PROJECTING A POINT ON TWO PROJECTION PLANES

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

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

In the space of the dihedral angle formed by two perpendicular planes - the frontal (vertical) plane of projections V and the horizontal plane of projections 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 designated by the letter x.

The V plane is depicted here as a rectangle, and the H plane as a parallelogram. The inclined 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 onto planes V and H. Points a" and a of the intersection of perpendiculars with the projection planes V and H are rectangular projections of point A. The figure Aaa x a" in space is a rectangle. The side aax of this rectangle in the visual image is reduced by 2 times.

Let's align the H planes with the V plane by rotating V around the line of intersection of the x planes. The result is a comprehensive 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).

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, a is the horizontal projection of point A.

Line a" a is called the vertical line of 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 the 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. A complex drawing of points A, B and C is shown in Fig. 86, b.

PROJECTING A POINT ON THREE PROJECTION PLANES

In cases where 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 projection plane W is introduced, 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 trihedral angle (the intersection of projection planes) are called projection axes and are designated 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’s drop a perpendicular from point A to the projection plane W and, marking the base of the perpendicular with the letter “a”, we obtain a profile projection of point A.

To obtain a complex drawing of point A, planes H and W are combined with plane V, 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 projecting 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 projection plane H. The coordinate y of 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 - the distance from point A to the profile plane of projections W.

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

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

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 laid-off segments - points a z and a y (Fig. 88, a) - draw straight lines parallel to the Ox axis, and lay them on segments equal to the x coordinate A. The resulting points a" and a are the frontal and horizontal projections of point A.

Using two projections a" and a of point A, you can construct its profile projection in three ways:

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

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

3) from the origin O, draw an auxiliary straight line at an angle of 45° to the Oy axis (Fig. 88, b), obtain point a y1, etc.

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 concepts, spatial thinking, cognitive interest and creative abilities of students;

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

Methods and techniques of teaching: 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-test.

Lesson type: lesson to consolidate knowledge.

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

During the classes

I. Organizational part.

State the topic and purpose of the lesson.

Let's carry out lesson-competition, for each task you will receive a certain number of points. Depending on the points scored, a grade for the lesson will be assigned.

II. Repetition of projection and its types.

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

Repetition is carried out using presentation.

1. Students are asked problematic situation . (Presentation 1)

Analyze the geometric shape of the part on the front 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? (Look at the part 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 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 profile projection – width and height.
  

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

Exercise 1

Fill in the missing words in the definition text.

1. There are _______________ and ______________ projection.

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

3. If ______________ rays are directed parallel, 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 a projection plane is obtained only with ______________ projection.

6. The 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 parts indicated by letters and write the answer in your notebook.

Fig.4

Task 3

An exercise to review knowledge of geometric bodies.

Using the verbal description, find a visual image of the part.

Description text.

The base of the part has the shape of a rectangular parallelepiped, the smaller faces of which have 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 there is a through cylindrical hole.

Rice. 5

Answer: part No. 3 (1 point)

Task 4

Find the correspondence between the technical drawings of the parts and their frontal projections (the direction of projection is marked with an arrow). Based on the scattered images of the drawing, make a drawing of each part, consisting of three images. Write your 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 No. 1. Research project

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

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

The points 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 No. 2. Creative work and verification of its implementation
(creative project)

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

(10 points)

IV. Summing up the lesson.

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

V. Homework assignment.

1. Research project.

Work according to the table: determine which drawing, designated by a number, corresponds to the drawing, designated by a letter.

Let's consider the projections of points onto two planes, for which we take two perpendicular planes (Fig. 4), which we will call horizontal frontal and planes. The line of intersection of these planes is called the projection axis. We project one point A onto the considered planes using a plane projection. To do this, it is necessary to lower the perpendiculars Aa and A from a given point onto the considered planes.

The projection onto the horizontal plane is called horizontal projection points A, and the projection A? on the frontal plane is called frontal projection.

Points to be projected are usually denoted in descriptive geometry using capital letters A, B, C. Small letters are used to indicate horizontal projections of points a, b, c... Frontal projections are indicated in small letters with a stroke at the top a?, b?, c?…

Points are also designated by Roman numerals I, II,... and for their projections - by Arabic numerals 1, 2... and 1?, 2?...

By rotating the horizontal plane by 90°, you can get a drawing in which both planes are in the same plane (Fig. 5). This picture is called diagram of a point.

Through perpendicular lines Ahh And Huh? Let's draw a plane (Fig. 4). The resulting plane is perpendicular to the frontal and horizontal planes because it contains perpendiculars to these planes. Therefore, this plane is perpendicular to the line of intersection of the planes. The resulting straight line intersects the horizontal plane in a straight line ahh x, and the frontal plane – in a straight line a?a X. Straight aah and a?a x are perpendicular to the axis of intersection of the planes. That is Aahaha? is a rectangle.

When combining horizontal and frontal projection planes A And A? will lie on the same perpendicular to the axis of intersection of the planes, since when the horizontal plane rotates, the perpendicularity of the segments ahh x and a?a x will not be broken.

We get that on the projection diagram A And A? some point A always lie on the same perpendicular to the axis of intersection of the planes.

Two projections a and A? of a certain point A can unambiguously determine its position in space (Fig. 4). This is confirmed by the fact that when constructing a perpendicular from projection a to the horizontal plane, it will pass through point A. In the same way, a perpendicular from projection A? to the frontal plane will pass through the point A, i.e. point A is simultaneously on two specific straight lines. Point A is their point of intersection, that is, it is definite.

Consider a rectangle Aaa X A?(Fig. 5), for which the following statements are true:

1) Point distance A from the frontal plane is equal to the distance of its horizontal projection a from the axis of intersection of the planes, i.e.

Huh? = ahh X;

2) point distance A from the horizontal plane of projections is equal to the distance of its frontal projection A? from the axis of intersection of the planes, i.e.

Ahh = a?a X.

In other words, even without the point itself on the diagram, using only its two projections, you can find out at what distance a given point is located from each of the projection planes.

The intersection of two projection planes divides space into four parts, which are called in quarters(Fig. 6).

The axis of intersection of the planes divides the horizontal plane into two quarters - the front and rear, and the frontal plane - into the upper and lower quarters. The upper part of the frontal plane and the anterior part of the horizontal plane are considered as the boundaries of the first quarter.

When receiving the diagram, the horizontal plane rotates and is aligned with the frontal plane (Fig. 7). In this case, the front part of the horizontal plane will coincide with the bottom part of the frontal plane, and the back part of the horizontal plane will coincide with the top part of the frontal plane.

Figures 8-11 show points A, B, C, D, located in different quarters of space. Point A is located in the first quarter, point B is in the second, point C is in the third and point D is in the fourth.

When the points are located in the first or fourth quarters of them horizontal projections are on the front part of the horizontal plane, and on the diagram they will lie below the axis of intersection of the planes. When a point is located in the second or third quarter, its horizontal projection will lie on the back of the horizontal plane, and on the diagram it will be located above the axis of intersection of the planes.

Frontal projections points that are located in the first or second quarters will lie on the upper part of the frontal plane, and on the diagram they will be located above the axis of intersection of the planes. When a point is located in the third or fourth quarter, its frontal projection is below the axis of intersection of the planes.

Most often, in real constructions, the figure is placed in the first quarter of space.

In some special cases, the point ( E) can lie on a horizontal plane (Fig. 12). In this case, its horizontal projection e and the point itself will coincide. The frontal projection of such a point will be located on the axis of intersection of the planes.

In the case when the point TO lies on the frontal plane (Fig. 13), its horizontal projection k lies on the axis of intersection of the planes, and the frontal k? shows the actual location of this point.

For such points, a sign that it lies on one of the projection planes is that one of its projections is on the axis of intersection of the planes.

If a point lies on the axis of intersection of the projection planes, it and both of its projections coincide.

When a point does not lie on the projection planes, it is called point of general position. In what follows, if there are no special marks, the point in question is a point in general position.

2. Lack of projection axis

To explain how to obtain projections of a point on a model perpendicular to the projection plane (Fig. 4), it is necessary to take a piece of thick paper in the shape of an elongated rectangle. It needs to be bent between projections. The fold line will represent the axis of intersection of the planes. If after this the bent piece of paper is straightened again, we will get a diagram similar to the one shown in the figure.

By combining two projection planes with the drawing plane, it is possible not to show the fold line, i.e., not to draw the axis of intersection of the planes on the diagram.

When plotting on a diagram, you should always place projections A And A? point A on one vertical line (Fig. 14), which is perpendicular to the axis of intersection of the planes. Therefore, even if the position of the axis of intersection of the planes remains uncertain, but its direction is determined, the axis of intersection of the planes can only be located on the diagram perpendicular to the straight line huh?.

If there is no projection axis on the diagram of a point, as in the first Figure 14 a, you can imagine the position of this point in space. To do this, draw anywhere perpendicular to the straight line huh? projection axis, as in the second figure (Fig. 14) and bend the drawing along this axis. If we restore perpendiculars at points A And A? before they intersect, you can get a point A. When changing the position of the projection axis, different positions of the point relative to the projection planes are obtained, but the uncertainty of the position of the projection axis does not affect the relative position of several points or figures in space.

3. Projections of a point onto three projection planes

Let's consider the profile plane of projections. Projections onto two perpendicular planes usually determine the position of a figure and make it possible to find out its real size and shape. But there are times when two projections are not enough. Then the construction of the third projection is used.

The third projection plane is drawn so that it is perpendicular to both projection planes simultaneously (Fig. 15). The third plane is usually called profile.

In such constructions, the common straight line of the horizontal and frontal planes is called axis X , the common straight line of the horizontal and profile planes – axis at , and the common straight line of the frontal and profile planes is axis z . Dot ABOUT, which belongs to all three planes, is called the origin point.

Figure 15a shows the point A and three of its projections. Projection onto the profile plane ( A??) are called profile projection and denote A??.

To obtain a diagram of point A, which consists of three projections a, a, a, it is necessary to cut the trihedron formed by all the planes along the y-axis (Fig. 15b) and combine all these planes with the plane of the frontal projection. The horizontal plane must be rotated about the axis X, and the profile plane is about the axis z in the direction indicated by the arrow in Figure 15.

Figure 16 shows the position of the projections huh, huh? And A?? points A, obtained by combining all three planes with the drawing plane.

As a result of the cut, the y-axis appears in two different places on the diagram. On a horizontal plane (Fig. 16) it takes a vertical position (perpendicular to the axis X), and on the profile plane – horizontal (perpendicular to the axis z).

There are three projections in Figure 16 huh, huh? And A?? points A have a strictly defined position on the diagram and are subject to unambiguous conditions:

A And A? should always be located on the same vertical line, perpendicular to the axis X;

A? And A?? should always be located on the same horizontal straight line, perpendicular to the axis z;

3) when carried out through a horizontal projection and a horizontal straight line, and through a profile projection A??– a vertical straight line, the constructed straight lines will necessarily intersect on the bisector of the angle between the projection axes, since the figure Oa at A 0 A n – square.

When constructing three projections of a point, you need to check whether all three conditions are met for each point.

4. Point coordinates

The position of a point in space can be determined using three numbers called its coordinates. Each coordinate corresponds to the distance of a point from some projection plane.

Determined point distance A to the profile plane is the coordinate X, wherein X = huh?Huh(Fig. 15), the distance to the frontal plane is coordinate y, and y = huh?Huh, and the distance to the horizontal plane is the coordinate z, wherein z = aA.

In Figure 15, point A occupies the width of a rectangular parallelepiped, and the measurements of this parallelepiped correspond to the coordinates of this point, i.e., each of the coordinates is represented in Figure 15 four times, i.e.:

x = a?A = Oa x = a y a = a z a?;

y = а?А = Оа y = а x а = а z а?;

z = aA = Oa z = a x a? = a y a?.

In the diagram (Fig. 16), the x and z coordinates appear three times:

x = a z a?= Oa x = a y a,

z = a x a? = Oa z = a y a?.

All segments that correspond to the coordinate X(or z), are parallel to each other. Coordinate at represented twice by an axis located vertically:

y = Oa y = a x a

and twice – located horizontally:

y = Oa y = a z a?.

This difference appears due to the fact that the y-axis is present on the diagram in two different positions.

It should be taken into account that the position of each projection is determined on the diagram by only two coordinates, namely:

1) horizontal – coordinates X And at,

2) frontal – coordinates x And z,

3) profile – coordinates at And z.

Using coordinates x, y And z, you can construct projections of a point on a diagram.

If point A is given by coordinates, their recording is defined as follows: A ( X; y; z).

When constructing point projections A the following conditions must be checked:

1) horizontal and frontal projections A And A? X X;

2) frontal and profile projections A? And A? must be located at the same perpendicular to the axis z, since they have a common coordinate z;

3) horizontal projection and also removed from the axis X, like profile projection A away from the axis z, since projections ah? and eh? have a common coordinate at.

If a point lies in any of the projection planes, then one of its coordinates is equal to zero.

When a point lies on the projection axis, two of its coordinates are equal to zero.

If a point lies at the origin, all three of its coordinates are zero.

Let's consider a system of three mutually perpendicular projection planes (Fig. 5): P 1 horizontal projection plane, P 2 frontal projection plane and P 3 profile projection plane.

Rice. 5. Projection planes:

x 12 = P 1 ∩ P 2 ;

y 13 = P 1 ∩ P 3 ;

z 23 = P 2 ∩ P 3

The intersection point of three planes O 123 is the origin of coordinates. The line of intersection of the horizontal and frontal planes is called the axis of projection x 12 = P 1 ∩ P 2, the line of intersection of the horizontal and profile planes is called the axis of projections y 13 = P 1 ∩ P 3, the line of intersection of the frontal and profile planes is called the axis of projections z 23 = P 2 ∩ P 3.

Since projection planes are infinite, three planes will divide the entire space into eight parts - octants. The order of counting octants (see Fig. 5): to the left of the plane P 3 (counterclockwise) from the first to the fourth, to the right - from the fifth to the eighth.

The direction of the x, y, z axes in the first octant is considered positive. The signs of axes extended beyond the origin are considered negative.

To obtain projections of point A onto three planes (Fig. 6) P 1, P 2 and P 3, projection rays are drawn through point A)