Constructing a projection of a point using coordinates. Construction of a complex drawing of a point

Construct traces of the plane given by ∆BCD and determine the distance from point A to the given plane using the right triangle method(for coordinates of points A, B, C and D, see Table 1 of the Tasks section);

1.2. Example of task No. 1

The first task presents a set of tasks on the following topics:

1. Orthogonal projection, Monge diagram, point, straight line, plane: by known coordinates of three points B, C, D construct horizontal and frontal projections of the plane given by ∆ BCD;

2. Traces of a straight line, traces of a plane, properties of belonging to a straight plane: construct traces of the plane given by ∆ BCD;

3. General and particular planes, intersection of a line and a plane, perpendicularity of a line and a plane, intersection of planes, right triangle method: determine the distance from a point A to plane ∆ BCD.

1.2.1. Based on the known coordinates of three points B, C, D let's construct horizontal and frontal projections of the plane given by ∆ BCD(Figure 1.1), for which it is necessary to construct horizontal and frontal projections of the vertices ∆ BCD, and then connect the projections of the vertices of the same name.

It is known that following the plane is a straight line obtained as a result of the intersection of a given plane with the projection plane .

A general plane has 3 traces: horizontal, frontal and profile.

In order to construct traces of a plane, it is enough to construct traces (horizontal and frontal) of any two straight lines lying in this plane and connect them to each other. Thus, the trace of the plane (horizontal or frontal) will be uniquely determined, since through two points on the plane (in this case, these points will be the traces of straight lines) a straight line can be drawn, and only one.

The basis for this construction is property of belonging to a straight plane: if a straight line belongs to a given plane, then its traces lie on similar traces of this plane .

The trace of a line is the point of intersection of this line with the projection plane. .

The horizontal trace of a straight line lies in the horizontal plane of projections, the frontal trace - in the frontal plane of projections.

Let's consider the construction horizontal trace direct D.B., for which you need:

1. Continue the frontal projection straight D.B. until it intersects with the axis X, point of intersection M 2 is a frontal projection of the horizontal trace;

2. From a point M 2 restore the perpendicular (projection connection line) until it intersects with the horizontal projection of the straight line D.B. M 1 and will be a horizontal projection of the horizontal trace (Figure 1.1), which coincides with the trace itself M.

The horizontal trace of the segment is constructed in a similar way NE straight: point M'.

To build frontal trace segment C.B. direct, you need:

1. Continue the horizontal projection of the straight line C.B. until it intersects with the axis X, point of intersection N 1 is a horizontal projection of the frontal trace;

2. From a point N 1 restore the perpendicular (projection connection line) until it intersects with the frontal projection of the straight line C.B. or its continuation. Intersection point N 2 and will be a frontal projection of the frontal trace, which coincides with the trace itself N.

Connecting the dots M′ 1 And M 1 straight line segment, we obtain a horizontal trace of the plane απ 1 . Point α x of intersection of απ 1 with the axis X called vanishing point . To construct the frontal trace of the απ 2 plane, it is necessary to connect the frontal trace N 2 with the vanishing point of traces α x

Figure 1.1 — Construction of plane traces

The algorithm for solving this problem can be presented as follows:

1. (D 2 B 2 ∩ OX) = M 2 ;
2. (MM 1 ∩ D 1 B 1) = M 1 = M;
3. (C 2 B 2 ∩ OX) = M′ 2 ;
4. (M′ 2 M′ 1 ∩ C 1 B 1) = M′ 1 = M′;
5. (CB∩ π 2) = N 2 = N;
6. (MM′) ≡ απ 1 ;
7. (α x N) ≡ απ 2 .

1.2.2. To solve the second part of the first task you need to know that:

• distance from point A to plane ∆ BCD determined by the length of the perpendicular restored from this point to the plane;
• any line is perpendicular to a plane if it is perpendicular to two intersecting lines lying in this plane;
• on the diagram, the projections of a straight line perpendicular to the plane are perpendicular to the inclined projections of the horizontal and frontal of this plane or the traces of the same name of the plane (Fig. 1.2) (see the Theorem on the perpendicular to the plane in the lectures).

To find the base of a perpendicular, it is necessary to solve the problem of the intersection of a line (in this problem, such a line is a perpendicular to a plane) with a plane:

1. Enclose the perpendicular in an auxiliary plane, which should be taken as a plane of particular position (horizontally projecting or frontally projecting; in the example, horizontally projecting γ is taken as an auxiliary plane, that is, perpendicular to π 1, its horizontal trace γ 1 coincides with a horizontal projection of a perpendicular);

2. Find the line of intersection of the given plane ∆ BCD with auxiliary γ ( MN in Fig. 1.2);

3. Find the point of intersection of the line of intersection of planes MN with a perpendicular (point TO in Fig. 1.2).

4. To determine the true distance from a point A to a given plane ∆ BCD should be used right triangle method: the true size of a segment is the hypotenuse of a right triangle, one leg of which is one of the projections of the segment, and the other is the difference in distances from its ends to the projection plane in which the construction is being carried out.

5. Determine the visibility of perpendicular sections using the competing point method. For example - points N And 3 to determine visibility on π 1, points 4 , 5 - to determine visibility on π 2.

Figure 1.2 - Construction of a perpendicular to the plane

Figure 1.3 — Example of design of control task No. 1

Video example of completing task No. 1

1.3. Task options 1

Lesson type: lesson of generalization and systematization of knowledge.

Methods: verbal, visual, paired, independent work, frontal questioning, control and evaluation

Equipment: interactive whiteboard, cards for independent work

Target: consolidate the skills of finding the coordinates of marked points and constructing points according to given coordinates.

Lesson objectives:

Educational:

• generalization of students’ knowledge and skills on the topic “Coordinate plane”;
• intermediate control of students' knowledge and skills.

Educational:

• development of students' computing skills;
• development of logical thinking;
• development of mathematically literate speech and outlook of students;
• development of independent work skills.

Educational:

• instilling discipline in organizing work in the classroom;
• fostering accuracy when performing constructions.

Lesson structure:

1. Organizational moment.
2. Checking homework.
3. Updating basic knowledge.
4. Diagnostics of students' knowledge and skills acquisition.
5. Summing up the lesson.
6. Homework.

PROGRESS OF THE LESSON

1. Organizational moment

Today we will repeat what we have covered over the course of several lessons. Remember what we did in class, what topics we studied, what interested you most, what you remember, what remained incomprehensible on the topic “Coordinate plane. Constructing a point from its coordinates." Our task: to repeat, generalize, systematize knowledge on the topic “Coordinate plane”.

2. Checking homework

Now let's check how you completed your homework. Using the given coordinates, you had to build a figure, connecting adjacent points to each other as you built. As a result of completing the work, you should have a figure:

3. Updating basic knowledge

The “Solve the crossword” task will help you remember the basic concepts on the topic “Coordinate plane.”
A crossword puzzle appears on the interactive whiteboard screen and students are asked to solve it.

1. Two coordinate lines form a coordinate ... (plane)
2. Coordinate lines are coordinate... (axes)
3. What angle is formed when coordinate lines intersect? (direct)
4. What is the name of a pair of numbers that determine the position of a point on a plane? (coordinate)
5. What is the name of the first coordinate? (abscissa)
6. What is the second coordinate called? (ordinate)
7. What is the name of the segment from 0 to 1? (unit)
8. How many parts is the coordinate plane divided into by coordinate lines? (four)

4. Diagnostics of students’ assimilation of knowledge and skills

Mark the points on the coordinate plane:

A(-3; 0); B(2; -3); C(-4; 2); D(0; 4); E(1; 3); O(0; 0)

Now let's move on to constructing a figure using points on the coordinate plane. The coordinates of the points are given. Construct a figure, connecting adjacent points to each other as you build.

Independent work.
(verification by mutual verification)

5. Summing up the lesson

Questions for students:

1) What is a coordinate plane?
2) What are the coordinate axes OX and OU called?
3) What angle is formed when the coordinate lines intersect?
4) What is the name of a pair of numbers that determine the position of a point on a plane?
5) What is the name of the first number?
6) What is the second number called?

6. Homework

1. P(-1.5; 10),
2. (-1,5; 11),
3. (-2; 12),
4. (-3; 12),
5. (-3,5; 11),
6. (-3,5; 10),
7. (-5; 12),
8. (-9; 14),
9. (-14; 15),
10. (-12; 10),
11. (-10; 8),
12. (-8; 7),
13. (-4; 6),
14. (-6; 6),
15. (-9; 5),
16. (-12; 3),
17. (-14; 0),
18. (-14; -2),
19. (-12; -2),
20. (-7; -1),
21. (-3; 3),
22. (-4; 1),
23. (-3; 0),
24. (-4; -1),
25. (-2,5; -2),
26. (-1; -1),
27. (-2; 0),
28. (-1; 1),

1. (-2; 3),
2. (2; -1),
3. (7; -2),
4. (9; -2),
5. (9; 0),
6. (7; 3),
7. (4; 5),
8. (1; 6),
9. (-1; 6),
10. (3; 7),
11. (5; 8),
12. (7; 10),
13. (9; 15),
14. (4; 14),
15. (0; 12),
16. (-1,5; 10).
17. P (-3.5; 10),
18. (-4; 6),
19. (-3; 3),
20. P (-1.5; 10),
21. (-1; 6),
22. (-2; 3).

1. (-2; 11),
2. (-3; 11)

Duration: 1 lesson (45 minutes).
Class: 6th grade
Technologies:

• multimedia presentation Microsoft Office PowerPoint, Notebook;
• use of an interactive whiteboard;
• handout for students created using Microsoft Office Word and Microsoft Office Excel.

Annotation:
In thematic planning, 6 hours are allocated for the topic “Coordinates”. This is the fourth lesson on the topic “Coordinates”. At the time of the lesson, students were already familiar with the concept of “coordinate plane” and the rules for constructing a point. Knowledge updating is carried out in the form of a frontal survey. In revision lessons, all students are involved in various activities. In this case, all channels of perception and reproduction of material are used.
The assimilation of the theory is also tested during oral work (the task of solving a crossword puzzle, in which quarter the dot is located). Additional tasks are provided for strong students.
The lesson uses multimedia equipment and an interactive whiteboard to demonstrate presentations and assignments in Microsoft Office PowerPoint and Notebook. To create test tasks and handouts, the following were used: Microsoft Office Excel, Microsoft Office Word.
Using an interactive whiteboard expands the possibilities of presenting material. In the Notebook program, students can independently move objects to the desired location. In Microsoft Office PowerPoint, it is possible to set the movement of objects, so physical exercises for the eyes are provided.

The lesson uses:

• checking homework;
• frontal work;
• individual work of students;
• presentation of the student's report;
• performing oral and written exercises;
• students’ work with an interactive whiteboard;
• independent work.

Lesson summary.

Target: consolidate the skills of finding the coordinates of marked points and constructing points according to given coordinates.
Lesson objectives:
educational:

• generalization of students’ knowledge and skills on the topic “Coordinate plane”;
• intermediate control of students' knowledge and skills;

developing:

• development of students' communicative competence;
• development of students' computing skills;
• development of logical thinking;
• developing students' interest in the subject through a non-traditional form of teaching;
• development of mathematically literate speech and outlook of students;
• developing the ability to independently work with a textbook and additional literature;
• development of students' aesthetic feelings;

educational:

• instilling discipline in organizing work in the classroom;
• fostering cognitive activity, a sense of responsibility, and a culture of communication;
• fostering accuracy when performing constructions.

Progress of the lesson.

• Organizational moment.

Greeting students. Introducing the topic and purpose of the lesson. Checking the class's readiness for the lesson. The task is set: to repeat, generalize, systematize knowledge on the announced topic.

2. Updating knowledge.

Oral counting.
1) Individual work: several people do the work on the cards.

2) Work with the class: calculate examples and make up a word. The table is on the screen of the interactive whiteboard, the letters are entered into the table with an electronic marker from the interactive whiteboard.

Students take turns going to the board and writing down the letters. The result is the word "Prometheus". One of the students, who prepared a report in advance, tells what this word means. (Ancient Greek astronomer Claudius Ptolemy, who used latitude and longitude as coordinates already in the 2nd century.)

Front work.

The “Solve the crossword” task will help you remember the basic concepts on the topic “Coordinate plane.”
The teacher shows a crossword puzzle on the screen of the interactive whiteboard and asks students to solve it. Students use electronic markers to write words in a crossword puzzle.
1. Two coordinate lines form a coordinate line....
2. Coordinate lines are coordinate….
3. What angle is formed when coordinate lines intersect?
4. What is the name of a pair of numbers that determine the position of a point on a plane?
5. What is the name of the first number?
6. What is the second number called?
7. What is the name of the segment from 0 to 1?
8. How many parts is the coordinate plane divided into by coordinate lines?

3. Consolidation of skills and abilities to build a geometric figure according to the given coordinates of its vertices.

Construction of geometric figures. Working with a textbook in notebooks.

• No. 1054a “Construct a triangle if the coordinates of its vertices are known: A(0;-3), B(6:2), C(5:2). Indicate the coordinates of the points where the sides of the triangle intersect the x-axis.”
• Construct a quadrilateral ABCD if A(-3;1), B(1;1), C(1;-2), D(-3;-2). Determine the type of quadrilateral. Find the coordinates of the intersection of the diagonals.

4. Exercise for the eyes.

On the slide, students must follow the movements of the object with their eyes. At the end of the physical session, a question is asked about the geometric shapes obtained as a result of eye movements.

5. Control over the ability to construct points on the coordinate plane according to given coordinates.

Independent work. Artists' competition.
The coordinates of the points are recorded on the slide. Cards are also printed for each student. If you correctly mark the points on the coordinate plane and connect them sequentially, you will get a drawing. Each student completes the task independently. After completing the work, the correct drawing opens on the screen. Each student receives a grade for independent work.

6. Homework.

• No. 1054b, No. 1057a.
• Creative task: draw a picture of points on a coordinate plane and write down the coordinates of these points.

7. Summing up the lesson.

Questions for students:

• What is a coordinate plane?
• What are the coordinate axes OX and OU called?
• What angle is formed when coordinate lines intersect?
• What is the name of a pair of numbers that determine the position of a point on a plane?
• What is the first number called?
• What is the second number called?

Literature and resources:

• G.V. Dorofeev, S.B. Suvorova, I.F. Sharygin “Mathematics. 6kl”
• Mathematics. Grade 6: Lesson plans (based on the textbook by G.V. Dorofeev and others)
• http://www.pereplet.ru/nauka/almagest/alm-cat/Ptolemy.htm

Projection planes V,H, W are taken as coordinate planes, and the projection axes X,Y,Z for coordinate axes, both positive and negative (Fig. 10).

The position of a point in space is specified by three coordinates - X,Y,Z. Projections of a point are specified by two coordinates: A(X, y),A'(X, z),A''(y, z).

Knowing the direction for the positive and negative values ​​of the coordinate axes, taking into account the properties of the projections of the point, it is possible to construct projections of the point from the coordinates. Let's consider several problems on this topic.

Task. Construct projections of a point A(–10; 40; –30) (Fig. 10).

Rice. 10. Construction of projections of a point A by coordinates

To construct a frontal projection A' points A to the right of the point ABOUT on the axis X set aside the value X= –10. Down from the point ABOUT along the axis Z set aside the value Z= –30. Intersection of perpendiculars from points and X And and Z,restored to the corresponding axes X And Z, determine the point A'.

To construct a horizontal projection A points A along the axis Y down from the point ABOUT set aside the value y= – 40. Through a point and Y draw a perpendicular until it intersects with the communication line a′a X. Mark a point A– horizontal projection of a point A. According to the location of the frontal and horizontal projections of the point A we determine that the point A located in the VΙΙΙ octant.

To construct a profile projection A'' points A through her frontal projection A' draw a line of communication a′a Z and on it, to the right of the point and Z, set aside the value y= 40. Mark the point A''– profile projection of a point A.

Task. Construct projections of points according to coordinates and indicate the octant in which each of them is located.

Initial data: A(10; –30; 40), IN(70; 50; –10), WITH(20; 15; 0), D(60; 35; 40), E(50; –10; –25).

Solution. The order of execution of the graphic part of the task (Fig. 11):

1. Draw the coordinate axes X,Y,Z. We indicate their positive and negative directions.

2. We construct points on a scale of 1:1.

Point A(10; –30; 40):

Frontal projection A' points A determined by coordinates X,Z; along the axis X set aside 10 mm, along the axis Z– 40 mm.

Horizontal projection A points A determined by coordinates X,(–Y), a distance of 30 mm is set aside along the axis (– Y Z.

Profile projection A'' points A determined by coordinates (– Y), Z. In this case, a distance of 30 mm is laid along the axis (– Y), coinciding with the positive direction of the axis X. Therefore, the point A is in the ΙΙ octant.

Point B(70; 50; –10):

Building a frontal projection b′(X= 70; Y= –10) points A. A distance of 10 mm must be set aside in the negative direction of the axis Z. Specify: frontal b′ and horizontal b point projection IN will be located on the communication line below the axis X. Profile projection b′′ points IN located to the right of the axis Z and below the axis X. Analyzing the coordinate signs (+ + –) and the location of the point’s projections, we conclude that the point IN is in the ΙV octant.

Point C(20; 15; 0):

When constructing this point, it is obvious that the frontal projection With' points WITH lies on the axis X, and its profile projection A'' lies on the axis Y, coinciding with the negative direction of the axis X. Delete a point WITH from the projection plane N equals zero ( y= 0), hence the point WITH lies in a plane N, at the border of Ι and ΙV octants.

Point D(60; 35; 40):

All coordinate values ​​are positive, therefore the point D is in the 1st octant.

Point E(50; –10; –25):

For negative values Y And Z the point is located in the ΙΙΙ octant. The projections of such a point are located:

Frontal projection e′ points E located below the axis X, to the left of the axis Y;

Horizontal projection e points E located above the axis X, to the left of the axis Z;

Profile projection e′′ points E located to the left of the axis Z, below the axis X.

Conclusion. The position of a point in space is completely defined if its three coordinates or any two orthogonal projections are known. As a consequence of this, using any two given orthogonal projections of a point, you can always construct its missing third orthogonal projection.

Rice. 11. Constructing points by coordinates indicating octants

Consider constructing a point from two given orthogonal projections.

Task. Using two given orthogonal projections, construct the missing projection of the point IN(Fig. 12).

Rice. 12. Graphical condition of the problem

Solution. We analyze the graphical condition of the problem: the frontal and profile projections of the point are given IN. This means that all three coordinates of the point are given IN. Therefore, it is necessary to construct its horizontal projection.

1. To construct a horizontal projection of a point IN need to know X B And U V. We find these coordinates in the drawing.

2. We measure У В = b Z b′′ and plot this coordinate along the connection line from the axis OH from point b X.

3. Construct a horizontal projection of a point IN(Fig. 13).

Rice. 13. Constructing the missing projection of a point IN

STRAIGHT LINE

In orthogonal projection on projection planes, a straight line is projected as a straight line. To construct projections of this straight line passing through given points A And IN, you need to construct projections of these points and draw straight lines through their projections of the same name (Fig. 14). We get:

ab– horizontal projection of a straight line segment;

a′b′– frontal projection of a straight line segment.

Rice. 14. Projections of a line segment passing through two points

Traces of a straight line

The straight line intersects the projection planes at points called traces direct.

Point of intersection of the line N with horizontal projection plane N(P 1) is called horizontal trace N H .

The point of intersection of the straight line with the frontal projection plane V(P 2) – frontal trace N V.

Point of intersection of the line N with profile projection plane W(P 3) – profile trace N W direct.

Conclusion:

· horizontal trace straight is a point that simultaneously belongs to a given line and lies in the horizontal plane of projections H(P 1);

· frontal trace straight is a point that simultaneously belongs to a given line and lies in the frontal plane of projections V(P 2);

· profile trace straight is a point that simultaneously belongs to a given line and lies in the profile plane of projections W(P 3).

Task. Construct line intersection points N from horizontal N(P 1) and frontal V(P 2) projection planes (Fig. 15 ab).

Analyzing the problem, we come to the conclusion that it is necessary to construct horizontal and frontal traces of a straight line.

1. Construction of the frontal trace N V .

N and frontal plane of projections. According to the material presented earlier, the horizontal projection of the desired point should:

- lie on the axis X;

– belong to the horizontal projection of the line N.

The order of execution of the graphic part of the task:

1.1. Mark the intersection point of the horizontal projection n direct N with axle X, we get the point n V– horizontal projection of the frontal trace.

1.2. Through the point n V X.

1.3. Finding the point of intersection of the communication line with the frontal projection n′ direct N, we get the point N V– frontal projection of the frontal trace. Through this point the straight line goes into the second quarter (Fig. 15 A) and in the third quarter (Fig. 15 b).

2. Construction of the horizontal trace N H .

It is necessary to construct a point belonging to a line N and horizontal projection plane N. According to the material presented earlier, the frontal projection of the desired point should:

- lie on the axis X;

– belong to the frontal projection of the straight line N.

The order of execution of the graphic part of the task:

2.1. Mark the intersection point of the frontal projection n′ straight N with axle X, we get the point nH– frontal projection of the horizontal trace.

2.2. Through the point nH draw a connection line perpendicular to the axis X.

2.3. Finding the point of intersection of the communication line with the horizontal projection n direct N, we obtain a frontal projection of the frontal trace. At this point the straight line intersects the horizontal plane and goes into the fourth quarter (Fig. 15 A,b).

Rice. 15. Constructing traces of a straight line N:

A– the straight line goes into the second quarter; b– the straight line goes into the third quarter

3.2. Position of the point relative to the projection planes

The position of a point in space relative to the projection planes is determined by its coordinates. The X coordinate determines the distance of a point from the P 3 plane (projection onto P 2 or P 1), the Y coordinate – the distance from the P 2 plane (projection onto P 3 or P 1), the Z coordinate – the distance from the P 1 plane (projection onto P 3 or P 2). Depending on the value of these coordinates, a point can occupy both a general and a particular position in space in relation to the projection planes (Fig. 3.1).

Rice. 3.1. Point classification

Tpointsgeneralprovisions. The coordinates of a generic point are not equal to zero ( x≠0, y≠0, z≠0 ), and depending on the sign of the coordinate, the point can be located in one of eight octants (Table 2.1).

In Fig. 3.2 provides drawings of points in general position. Analysis of their images allows us to conclude that they are located in the following octants of space: A(+X;+Y; +Z(  Ioctant;B(+X;+Y;-Z(  IVoctant;C(-X;+Y; +Z(  Voctant;D(+X;+Y; +Z(  IIoctant.

Points of special position. One of the coordinates at a point of particular position is equal to zero, so the projection of the point lies on the corresponding projection field, the other two - on the projection axes. In Fig. 3.3 such points are points A, B, C, D, G.A  P 3, then point X A = 0; IN  P 3, then point X B = 0; WITH  П 2, then pointY C =0;D  P 1, then point Z D = 0.

A point can belong to two projection planes at once if it lies on the line of intersection of these planes - the projection axis. For such points, only the coordinate on this axis is not zero. In Fig. 3.3 such a point is the point G(G  OZ, then point X G =0,Y G =0).

3.3. Relative position of points in space

Let's consider three options for the relative arrangement of points depending on the ratio of coordinates that determine their position in space.

In Fig. 3.4 points A and B have different coordinates.

Their relative position can be assessed by their distance to the projection planes: Y A >Y B, then point A is located further from the P 2 plane and closer to the observer than point B; Z A >Z B, then point A is located further from plane P 1 and closer to the observer than point B; X A

In Fig. 3.5 shows points A, B, C, D, for which one of the coordinates is the same, and the other two are different.

Their relative position can be assessed by their distance to the projection planes as follows:

Y A =Y B =Y D, then points A, B and D are equidistant from the plane P 2, and their horizontal and profile projections are located, respectively, on the straight lines [A 1 B 1 ]llОХ and [A 3 B 3 ]llOZ. The geometric location of such points is a plane parallel to P2;

Z A =Z B =Z C, then points A, B and C are equidistant from the plane P 1, and their frontal and profile projections are located, respectively, on the straight lines [A 2 B 2 ]llОХ and [A 3 C 3 ]llOY. The geometric location of such points is a plane parallel to P 1;

X A =X C =X D, then points A, C and D are equidistant from the plane P 3 and their horizontal and frontal projections are located, respectively, on the straight lines [A 1 C 1 ]llOY and [A 2 D 2 ]llOZ. The geometric location of such points is a plane parallel to P3.

3. If points have equal two coordinates of the same name, then they are called competing. Competing points are located on the same projection line. In Fig. 3.3 there are three pairs of such points for which: X A = X D ; Y A = Y D ; Z D > Z A; X A = X C ; Z A = Z C ; Y C > Y A ; Y A = Y B ; Z A = Z B ; X B > X A .

There are horizontally competing points A and D, located on the horizontally projecting line AD, frontally competing points A and C, located on the frontally projecting line AC, profile competing points A and B, located on the profile projecting line AB.

Conclusions on the topic

1. A point is a linear geometric image, one of the basic concepts of descriptive geometry. The position of a point in space can be determined by its coordinates. Each of the three projections of a point is characterized by two coordinates, their names correspond to the names of the axes that form the corresponding projection plane: horizontal – A 1 (XA; YA); frontal – A 2 (XA; ZA); profile – A 3 (YA; ZA). Translation of coordinates between projections is carried out using communication lines. Using two projections, you can construct projections of a point either using coordinates or graphically.

3. A point in relation to the projection planes can occupy both a general and a particular position in space.

4. A point in general position is a point that does not belong to any of the projection planes, i.e., lying in the space between the projection planes. The coordinates of a generic point are not equal to zero (x≠0,y≠0,z≠0).

5. A point of particular position is a point belonging to one or two projection planes. One of the coordinates at a point of particular position is equal to zero, so the projection of the point lies on the corresponding field of the projection plane, the other two - on the projection axes.

6. Competing points – points whose coordinates of the same name coincide. There are horizontally competing points, frontally competing points, profile competing points.

Keywords

Point coordinates

General point

Private point

Competing points

Methods of activity necessary to solve problems

– construction of a point according to given coordinates in a system of three projection planes in space;

– construction of a point according to given coordinates in a system of three projection planes on a complex drawing.

Self-test questions

1. How is the connection established between the location of coordinates on a complex drawing in the system of three projection planes P 1 P 2 P 3 with the coordinates of point projections?

2. What coordinates determine the distance of points to the horizontal, frontal, profile projection planes?

3. What coordinates and projections of the point will change if the point moves in the direction perpendicular to the profile plane of the projections P 3?

4. What coordinates and projections of a point will change if the point moves in a direction parallel to the OZ axis?

5. What coordinates determine the horizontal (frontal, profile) projection of a point?

7. In what case does the projection of a point coincide with the point in space itself and where are the other two projections of this point located?

8. Can a point belong to three projection planes simultaneously and in what case?

9. What are the names of points whose projections of the same name coincide?

10. How can you determine which of two points is closer to the observer if their frontal projections coincide?

Tasks for independent solution

2. Construct projections of points A and B according to their coordinates on a visual image and a complex drawing: A(13.5; 20), B(6.5; –20). Construct a projection of point C, located symmetrically to point A relative to the frontal plane of projections P 2.

3. Construct projections of points A, B, C according to their coordinates on a visual image and a complex drawing: A(–20; 0; 0), B(–30; -20; 10), C(–10, –15, 0 ). Construct point D, located symmetrically to point C relative to the OX axis.

An example of solving a typical problem

Task 1. The X, Y, Z coordinates of points A, B, C, D, E, F are given (Table 3.3)