere", for example: "Sphere diameter 45".

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

After examining the images, you can establish that a cone, a cylinder and a rectangular parallelepiped are given on it. They are located differently with respect to the projection planes and each other. How exactly?

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

The frontal projection of the cone touches the projection of the parallelepiped. However, based on the horizontal projection, the parallelepiped is not touching the cone. The cone is located to the left of the cylinder and parallelepiped. On a profile projection, it partially covers them. Therefore, the 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. Checkers are on the table, as shown in Figure 84, a. Count on the drawing 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, first try to take and put the checkers in columns using the drawing. Now try to complete the tasks correctly.

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

Figure: 76. Cube and parallelepiped: a - projection; b, d drawings in the system of rectangular projections; c, d - isometric projections

Figure: 77. Image of a part in one form

Figure: 78. Prisms:
a, d - projection; b, e - drawings in the system of rectangular projections; c, f - isometric projections

To perform an isometric projection of any part, you need to know the rules for constructing isometric projections of flat and volumetric geometric shapes.

Rules for constructing isometric projections of geometric shapes. The construction of any flat figure should begin with drawing the axes of isometric projections.

When constructing an isometric projection of a square (Fig. 109) from point O along the axonometric axes, half the length of the side of the square is laid on both sides. Straight lines parallel to the axes are drawn through the obtained serifs.

When constructing an isometric projection of a triangle (Fig. 110) along the X-axis from point 0 in both directions, set aside segments equal to half the side of the triangle. The height of the triangle is plotted along the Y-axis from point O. Connect the resulting serifs with straight line segments.

Figure: 109. Rectangular and isometric projections of a square

Figure: 110. Rectangular and isometric projections of a triangle

When constructing an isometric projection of a hexagon (Fig. 111) from point O along one of the axes, the radius of the circumscribed circle is laid (in both directions), and on the other - H / 2. Through the obtained serifs, straight lines are drawn parallel to one of the axes, and the length of the side of the hexagon is laid on them. Connect the resulting serifs with straight line segments.

Figure: 111. Rectangular and isometric projections of the hexagon

Figure: 112. Rectangular and isometric projections of a circle

When constructing an isometric projection of a circle (Fig. 112) from point O along the coordinate axes, segments equal to its radius are laid. Straight lines parallel to the axes are drawn through the received serifs, obtaining an axonometric projection of the square. From vertices 1, 3 draw arcs CD and KL with a radius of 3C. Points 2 are connected with 4, 3 with C and 3 with D. At the intersections of the straight lines, centers a and b of small arcs are obtained, after which they get an oval, replacing the axonometric projection of the circle.

Using the described constructions, it is possible to perform axonometric projections of simple geometric bodies (Table 10).

10. Isometric projections of simple geometric bodies

Methods for constructing an isometric projection of a part:

1. The method of constructing an isometric projection of a part from a shaping face is used for parts whose shape has a flat face, called a shaping face; the width (thickness) of the part is the same throughout, there are no grooves, holes and other elements on the side surfaces. The sequence for constructing an isometric projection is as follows:

1) building the axes of an isometric projection;

2) construction of an isometric projection of the forming face;

3) construction of projections of the remaining faces by means of the image of the edges of the model;

Figure: 113. Creation of an isometric projection of a part, starting from the formative face

4) outline isometric projection (fig. 113).

The method of constructing an isometric projection based on the sequential removal of volumes is used in cases where the displayed shape is obtained as a result of removing any volumes from the original shape (Fig. 114).
The method of constructing an isometric projection based on sequential increment (addition) of volumes is used to perform an isometric image of a part, the shape of which is obtained from several volumes connected in a certain way with each other (Fig. 115).
The combined method of constructing an isometric projection. An isometric projection of a part, the shape of which is obtained as a result of a combination of various shaping methods, is performed using a combined construction method (Fig. 116).

An axonometric projection of a part can be performed with an image (Fig. 117, a) and without an image (Fig. 117, b) of invisible parts of the form.

Figure: 114. Construction of an isometric projection of a part based on sequential removal of volumes

Figure: 115 Building an isometric projection of a part based on sequential volume increments

Figure: 116. Using a combined method of constructing an isometric projection of a part

Figure: 117. Variants of the image of isometric projections of the part: a - with the image of invisible parts;
b - without the image of invisible parts

Topic "Projections of a group of geometric bodies."

Goal:Teaching students graphic literacy, the development of spatial thinking, to identify the level of formation of intellectual qualities in students.

Tasks:

I. Educational: Create conditions for the development of visual memory, spatial imagination and imaginative thinking; to teach how to define projections of the simplest geometric bodies on a drawing and define them mutual arrangement; develop logical thinking and the ability to express your thoughts in graphic language.

II. Developing: : develop spatial representation and spatial thinking, rationality, taking into account individual abilities. Continue the formation of general educational competencies of students.

III. Educational: To educate accuracy and precision when performing graphic works; to educate the beginnings of aesthetic perception of the object environment surrounding him.

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

Lesson type: combined

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

I... Organizational stage.Greeting. Checking readiness for the lesson. Organization of attention. Disclosure of the lesson plan.

II. Homework check: establish the correctness, completeness and awareness of homework. What line will turn out in the intersection of the cylinder with an inclined plane, intersecting all its generators? (If the cylinder is cut with an inclined plane so that all its generators intersect, then the line of intersection of the lateral surface with this plane will be an ellipse, the size and shape of which depend on the angle of inclination of the secant plane to the planes of the cylinder bases).

III... Repetition of covered topics(test).

Question 1: What geometric bodies have we studied? (polyhedra and bodies of revolution).

Question 2: What are 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: which shape we rotated, we got a cylinder.

1. Trapezoid

2. Rectangle

3. Triangle

Question 6: The geometric body has 2 bases, the side faces are trapeziums, name it:

1. Truncated cone

2. Truncated pyramid

Question 7: What is the size of the hexagonal prism?

1. Height and width

2. The height and side of the hexagon

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

Question 8: What is 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... Actualization of the subjective experience of students:

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

B) Creation of a problem situation.A drawing of a group of geometric bodies is offered. A critical point is created: we can - we do not know how.

C) Message of the topic of the lesson... Forming 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... The stage of learning new material... Providing perception, comprehension and primary memorization of new material by students.

Consider the drawing images 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 and is depicted by an isosceles triangle, and on the projection plane H - by a circle. Only a cone has such projections. The axis of the cone is perpendicular to the horizontal projection plane.

The second geometric body was mapped onto two projection planes (H, two rectangles, and on the frontal one - in a circle. Such projections are inherent in a cylinder whose axis is perpendicular to the frontal projection plane. are parallel to the projection planes.Thus, we can come to the conclusion that the drawing shows a group of geometric bodies made up 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 the analysis of the images, we come to the conclusion that the parallelepiped and the cylinder are closer to us, and the cone is located behind them (Fig. 120). This is how the drawings of a group of geometric bodies are read.
VI... The stage of primary testing of new knowledge. To establish the correctness and awareness of the studied material by students. Identify gaps in primary 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 in turn.

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 on the plane P is shown as a rectangle, and on the plane P3 - as a circle?
- how is the base of the pyramid located on the plane P2?
- which body was mapped onto the P3 plane as a square, and on the P1 plane by a rectangle and P2 - by rectangles?
- how is the axis of the cylinder located to the planes P1, P2, P3?
- which body was reflected on three planes in different shapes?
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 in the group? Conclusion. Closer to us are the prism and, the cylinder and the pyramid are located behind them.

V. Securing new material:to ensure the consolidation of students' knowledge and methods of action that they need to work . Checking the completeness and awareness of the assimilation of new knowledge by students. Identifying gaps in primary comprehension. Elimination of ambiguity of comprehension.

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

VI. Homework:paragraph of the tutorial 3.6, prepare A3 format, prepare drawing tools for work.

Vii. The stage of summarizing the lesson:assess the performance 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?

\u003e\u003e Drafting: Projections of a group of geometric bodies

The second geometric body was mapped onto two projection planes (H, two rectangles, and on the frontal one - in a circle. Such projections are inherent in a cylinder whose axis is perpendicular to the frontal projection plane. are 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 taper. The assumption is confirmed by other projections. The front face of the 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.

Questions and tasks
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 in turn.
Figure 2. 122 is a drawing of a group of geometric bodies. Consider it carefully and answer the questions:
- How manygeometries are shown in the drawing? Name them.

- What geometric bodies touch each other? How did you define it?
- Are there rotation bodies in the drawing? If so, name them.
- What does the dashed line mean in the left view? What do the dot-and-dash lines mean?
- What dimensions does each geometric body have? Take measurements in the drawing.
3. Using the drawing shown in fig. 123, finish drawing a frontal projection and build a profile projection of a group of geometric bodies. Complete her technical drawing.
4 Fig. 124 technical drawings of three groups of geometric bodies are given. Draw one of the groups of geometric bodies in the system of three projections.

N.A.Gordeenko, V.V.Stepakova - Drawing., Grade 9
Submitted by readers from internet sites

Lesson content lesson outline support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photos, pictures charts, tables, schemes humor, jokes, jokes, comics parables, sayings, crosswords, quotes Supplements abstracts articles chips for the curious cheat sheets textbooks basic and additional vocabulary of terms others Improving textbooks and lessons bug fixes in the tutorial updating a fragment in the textbook elements of innovation in the lesson replacing outdated knowledge with new ones For teachers only perfect lessons calendar plan for the year guidelines discussion agenda Integrated lessons

Lesson objectives:

to consolidate knowledge about geometric bodies, skills and abilities to build drawings of polyhedra;
develop spatial representations and spatial thinking;
to form a graphic culture.

Lesson type: combined.

Lesson equipment: MIMIO interactive whiteboard, multimedia projector, computers, mimo project for an interactive whiteboard, multimedia presentation, Compass-3D LT program.

DURING THE CLASSES

I. Organizational moment

1. Greetings;

2. Checking the attendance of students;

3. Checking the readiness for the lesson;

4. Completing the class journal (and electronic)

II. Repetition of previously learned material

The mimo project is open on the interactive whiteboard

Sheet 1. In your math class, you studied geometric solids. You see several bodies on the screen. Let's remember their names. Students give names to geometric bodies, if there are difficulties, I help. (Fig. 1).

1 - quadrangular prism
2 - truncated cone
3 - triangular prism
4 - cylinder
5 - hexagonal prism
6 - cone
7 - cube
8 - truncated hexagonal pyramid

Sheet 4... Task 2. Geometric bodies and names of geometric bodies are given. We call the student to the board and together with him we drag polyhedrons and bodies of revolution under the names, and then we drag the names of the geometric bodies (Fig. 2).

We conclude that all bodies are divided into polyhedrons and bodies of revolution.

We turn on the presentation "Geometric Solids" ( application ). The presentation contains 17 slides. You can use the presentation in several lessons, it contains additional material (slides 14-17). From slide 8 there is a hyperlink to Presentation 2 (cube sweeps). Presentation 2 contains 1 slide, which shows 11 cube unfolded (they are links to videos). The lesson used an interactive whiteboard MIMIO, and students work on computers (doing practical work).

Slide 2. All geometric bodies are divided into polyhedrons and bodies of revolution. Polyhedra: prism and pyramid. Bodies of revolution: cylinder, cone, ball, torus. Students trace the diagram into a workbook.

III. Explanation of the new material

Slide 3.Consider a pyramid. We write down the definition of the pyramid. The top of the pyramid is the common top of all faces, denoted by the letter S. The height of the pyramid is the perpendicular, lowered from the top of the pyramid (Fig. 3).

Slide 4.Correct pyramid. If the base of the pyramid is a regular polygon, and the height drops to the center of the base, then the pyramid is correct.
In a regular pyramid, all side edges are equal, all side edges are equal isosceles triangles.
The height of the triangle of the side face of a regular pyramid is called - apothem of the regular pyramid.

Slide 5. Animation of building a regular hexagonal pyramid with the designation of its main elements (Fig. 4).

Slide 6... We write down the definition of a prism in a notebook. A prism is a polyhedron with two bases (equal, parallel polygons), and the side faces of a parallelogram. The prism can be quadrangular, pentagonal, hexagonal, etc. The prism is named after the figure lying at the base. Animation of building a regular hexagonal prism with the designation of its main elements (Fig. 5).

Slide 7.A regular prism is a straight prism with a regular polygon at its base. The parallelepiped is a regular quadrangular prism (Fig. 6).

Slide 8.A cube is a parallelepiped, all the faces of which are squares (Fig. 7).

(Additional material: there is a hyperlink to the presentation with cube sweeps on the slide, 11 different sweeps in total).
Slide 9.To write down the definition of a cylinder, a body of revolution is a cylinder formed by rotating a rectangle around an axis passing through one of its sides. Cylinder retrieval animation (Fig. 8).

Slide 10.A cone is a body of revolution formed by the rotation of a right-angled triangle around an axis passing through one of its legs (Fig. 9).

Slide 11.A truncated cone is a body of revolution formed by the rotation of a rectangular trapezoid around an axis passing through its height (Fig. 10).

Slide 12.A ball is a body of revolution formed by the rotation of a circle around an axis passing through its diameter (Fig. 11).

Slide 13.A torus is a body of revolution formed by the rotation of a circle around an axis parallel to the diameter of the circle (Fig. 12).

Students write down the definitions of geometric bodies in a notebook.

IV. Practical work "Building a drawing of the correct prism"

Switching to the mimio project

Sheet 7... A triangular regular prism is given. A regular triangle lies at the base. Prism height \u003d 70mm and base side \u003d 40mm. We consider the prism (the direction of the main view is shown by the arrow), we determine flat figureswhich we will see in the front, top and left views. We take out the images of the views and place them on the drawing field (Fig. 13).

Students independently draw a regular hexagonal prism in the "Compass - 3D" program. Prism dimensions: height - 60 mm, diameter of the circumscribed circle around the base - 50 mm.
Building a drawing from a top view (Fig. 14).

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