Direct projections. Presentation "projection types" Projection three projection planes presentation

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The straight line is perpendicular to the frontal plane of the projections P2 and parallel to P1 and P3. The frontal projection A2 B2 degenerates into a point. On P1 and P3, the straight line is projected in full size. Projection A1 B1 is perpendicular to the x coordinate axis Spatial picture Complex drawing A B x Frontally projecting line (P2) P 1

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x Spatial picture Complex drawing A B Horizontally projecting line (P1) The line is perpendicular to P1, so its horizontal projection A1 B1 degenerates into a point. With respect to P2 and P3, the straight line is parallel and is depicted on these projection planes in full size. The projection A2 B2 is perpendicular to the coordinate axis x P 2 1 P 1

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All points of the straight line AB are equidistant from the profile plane of the projections P3 and have the same coordinate x (x = const). Horizontal A1 B1 and frontal A2 B2 projections of a straight line are perpendicular to the x-axis. Profile projection A3 B3, angles and have a natural size on P3 Spatial picture Complex drawing z O x y1 y3 B A p Level lines: profile straight line (p P3) B 3 z y

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Spatial picture Complex drawing x B f Level lines: frontal (f P2) A All points of line AB are equidistant from the frontal plane of projections P2 and have the same coordinate y (y= const). The horizontal projection of the front A1 B1 is parallel to the x-axis. Frontal projection of the front A2 B2 , angles and are depicted in full size on P2 y=const y=const

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All points of the straight line AB are equidistant from the horizontal plane of projections P1 and have the same applicate z= const. The frontal projection of the horizontal A2 B2 is parallel to the x-axis. Horizontal projection of the horizontal A1 B1, angles and are depicted in full size on P1 Spatial picture Complex drawing x h B A Level lines: horizontal (h P1) z=const

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In the drawing, the projections of a line segment in general position have distorted metric characteristics, none of its projections is parallel to the coordinate axes and is not perpendicular to them. The line of general position is inclined to all projection planes. The line of general position k

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For a straight line of private position on a complex drawing, the natural values ​​​​of any of its characteristics are determined. The level line is projected without distortion onto the projection plane to which it is parallel. One of the projections of the projecting line degenerates into a point The line of particular position is parallel or perpendicular to one of the projection planes The line parallel to one of the projection planes is called the level line: Horizontal level line (horizontal) h P1 Frontal level line (frontal) f P2 Profile line p P3 A line perpendicular to one of the projection planes is called a projecting line: Horizontally projecting line P1 Frontally projecting line P2 Profile projecting line P3 Direct lines of particular position

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Metric characteristics of the segment: n.v. - natural size of the segment; is the angle of inclination of the segment to the plane П1; is the angle of inclination of the segment to the plane P2; - angle of inclination of the segment to the plane P3 B A Position of the straight line relative to the projection planes N.v. A 2 B 1 B 2 A 1 B 3 A 3 z y

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To build a profile projection of a straight line on an axisless drawing, a drawing constant k is drawn at an angle of 45 . With its help, along the communication lines, a profile projection of the straight line A3 B3 is obtained, the position of which is determined by the differences in the coordinates z and y k 45 An axisless drawing is a drawing in which there are no projection axes An axisless drawing 45 z B 1

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The projections of the line m pass through pairs of corresponding projections of points: the horizontal projection of the line m1 - through A1 and B1; frontal projection of the straight line m2 - through A2 and B2 x Spatial picture Complex drawing Projections of the straight line x O A B m

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The position of the line m in space is determined by two arbitrary points A and B lying on this line. This is the most convenient way to set a straight line. The straight line m is considered given if the projections of its two points A and B are constructed on the complex drawing Spatial picture Projections of the straight line O A B m

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Metric problems Task 1. Determine the distance from point A to the straight line l by changing the projection planes P4 P1 P4 l 2. P5 P4 P5 l AK- the desired distance In the second transformation, we introduce a new projection plane P5 perpendicular to the straight line l so that the straight line takes the projecting position. On P5 we determine the actual size A5 K5 of the perpendicular AK P1 P2 x l2 A1 l1 A2 P4 P5 x2 l4 P1 P4 x1 K1 K2

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Metric tasks Task 1. Determine the distance from point A to the straight line l by changing the projection planes The required distance is a perpendicular. Let us introduce a new projection plane P4 parallel to the straight line l so that the straight line takes a particular position of the level. According to the right angle projection theorem, the projection of the desired distance А4К4 l4 is determined on the plane of projections П4 П4 П1 П4 l П1 П2 x l2 А1 l1 А2 l4 П1 П4 x1

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Mutual position of two lines. Crossing lines do not intersect and are not parallel to each other. Projections of skew lines can be parallel, because straight lines m and n lie in parallel planes. Projections of intersecting lines may have an intersection, because lines m and n are not parallel to each other. 1 and 2 are competing points belonging to different lines m n m1 n1 m2 n2 x m 1 m n x n 1 2

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Mutual position of two lines Parallel lines have no common points Projections of parallel lines do not intersect. Projections of the same name are parallel or coincide if the parallel lines lie in the projecting plane n m x n 1 m n m1 n1 m2 n2 m 1 n 1 m 2 n 2 m 2 n 2 m 1

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Mutual position of two lines Intersecting lines have one common point B A D C K x C 2 AB CD \u003d K (K1, K2) A1 B1 C1 D1 \u003d K1 A2 B2 C2 D2 \u003d K2 The intersection point of K lines AB and CD is projected into the intersection points of the corresponding projections straight lines: on P1 - this is the point K1; on P2 - point K2. The intersection points K1 and K2 of the same-name projections of lines lie on the same line of communication B 1 A 1 A 2 B 2 D 1 D 2 C 2 C 1 A 1 A 2 B 2 B 1 D 2 C 1 D 1

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Determination of the natural size of the segment and its angles of inclination to the projection planes New projections of points A1 and B1 are located on the corresponding traces of the level frontal planes Ф(Ф1) and Ф (Ф1) . On P1 we have n.v. cut and angle

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Determination of the natural size of the segment and its angles of inclination to the projection planes x Scheme: D2 The horizontal projection of the straight line (A1 B1 A1 B1) is placed parallel to the x axis. The frontal projection (determining the n.v. of the segment and angle) is set by new projections of points A2 and B2 located on the corresponding traces of the horizontal level planes Г(Г2) and Г(Г2)

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Determination of the natural size of the segment and its angles of inclination to the projection planes. This segment AB occupies general position, we transform it into a frontal line of the level by moving the ends of the segment along the horizontal planes of the level according to the scheme

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Determination of the natural size of the segment and its angles of inclination to the projection planes Scheme: To determine the angle, the straight line AB must be rotated around the i-axis P2 to the horizontal position. The axis passes through point A, which is stationary. Point B2 rotates along an arc of a circle centered at point i2 to position B2 A2 of the x-axis. On P1, the angle and segment AB are not distorted

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Determination of the natural size of the segment and its angles of inclination to the projection planes Scheme: To simplify, the horizontally projecting axis of rotation l is drawn through point B, which remains motionless. Point A1 describes an arc of a circle centered at point l1 so that B1 A1 of the x axis. Then the line AB will take the position of the front. On P2, the angle and segment AB are not distorted

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Determination of the natural size of the segment and its angles of inclination to the projection planes x A1 B1 A2 B2 P2 P1 x1 P4 P1 A4 B4 The x2 axis of the new projection plane P5 will be drawn parallel to the frontal projection of the segment A2 B2. This transformation preserves the y coordinates of the points. On P5, the natural size of the segment and its angle of inclination to the projection plane P2 x2 P2 P5 A5 V5 are determined. Scheme:

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Definition of n.v. of the segment and its angles of inclination to the projection planes (method of replacing the projection planes) Let's draw the x1 axis of the new projection plane P4 parallel to the horizontal projection of the segment A1 B1. This transformation preserves the z-coordinates of the points. On P4, the actual size of the segment and its angle of inclination to the projection plane P1 x1 P4 P1 A4 V4 are determined.

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Slides captions:

Projection types of projection, projection onto one projection plane

Projection is the process of constructing an image of an object on a plane. The resulting image is called the projection of the object. The word projection comes from the Latin projection - throwing forward. In this case, we look (cast a glance) and display what we see on the plane of the sheet. PROJECTION

PROJECTION OF POINT a A H Projection plane (H) Projection beam (Aa) Projected point (A) Projection of point A on the plane (a)

PROJECTION Projection is the process of constructing a projection of an object. Projection plane - the plane on which the projection is obtained. A projecting beam is a straight line with which a projection of vertices, faces, edges is built.

TYPES OF PROJECTION

CENTRAL PROJECTION If the projecting rays come from one point, then such a projection is called central. The point from which the projection comes out is the center of the projection. EXAMPLE: photographs and film frames, shadows cast from an object by the rays of an electric light bulb.

PARALLEL PROJECTION If the projecting beams are parallel to each other, then such a projection is called parallel. An example of a parallel projection can be conditionally considered the sun's shadows of objects, as well as rain streams.

PARALLEL PROJECTION Oblique projection - the projecting beams are parallel and fall on the projection plane at an acute angle. Rectangular projection - the projecting beams are parallel and fall on the projection plane at an angle of 90 degrees.

PROJECTION ON THE ONE PLANE OF PROJECTIONS The plane located in front of the viewer is called the frontal one and is denoted by the letter V. The object is placed in front of the plane so that its two surfaces are parallel to this plane and projected without distortion.

DRAWING OF THE DETAILS Based on the resulting projection, we can judge the height, length and diameter of the hole. What is the thickness of the object? s6

What "projection" did the jets of water give in each case? Bucket under shower Bucket under sheer rain

EXERCISE FOR CONSOLIDATION № p / n New concepts Definition 1 Image on the plane. 2 The plane on which the projection is obtained. 3 Straight line, with which the object is projected onto the plane. 4 Projection, in which the projecting rays come out from one point. 5 Projection in which the projecting beams are parallel to each other. 6 Projection, in which the projecting rays fall on the projection plane at a right angle. 7 Projection in which the projecting rays do not fall on the projection plane at a right angle. Projection beam, central projection, projection, oblique projection, projection plane, parallel projection, rectangular projection. Projection. projection plane. projecting beam. central projection. Parallel projection. Rectangular projection. oblique projection.

Sections: Technology

Goals and objectives of the lesson:

educational: show students how to use the rectangular projection method when making a drawing;

The need to use three projection planes;

Create conditions for the formation of skills to project an object onto three projection planes;

developing: develop spatial representations, spatial thinking, cognitive interest and Creative skills students;

nurturing: responsible attitude to drawing, to cultivate a culture of graphic work.

Methods, teaching techniques: explanation, conversation, problem situations, research, exercises, frontal work with the class, creative work.

Material support: computers, presentation "Rectangular projection", tasks, exercises, exercise cards, presentation for self-examination.

Type of lesson: a lesson for consolidating knowledge.

Vocabulary work: horizontal plane, projection, projection, profile, research, project.

During the classes

I. Organizational part.

Message about the topic and purpose of the lesson.

Let's spend competition lesson, for each task you will receive a certain number of points. The class will be graded based on the points scored.

II. A review of projection and its types.

Projection is the mental process of constructing images of objects on a plane.

Repetition is carried out using the presentation.

1. The students are given problem situation . (Presentation 1)

Analyze the geometric shape of the detail on the frontal projection and find this detail among the visual images.

From this situation, it is concluded that all 6 parts have the same frontal projection. This means that one projection does not always give a complete picture of the shape and design of the part.

What is the way out of this situation? (Look at the detail from the other side).

2. There was a need to use another projection plane. (Horizontal projection).

3. The need for a third projection arises when even two projections are not enough to determine the shape of an object.

Sizing:

• in frontal projection length and height;
• on a horizontal projection - lenght and width;
• on a profile projection - width and height.
  

Conclusion: it means that in order to learn how to make drawings, you need to be able to project objects onto a plane.

Exercise 1

Insert the missing words in the definition text.

1. There is _______________ and ______________ projection.

2. If ______________ rays come out from one point, the projection is called ______________.

3. If ______________ rays are directed in parallel, the projection is called _____________.

4. If ______________ rays are directed parallel to each other and at an angle of 90 ° to the projection plane, then the projection is called ______________.
5. A natural image of an object on the projection plane is obtained only with ______________ projection.

6. Projections are located relative to each other______________________________.

7. The founder of the rectangular projection method is _______________

Task 2. Research project

Match the main types, indicated by numbers, with the details, indicated by letters, and write down the answer in a notebook.

Fig.4

Task 3

An exercise to repeat the knowledge of geometric bodies.

By verbal description find a visual representation of the part.

Description text.

The base of the part has the shape of a rectangular parallelepiped, in the smaller faces of which grooves are made, having the shape of a regular quadrangular prism. A truncated cone is located in the center of the upper face of the parallelepiped, along the axis of which there is a through cylindrical hole.

Rice. 5

Answer: part number 3 (1 point)

Task 4

Find the correspondence between the technical drawings of the parts and their frontal projections (the projection direction is marked with an arrow). From the scattered images of the drawing, make a drawing of each part, consisting of three images. Write down the answer in the table (Fig. 129).

Rice. 6

Technical drawings	frontal projection	Horizontal projection	Profile projection
A	4	13	10
B	12	9	2
IN	14	5	1
G	6	15	8
D	11	3	7

III. Practical work.

Task number 1. research project

Find the frontal and horizontal projections to this visual image. Write down the answer in a notebook.

Evaluation of work in the classroom. Self-test. (Presentation 2)

The scores for grading the first part of the work are written on the board:

23-26 points “5”

19-22 points “4”

15 -18 points “3”

Task number 2. creative work and checking its implementation
(creative project)

Redraw the frontal projection in a workbook.
Draw a horizontal projection, changing the shape of the part in order to reduce its mass.
If necessary, make changes on the frontal projection.
To check the completion of the task, call one or two students to the board in order to explain their version of solving the problem.

(10 points)

IV. Summing up the lesson.

1. Evaluation of work in the lesson. (Checking the practical part of the work)

V. Homework.

1. Research project.

Work on the table: determine which drawing, indicated by a number, corresponds to the drawing indicated by a letter.

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Slides captions:

RECTANGULAR PROJECTION

RECTANGULAR PROJECTION V The vertical projection plane (V) located in front of the viewer is called the frontal plane. To build a projection of an object, we draw projecting rays through the vertices and points of the holes of the object, perpendicular to the plane V

FRONTAL PROJECTION V S 6 According to the resulting projection, we can judge the two dimensions of an object - height and width. In order to judge the shape of a flat part from such an image, it is supplemented with an indication of the thickness (S) of the part

Analyze the geometric shape of the detail on the frontal projection and find this detail among the visual images.

A drawing, represented by three projections or views, gives the most complete idea of ​​the shape and design of the object and is called COMPLEX DRAWING Front view Profile view from the left Horizontal view from above

X One projection does not always determine the geometric shape of an object. In this case, you can build two rectangular projections of the object on two mutually perpendicular planes: frontal (V) and horizontal (H). The line of intersection of the planes (X) is called the axis of projections

RECTANGULAR PROJECTION V Н The constructed projections turned out to be located in space in different planes (vertical and horizontal). To obtain a drawing of an object, both planes are combined into one

RECTANGULAR PROJECTION V H

RECTANGULAR PROJECTION V H

Analyze the geometric shape of the part on the frontal and horizontal projections and find this part among the visual images.

Determine which part corresponds to this drawing

RECTANGULAR PROJECTION V H W In order to reveal the shape of an object, two projections are not always enough. In this case, you need to build another plane. The third plane of projections is called the profile, and the projection obtained on it is called the profile projection of the object. It is denoted by the letter W.

To obtain a drawing of an object, the W plane is rotated 90 0 to the right, and the H plane 90 0 down

RECTANGULAR PROJECTION H W V

RECTANGULAR PROJECTION H W V

RECTANGULAR PROJECTION The resulting drawing contains three rectangular projections of the object: frontal, horizontal and profile. Projection axes and projecting rays in the drawing do not show

RECTANGULAR PROJECTION 76 78 18 30 58 60 Ф 30 26 18 Chertil Petrov V. Checked School No. 1274 class. 9 B steel 1:1 Rack In the projection drawing, they are placed in a projection connection. A drawing consisting of several rectangular projections is called a drawing in the system of rectangular projections.

TASK №3 The arrows show the direction of projection. The projection of the part is indicated by numbers. a) which projection (indicated by a number) corresponds to each direction of projection (indicated by a letter) b) name the projections 1,2,3.

Three details are given, different in shape, which are projected onto two projection planes in exactly the same way. In this case, the profile projection of the part makes it possible to accurately determine the shape of each of them.

QUESTIONS FOR VERIFICATION Is it always enough in the drawing of one projection of the object? What are projection planes called? How are they designated? What are the projections obtained by projecting an object onto three projection planes called? How are these planes arranged relative to each other?

TYPES OF PROJECTION

Drawing presentation

General information about projection .

• Images of items on the drawings in accordance with the rules state standard perform according to the method (method) of rectangular projection. Projection is the process of constructing a projection of an object. How are projections made? Consider this example.
• Let us take an arbitrary point A in space and some plane H (Fig. 37). Let us draw a straight line through the point A so that it intersects the plane H at some point a. Then point a will be the projection of point A. The plane on which the projection is obtained is called the plane of projections. The straight line Aa is called the projecting ray. With its help, point A is projected onto the plane H. In this way, projections of all points of any spatial figure can be constructed.

Rice. 37. Obtaining projections of a point

Rice. 38. Figure projection

• We will further designate the points taken on the subject in capital letters, and their projections in lowercase. The projection of point A onto a given plane will be point 0 as a result of the intersection of the projecting beam Aa with the projection plane. The projections of points B and C will be points b and c. Connecting the points a, b and with line segments on the plane, we get the figure abc, which will be the projection of the given figure ABC.
• An idea of ​​the projection can be obtained by examining the shadows of objects. Take, for example, a wire model of a prism (Fig. 39). Let this model, when illuminated by the sun's rays, cast a shadow on the wall. The shadow obtained in this way can be taken as a projection of a given object.

Rice. 39. Getting the shadow of the model

Center and side projection

• If the projecting rays, with the help of which the projection of an object is built, come from one point, the projection is called central (Fig. 40). The point from which the rays emanate is called the center of projection. The resulting projection is called central .

Rice. 40. Central projection

• Central projection is often called perspective. Examples of central projection are photographs and film frames, shadows cast from an object by the rays of an electric light bulb, etc. Central projections are used in drawing from nature.
• If the projecting rays are parallel to each other (Fig. 41), then the projection is called parallel. and the resulting projection is parallel. An example of a parallel projection can be conditionally considered the solar shadows of objects (Fig. 39).

• It is easier to build an image of an object in a parallel projection than in a central one. In drawing, such projections are used to build drawings and visual images.
• In parallel projection, all rays fall on the projection plane at the same angle. If it is any acute angle, as in Figure 41, then the projection is called oblique .

Rice. 41. Oblique projection

• In the case when the projecting rays are perpendicular to the projection plane (Fig. 42), that is, they make an angle of 90 ° with it, the projection is called rectangular. The resulting projection is called rectangular.

Rice. 42. Rectangular projection

• What is called projection? Give examples of projections.
• How to build a projection of a point on a plane? figure projection?
• Which projection is called central, parallel, rectangular, oblique?
• What projection method is used when constructing a drawing and why?