Orthogonal projection onto three projection planes. Point projection Three-plane projection

One image of the original (Fig. 8) cannot be used to judge its shape, size and position in space.

Drawing reversibility - restoration of the original from its projection images can be provided by projection onto two (three) non-parallel projection planes.

For the convenience of projection, choose two (three) mutually perpendicular projection planes (Fig. 9).

P 1 - horizontal plane of projections.

P 2 - frontal plane of projections.

P 3 - profile plane of projections.

The lines of intersection of the projection planes form the coordinate axes. X-axis - called the axis abscissa, Y-axis - axis ordinate and the Z axis – axis applicator.

Coordinate planes divide the space into eight parts - octants. In (Table 1) the signs of coordinates for four octants (quarters) are presented.

Table 1.

quarter	Coordinate signs

Point A belongs to the first quarter. From this point, three projection beams are drawn to the projection planes P 1, P 2, P 3. As a result, three projections of the point are obtained (Fig. 10).

AND 1 - horizontal projection of point A.

AND 2 - frontal projection of point A.

AND 3 - profile projection of point A.

Point position AND in space is determined by three coordinates AND (X, Y, Z), showing the values \u200b\u200bof the distances at which the point is removed from the projection planes.

Distance from point AND to the projection plane P 3 is determined by the abscissa X:

АА 3  \u003d А X 0 \u003d X

Distance from point AND to the plane of projections P 2 is determined by the ordinate Y:

АА 2  \u003d А 1 А X  \u003d Y

Distance from point AND to the plane of projections П 1 is determined by the applicate Z:

АА 1  \u003d А Z 0 \u003d Z

1.4 Complex drawing of a point (Monge diagrams)

It is inconvenient to use the spatial model (Fig. 10) to display orthogonal projections of geometric figures due to its cumbersomeness, and also due to the fact that the shape and size of the projected figure is distorted on the projection planes.

Therefore, the spatial model is converted to a planar view - a complex drawing.

Complex drawingis the image of a geometric object in two (three) projections on aligned projection planes.

To do this, rotate the plane P 1 90 0 around the X axis in the direction of the clockwise movement until it is aligned with the frontal plane of the projections (Fig. 11).

Plane P 3 is rotated 90 0 counterclockwise around the Z axis, until aligned with the frontal plane of the projections (Fig. 12).

The horizontal and frontal projections of the point lie on one line perpendicular to the X axis, called vertical communication line.

Frontal and profile projections of a point lie on horizontal communication line, perpendicular to the Z axis.

To build a complex drawing of a point AND(Fig. 13) along the X, Y and Z coordinates, the algorithm must be performed.

PROJECTING A POINT ON TWO PROJECTION PLANES

The formation of a segment of a straight line AA 1 can be represented as a result of moving point A in any plane H (Fig. 84, a), and the formation of a plane - as a movement of a segment of a straight line AB (Fig. 84, b).

A point is the main geometric element of a line and a surface, therefore, the study of rectangular projection of an object begins with the construction of rectangular projections of a point.

In the space of a dihedral angle formed by two perpendicular planes - the frontal (vertical) projection plane V and the horizontal projection plane H, we place point A (Fig. 85, a).

The line of intersection of the projection planes is a straight line, which is called the projection axis and is denoted by the letter x.

The plane V is shown here as a rectangle, and the plane H is shown as a parallelogram. The oblique side of this parallelogram is usually drawn at an angle of 45 ° to its horizontal side. The length of the inclined side is taken equal to 0.5 of its actual length.

From point A, perpendiculars are lowered on the plane V and H. Points a "and a intersection of perpendiculars with the projection planes V and H are rectangular projections of point A. Figure Aaa x a" in space is a rectangle. The side aax of this rectangle is reduced by 2 times in the visual image.

Align the H plane with the V plane by rotating V around the intersection of the x planes. The result is a complex drawing of point A (Fig. 85, b)

To simplify the complex drawing, the boundaries of the projection planes V and H are not indicated (Fig. 85, c).

The perpendiculars drawn from point A to the projection planes are called projection lines, and the bases of these projection lines - points a and a "- are called projections of point A: a" is the frontal projection of point A, and is the horizontal projection of point A.

Line a "a" is called the vertical line of the projection connection.

The location of the projection of a point in a complex drawing depends on the position of this point in space.

If point A lies on the horizontal plane of projections H (Fig. 86, a), then its horizontal projection a coincides with a given point, and the frontal projection a "is located on the axis. When point B is located on the frontal plane of projections V, its frontal projection coincides with this point, and the horizontal projection lies on the x-axis.The horizontal and frontal projections of a given point C lying on the x-axis coincide with this point.The complex drawing of points A, B and C is shown in Fig. 86, b.

PROJECTING A POINT ON THREE PROJECTION PLANES

In those cases when it is impossible to imagine the shape of an object from two projections, it is projected onto three projection planes. In this case, a profile plane of projections W is introduced, which is perpendicular to the planes V and H. A visual representation of the system of three projection planes is given in Fig. 87, a.

The edges of a triangular corner (intersection of projection planes) are called projection axes and are denoted by x, y, and z. The intersection of the projection axes is called the beginning of the projection axes and is denoted by the letter O. Let us drop the perpendicular from point A onto the projection plane W and, having marked the base of the perpendicular with the letter a ", we obtain a profile projection of point A.

To obtain a complex drawing, points A of the H and W plane are aligned with the V plane, rotating them around the Ox and Oz axes. A comprehensive drawing of point A is shown in Fig. 87, b and c.

The segments of the projection lines from point A to the projection planes are called the coordinates of point A and are designated: x A, y A and z A.

For example, the coordinate z A of point A, equal to the segment a "a x (Fig. 88, a and b), is the distance from point A to the horizontal plane of projection H. The coordinate at point A, equal to the segment aa x, is the distance from point A to the frontal plane of projections V. Coordinate x A equal to the segment aa y is the distance from point A to the profile plane of projections W.

Thus, the distance between the point's projection and the projection axis defines the point's coordinates and is the key to reading its complex drawing. From two projections of a point, all three coordinates of a point can be determined.

If the coordinates of point A are given (for example, x A \u003d 20 mm, y A \u003d 22 mm and z A \u003d 25 mm), then three projections of this point can be built.

To do this, from the origin of coordinates O in the direction of the Oz axis, the coordinate z A is laid up and the coordinate y A is laid down. From the ends of the deferred segments - points az and a y (Fig. 88, a), straight lines are drawn parallel to the Ox axis, and they are laid on segments equal to the x coordinate A. The obtained points a "and a are the frontal and horizontal projections of point A.

On two projections a "and a point A, you can build its profile projection in three ways:

1) from the origin of coordinates O draw an auxiliary arc with a radius of Oa y equal to the coordinate (Fig. 87, b and c), from the obtained point a y1 draw a straight line parallel to the axis Oz, and lay a segment equal to z A;

2) from the point a y draw an auxiliary straight line at an angle of 45 ° to the axis Oy (Fig. 88, a), get the point a y1, etc .;

3) from the origin of coordinates O, an auxiliary straight line is drawn at an angle of 45 ° to the axis Oy (Fig. 88, b), point a y1 is obtained, and so on.

It can be considered as a special case of the central one, in which the projection center is removed to infinity.

Apply parallel projecting lines drawn in a given direction.

If the projection direction is perpendicular to the projection plane, then the projection is called rectangular or orthogonal.

With parallel projection, all the properties of the central one are preserved, and the following properties also appear:

and). The projections are // mutually direct //, and the ratio of the lengths of segments of such straight lines is equal to the ratio of the lengths of their projections

b). Plane figure, // the projection plane is projected onto this plane in full size

in). If a straight line is perpendicular to the direction of projection, then its projection is a point

If there is a center of parallel projection, we will not be able to determine the position of the point in space.

D aspar Monge suggested taking two mutually perpendicular projection planes (horizontal P 1 and frontal P 2) and using the method of rectangular projection to direct the projection rays perpendicular to the planes.

P 1 - horizontal projection plane

П 2 - frontal plane of projections

X- axis of projections- line of intersection of planes P 1 and P 2 or P 1 / P 2

A x A 1 and A x A 2 - perpendicular to the X-axis - communication lines

If there is a point A in space, then we lower the perpendicular from it to P 1 (horizontal projection of point A - A 1) and onto the plane P2 (frontal projection of point A - A 2)

But this visual image of a point in the P 1 / P 2 system is inconvenient for drawing purposes.

We transform it so that the horizontal plane of the projections coincides with the frontal one, forming one plane of the drawing.

This transformation is carried out by rotating the plane P 1 around the X axis at an angle of 90 ° downward. In this case, A x A 2 and A x A 1 form one segment located on the perpendicular to the projection axis X, called communication line.

Received a drawing called Monge diagrams.

The horizontal and frontal projections always lie on the same communication line, perpendicular to the axis.

Depending on the complexity, three or more images are needed to fully reveal the shapes of parts. Therefore, three or more projection planes are introduced.

Projecting a point onto three projection planes. Point complex drawing.

Received a Monge plot for three planes or a complex drawing of point A

H (P 1) - horizontal projection plane

V (P 2) - frontal projection plane

W (P 3) - profile plane of projections

А 1 - horizontal projection of point А

A 2 - frontal projection of point A

А 3 - profile projection of point А

P 1 and P 2 - form the X axis

P 2 and P 3 - form the Z axis

P 1 and P 3 - form the Y axis

Two projections of a point lie on the same link, perpendicular to the axis.

Sections of projection lines from point A to projection planes - point coordinates (X AND, Have AND , Z AND ). Are given by numbers.

ОА х - abscissa of point А – coordinate Х А - distance from А to П 3. ОА х \u003d А 1 А у \u003d А z А 2

ОА у - ordinate of point А – coordinate У А - distance from А to П 2. ... OA y \u003d A x A 1

ОА z - applicate of point А – coordinateZ А - distance from А to П 1. OA z \u003d A x A 2

Self-test questions

What projection methods are there?

What are the properties of central projection?

What are the properties of parallel projection?

How to get projections of a point on two projection planes?

How to get projections of a point on three projection planes?

Goals and objectives of the lesson:

educational: show students the use of the rectangular projection method when performing 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 creativity of students;

educating: responsible attitude to drawing, foster a culture of graphic work.

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

Material support: computers, rectangular projection presentation, tasks, exercises, exercise cards, self-test presentation.

Lesson type: knowledge consolidation lesson.

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

During the classes

I. Organizational part.

Communication of the topic and purpose of the lesson.

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

II. Repetition about projection and its types.

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

Repetition is done using a presentation.

1. Students are given problem situation ... (Presentation 1)

Analyze the geometric shape of the part on the frontal projection and find this part 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? (View the detail from the other side).

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

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

Sizing:

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

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

Exercise 1

Insert the missing words into the text of the definitions.

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 method of rectangular projection is _______________

Task 2. Research project

Establish the correspondence of the main types, indicated by numbers, to the details indicated by letters, and write down the answer in a notebook.

Fig. 4

Assignment 3

Exercise to review the knowledge of geometric bodies.

Find a visual image of the part by verbal description.

Description text.

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

Figure: five

Answer: part number 3 (1 point)

Assignment 4

Find the correspondence between the technical drawings of the parts and their front projections (the projection direction is indicated by an arrow). From the scattered images of the drawing, draw up a drawing of each part, consisting of three images. Write down the answer in the table (fig. 129).

Figure: 6

Technical Drawings	Front projection	Horizontal projection	Profile projection
AND	4	13	10
B	12	9	2
IN	14	5	1
D	6	15	8
D	11	3	7

III. Practical work.

Task number 1. Research project

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

Assessment of work in the lesson. Self-test. (Presentation 2)

The scores for the assessment of 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 verification of its completion
(creative project)

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

(10 points)

IV. Summing up the lesson.

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

V. Assignment at home.

1. Research project.

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

Instruction:

- introductory:

sequence of work:

1. Analysis of the geometric shape of the object;

2. Determination of the main species;

3. Layout on the sheet;

4. Construction of the drawing (thin lines);

5. Drawing the dimensions of the structural elements of the part, taking into account their readability and uniform distribution over all types of the drawing;

6. Application of the overall dimensions of the part (length, width and height);

7. Checking the correctness and availability of all dimensions sufficient for the manufacture and control of the part;

8. Final design of the drawing (checking compliance with all lines of the drawing);

-current:

correction and correction of the current mistakes of students in the course of the practical assignment;

-final:

Look again at the board and in your notebooks and compare the drawings, was everything done correctly?

Now each of you will receive a card with a task on which we will work. I will ask me to help the guys at the first desks to distribute them.

In notebooks, open a sheet with a frame and a title block and draw perpendicular to the projection axes X, Y, Z.

One person goes out to work at the board (optional), draws the axes, marks them, marks the main projection planes, indicates the location of the views, and earns a grade.

(Assessment of the student).

Look at the cards you received and answer the questions.

What is commonly understood by the term view?

This is an image of the surface of the part facing the observer.

What kind is called main or front view?

This is the view that gives the most complete picture of the shape of the object.

Look at the thumbnail image of the part and try to identify the main look.

Indeed, this species can be taken as the main one.

Where will we place it?

On the frontal projection plane.

As in previous lessons, we begin to build a drawing from the main overall dimensions, and then we build constructive (small) elements.

We built the main view, draw the lines of the projection connection on the horizontal and profile projection planes. Then, on the horizontal projection plane, we build a view from the top. To do this, draw a horizontal line parallel to the X axis.Do not forget to step back from the X axis by distance 15 mm, same as in the main view.Then we put it down 75 mm and draw another parallel line. From the central line of the projection connection (it will also be our axis of symmetry), set aside 5 mm from the bottom, and we get a cutout. And after putting aside 15 mm from the bottom edge, we get the center point of the circle. Let's draw the axes of symmetry and draw a circle. Draw a horizontal line from above at a distance of 15 mm. The top view is ready. Who can complete the two views on the left and get an estimate for it?

(The student completes the left view and receives a grade.)

It is very important to show the hidden lines of the part drawing in the left view. It is very easy to determine their location if you draw all the lines of projection communication.

How the dimensions are applied.

To determine the size of the depicted product or any part of it according to the drawing, dimensions are applied on it.

Linear dimensions in the drawings are indicated in millimeters, but unit designation is not applied. The total number of dimensions in the drawing should be the smallest, but sufficient for the manufacture and control of the product. Dimensioning rules are established by the standard. Here are some of them :

1. Dimensions in the drawings are indicated by dimension numbers and dimension lines. To do this, first draw extension lines perpendicular to the segment, the size of which is then indicated at a distance of 10 mm from the contour of the part, draw a dimension line parallel to it. The dimension line is limited by arrows on both sides. The length of the arrowhead is 5 mm. The extension lines extend beyond the ends of the arrows of the dimension line by 1 (1 ... 5) mm. The extension and dimension lines are drawn with a solid thin line. Above the dimension line, closer to its middle, the dimension number is applied.

2. Dimension lines are applied outside the contour of the image, but it is allowed to apply them inside the contour, if the legibility of the drawing is not disturbed. The distance of the dimension line from the contour line parallel to it must be at least 10 mm, and the distance between the parallel dimension lines must be within 7 ... 10 mm. Avoid crossing dimension and extension lines. Dimension lines with smaller numerical values \u200b\u200bare located first from the contour.

4. To designate the diameter, a special sign is applied in front of the dimension number - a circle crossed out with a line. If the dimension number does not fit inside the circle, it is moved outside the circle using the leader shelf, while the arrows are also taken out, and their ends are directed to the center of the circle.

When dimensioning views, it is very important to keep in mind that they are evenly spaced and readable.