This article is the answer to two questions: "What is" and "How to find coordinates of the projection of a point on a plane"? First, the necessary information about projection and its types is given. Next, the definition of the projection of a point onto a plane is given and a graphic illustration is given. After that, a method was obtained for finding the coordinates of the projection of a point onto a plane. In conclusion, solutions of examples are analyzed in which the coordinates of the projection of a given point onto a given plane are calculated.

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Projection, types of projection - necessary information.

When studying spatial figures, it is convenient to use their images in the drawing. The drawing of a spatial figure is a so-called projection this figure to the plane. The process of constructing an image of a spatial figure on a plane occurs according to certain rules. So the process of constructing an image of a spatial figure on a plane, together with a set of rules by which this process is carried out, is called projection figures on this plane. The plane in which the image is built is called projection plane.

Depending on the rules by which the projection is carried out, there are central and parallel projection. We will not go into details, as this is beyond the scope of this article.

In geometry, a special case of parallel projection is mainly used - perpendicular projection, which is also called orthogonal. In the name of this type of projection, the adjective "perpendicular" is often omitted. That is, when in geometry they talk about the projection of a figure onto a plane, they usually mean that this projection was obtained using perpendicular projection (unless otherwise specified, of course).

It should be noted that the projection of a figure onto a plane is a set of projections of all points of this figure onto the projection plane. In other words, in order to get the projection of a certain figure, it is necessary to be able to find the projections of the points of this figure onto the plane. The next paragraph of the article just shows how to find the projection of a point onto a plane.

Projection of a point onto a plane - definition and illustration.

We emphasize once again that we will talk about the perpendicular projection of a point onto a plane.

Let's make constructions that will help us define the projection of a point onto a plane.

Let in three-dimensional space we are given a point M 1 and a plane. Let's draw a straight line a through the point M 1, perpendicular to the plane. If the point M 1 does not lie in the plane, then we denote the intersection point of the line a and the plane as H 1. Thus, by construction, the point H 1 is the base of the perpendicular dropped from the point M 1 to the plane.

Definition.

Projection of point M 1 onto a plane is the point M 1 itself, if , or the point H 1, if .

The following definition is equivalent to this definition of the projection of a point onto a plane.

Definition.

Projection of a point onto a plane- this is either the point itself, if it lies in a given plane, or the base of the perpendicular dropped from this point to a given plane.

In the drawing below, the point H 1 is the projection of the point M 1 onto the plane; point M 2 lies in the plane, therefore M 2 is the projection of the point M 2 itself onto the plane.

Finding the coordinates of the projection of a point on a plane - solving examples.

Let Oxyz be introduced in three-dimensional space, a point and plane. Let's set ourselves the task: to determine the coordinates of the projection of the point M 1 onto the plane.

The solution of the problem follows logically from the definition of the projection of a point onto a plane.

Denote the projection of the point M 1 onto the plane as H 1 . By definition, the projection of a point onto a plane, H 1 is the intersection point of a given plane and a straight line a passing through the point M 1 perpendicular to the plane. Thus, the desired coordinates of the projection of the point M 1 onto the plane are the coordinates of the point of intersection of the line a and the plane.

Hence, to find the projection coordinates of a point on the plane you need:

Let's consider examples.

Example.

Find the projection coordinates of a point to the plane .

Decision.

In the condition of the problem, we are given a general equation of the plane of the form , so it does not need to be compiled.

Let's write the canonical equations of the straight line a, which passes through the point M 1 perpendicular to the given plane. To do this, we obtain the coordinates of the directing vector of the straight line a. Since the line a is perpendicular to the given plane, the direction vector of the line a is the normal vector of the plane . I.e, - directing vector of straight line a . Now we can write the canonical equations of a straight line in space that passes through the point and has a direction vector :
.

To obtain the required coordinates of the projection of a point onto a plane, it remains to determine the coordinates of the point of intersection of the line and plane . To do this, from the canonical equations of the straight line, we pass to the equations of two intersecting planes, we compose a system of equations and find its solution. We use:

So the projection of the point to the plane has coordinates.

Answer:

Example.

In a rectangular coordinate system Oxyz in three-dimensional space, points and . Determine the coordinates of the projection of the point M 1 onto the plane ABC.

Decision.

Let us first write the equation of a plane passing through three given points:

But let's look at an alternative approach.

Let's get the parametric equations of the straight line a , which passes through the point and perpendicular to the plane ABC. The normal vector of the plane has coordinates , therefore, the vector is the direction vector of the line a . Now we can write the parametric equations of a straight line in space, since we know the coordinates of a point on a straight line ( ) and the coordinates of its direction vector ( ):

It remains to determine the coordinates of the point of intersection of the line and planes. To do this, we substitute into the equation of the plane:
.

Now by parametric equations calculate the values ​​of the variables x , y and z at :
.

Thus, the projection of the point M 1 onto the plane ABC has coordinates.

Answer:

In conclusion, let's discuss finding the coordinates of the projection of some point on the coordinate planes and planes parallel to the coordinate planes.

point projections to the coordinate planes Oxy , Oxz and Oyz are the points with coordinates and correspondingly. And the projections of the point on the plane and , which are parallel to the coordinate planes Oxy , Oxz and Oyz respectively, are points with coordinates and .

Let us show how these results were obtained.

For example, let's find the projection of a point onto the plane (other cases are similar to this).

This plane is parallel to the coordinate plane Oyz and is its normal vector. The vector is the direction vector of the line perpendicular to the Oyz plane. Then the parametric equations of the straight line passing through the point M 1 perpendicular to the given plane have the form .

Find the coordinates of the point of intersection of the line and the plane. To do this, first we substitute into the equation of equality: , and the projection of the point

Bugrov Ya.S., Nikolsky S.M. Higher Mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.

Ilyin V.A., Poznyak E.G. Analytic geometry.

A point, as a mathematical concept, has no dimensions. Obviously, if the object of projection is a zero-dimensional object, then it is meaningless to talk about its projection.

Fig.9 Fig.10

In geometry under a point, it is advisable to take a physical object that has linear dimensions. Conventionally, a ball with an infinitely small radius can be taken as a point. With this interpretation of the concept of a point, we can talk about its projections.

When constructing orthogonal projections of a point, one should be guided by the first invariant property of orthogonal projection: the orthogonal projection of a point is a point.

The position of a point in space is determined by three coordinates: X, Y, Z, showing the distances at which the point is removed from the projection planes. To determine these distances, it is enough to determine the meeting points of these lines with the projection planes and measure the corresponding values, which will indicate the values ​​of the abscissa, respectively. X, ordinates Y and appliques Z points (Fig. 10).

The projection of a point is the base of the perpendicular dropped from the point to the corresponding projection plane. Horizontal projection points a call the rectangular projection of a point on the horizontal plane of projections, frontal projection a /- respectively on the frontal plane of projections and profile a // – on the profile projection plane.

Direct Aa, Aa / and Aa // are called projecting lines. At the same time, direct Ah, projecting point BUT on the horizontal plane of projections, called horizontally projecting line, Аa / and Aa //- respectively: frontally and profile-projecting straight lines.

Two projecting lines passing through a point BUT define the plane, which is called projecting.

When converting the spatial layout, the frontal projection of the point A - a / remains in place as belonging to a plane that does not change its position under the considered transformation. Horizontal projection - a together with the horizontal projection plane will turn in the direction of clockwise movement and will be located on one perpendicular to the axis X with front projection. Profile projection - a // will rotate together with the profile plane and by the end of the transformation will take the position indicated in Figure 10. At the same time - a // will be perpendicular to the axis Z drawn from the point a / and will be removed from the axis Z the same distance as the horizontal projection a away from axis X. Therefore, the connection between the horizontal and profile projections of a point can be established using two orthogonal segments aa y and a y a // and a conjugating arc of a circle centered at the point of intersection of the axes ( O- origin). The marked connection is used to find the missing projection (for two given ones). The position of the profile (horizontal) projection according to the given horizontal (profile) and frontal projections can be found using a straight line drawn at an angle of 45 0 from the origin to the axis Y(this bisector is called a straight line) k is the Monge constant). The first of these methods is preferable, as it is more accurate.

Therefore:

1. Point in space removed:

from the horizontal plane H Z,

from the frontal plane V by the value of the given coordinate Y,

from profile plane W by the value of the coordinate. x.

2. Two projections of any point belong to the same perpendicular (one connection line):

horizontal and frontal - perpendicular to the axis x,

horizontal and profile - perpendicular to the Y axis,

frontal and profile - perpendicular to the Z axis.

3. The position of a point in space is completely determined by the position of its two orthogonal projections. Therefore - from any two given orthogonal projections of a point, it is always possible to construct its missing third projection.

If a point has three definite coordinates, then such a point is called point in general position. If a point has one or two coordinates equal to zero, then such a point is called private position point.

Rice. 11 Fig. 12

Figure 11 shows a spatial drawing of points of particular position, Figure 12 shows a complex drawing (diagrams) of these points. Dot BUT belongs to the frontal projection plane, the point AT– horizontal plane of projections, point With– profile plane of projections and point D– abscissa axis ( X).

Auxiliary line of multidrawing

In the drawing shown in fig. 4.7, a, projection axes are drawn, and the images are interconnected by communication lines. Horizontal and profile projections are connected by communication lines using arcs centered at a point O axis intersections. However, in practice, another implementation of the integrated drawing is also used.

On axisless drawings, images are also placed in a projection relationship. However, the third projection can be placed closer or further away. For example, a profile projection can be placed to the right (Fig. 4.7, b, II) or to the left (Fig. 4.7, b, I). This is important for saving space and ease of sizing.

Rice. 4.7.

If in a drawing made according to an axisless system it is required to draw communication lines between the top view and the left view, then an auxiliary straight line of the complex drawing is used. To do this, approximately at the level of the top view and slightly to the right of it, a straight line is drawn at an angle of 45 ° to the drawing frame (Fig. 4.8, a). It is called the auxiliary line of the complex drawing. The procedure for constructing a drawing using this straight line is shown in fig. 4.8, b, c.

If three views have already been built (Fig. 4.8, d), then the position of the auxiliary line cannot be chosen arbitrarily. First you need to find the point through which it will pass. To do this, it suffices to continue until the mutual intersection of the axis of symmetry of the horizontal and profile projections and through the resulting point k draw a straight line segment at an angle of 45 ° (Fig. 4.8, d). If there are no axes of symmetry, then continue until the intersection at the point k 1 horizontal and profile projections of any face projected as a straight line (Fig. 4.8, d).

Rice. 4.8.

The need to draw communication lines, and, consequently, an auxiliary straight line, arises when constructing missing projections and when performing drawings on which it is necessary to determine the projections of points in order to clarify the projections of individual elements of the part.

Examples of the use of the auxiliary line are given in the next paragraph.

Projections of a point lying on the surface of an object

In order to correctly build projections of individual elements of a part when making drawings, it is necessary to be able to find projections of individual points on all drawing images. For example, it is difficult to draw a horizontal projection of the part shown in Fig. 4.9 without using the projections of individual points ( A, B, C, D, E and etc.). The ability to find all the projections of points, edges, faces is also necessary for recreating in the imagination the shape of an object according to its flat images in the drawing, as well as for checking the correctness of the completed drawing.

Rice. 4.9.

Let's consider ways of finding the second and third projections of a point given on the surface of an object.

If one projection of a point is given in the drawing of an object, then first it is necessary to find the projections of the surface on which this point is located. Then choose one of the two methods described below for solving the problem.

First way

This method is used when at least one of the projections shows the given surface as a line.

On fig. 4.10, a a cylinder is shown, on the frontal projection of which the projection is set a" points BUT, lying on the visible part of its surface (given projections are marked with double colored circles). To find the horizontal projection of a point BUT, they argue as follows: the point lies on the surface of the cylinder, the horizontal projection of which is a circle. This means that the projection of a point lying on this surface will also lie on the circle. Draw a line of communication and mark the desired point at its intersection with the circle a. third projection a"

Rice. 4.10.

If the point AT, lying on the upper base of the cylinder, given by its horizontal projection b, then the communication lines are drawn to the intersection with straight line segments depicting the frontal and profile projections of the upper base of the cylinder.

On fig. 4.10, b shows the detail - emphasis. To construct projections of a point BUT, given by its horizontal projection a, find two other projections of the upper face (on which lies the point BUT) and, drawing the connection lines to the intersection with the line segments depicting this face, determine the desired projections - points a" and a". Dot AT lies on the left side vertical face, which means that its projections will also lie on the projections of this face. So from a given point b" draw communication lines (as indicated by arrows) until they meet with line segments depicting this face. frontal projection with" points WITH, lying on an inclined (in space) face, are found on the line depicting this face, and the profile with"- at the intersection of the connection line, since the profile projection of this face is not a line, but a figure. Construction of point projections D shown by arrows.

Second way

This method is used when the first method cannot be used. Then you should do this:

• draw through the given projection of the point the projection of the auxiliary line located on the given surface;
• find the second projection of this line;
• to the found projection of the line, transfer the given projection of the point (this will determine the second projection of the point);
• find the third projection (if required) at the intersection of communication lines.

On fig. 4.10, frontal projection is given a" points BUT, lying on the visible part of the surface of the cone. To find the horizontal projection through a point a" carry out a frontal projection of an auxiliary straight line passing through the point BUT and the top of the cone. Get a point V is the projection of the meeting point of the drawn line with the base of the cone. Having frontal projections of points lying on a straight line, one can find their horizontal projections. Horizontal projection s the top of the cone is known. Dot b lies on the circumference of the base. A line segment is drawn through these points and a point is transferred to it (as shown by the arrow). a", getting a point a. Third projection a" points BUT located at the crossroads.

The same problem can be solved differently (Fig. 4.10, G).

As an auxiliary line passing through a point BUT, they take not a straight line, as in the first case, but a circle. This circle is formed if at the point BUT intersect the cone with a plane parallel to the base, as shown in the visual representation. The frontal projection of this circle will be depicted as a straight line segment, since the plane of the circle is perpendicular to the frontal projection plane. The horizontal projection of a circle has a diameter equal to the length of this segment. Describing a circle of the specified diameter, draw from a point a" connection line to the intersection with the auxiliary circle, since the horizontal projection a points BUT lies on the auxiliary line, i.e. on the constructed circle. third projection as" points BUT found at the intersection of communication lines.

In the same way, you can find the projections of a point lying on a surface, for example, a pyramid. The difference will be that when it is crossed by a horizontal plane, not a circle is formed, but a figure similar to the base.

Goals:

• Studying the rules for constructing projections of points on the surface of an object and reading drawings.
• Develop spatial thinking, the ability to analyze the geometric shape of an object.
• To cultivate industriousness, the ability to cooperate when working in groups, interest in the subject.

DURING THE CLASSES

I STAGE. MOTIVATION OF LEARNING ACTIVITIES.

II STAGE. FORMATION OF KNOWLEDGE, SKILLS AND SKILLS.

HEALTH-SAVING PAUSE. REFLECTION (MOOD)

STAGE III. INDIVIDUAL WORK.

I STAGE. MOTIVATION OF LEARNING ACTIVITIES

1) Teacher: Check your workplace, is everything in place? Is everyone ready to go?

BREATHED DEEPLY, HOLD THE BREATH ON EXHAUST, EXHALED.

Determine your mood at the beginning of the lesson according to the scheme (such a scheme is on the table for everyone)

I WISH YOU GOOD LUCK.

2)Teacher: Practical work on the topic “ Projections of Vertices, Edges, Faces” showed that there are guys who make mistakes when projecting. They get confused which of the two matching points in the drawing is the visible vertex and which is the invisible one; when the edge is parallel to the plane, and when it is perpendicular. Same thing with edges.

To avoid repeating mistakes, complete the necessary tasks using the consulting card and correct mistakes in practical work (by hand). And as you work, remember:

“EVERYBODY CAN MAKE MISTAKES, STAY AT HIS MISTAKE - ONLY THE CRAZY”.

And those who have mastered the topic well will work in groups with creative tasks (see. Appendix 1 ).

II STAGE. FORMATION OF KNOWLEDGE, SKILLS AND SKILLS

1)Teacher: In production, there are many parts that are attached to each other in a certain way.
For example:
The desktop cover is attached to the vertical posts. Pay attention to the table at which you are, how and with what the lid and racks are attached to each other?

Answer: Bolt.

Teacher: What is required for a bolt?

Answer: Hole.

Teacher: Really. And in order to make a hole, you need to know its location on the product. When making a table, the carpenter cannot contact the customer every time. So, what is the need to provide a carpenter?

Answer: Drawing.

Teacher: Drawing!? What do we call a drawing?

Answer: A drawing is an image of an object by rectangular projections in a projection connection. According to the drawing, you can represent the geometric shape and design of the product.

Teacher: We have completed rectangular projections, and then? Will we be able to determine the location of the holes from one projection? What else do we need to know? What to learn?

Answer: Build points. Find projections of these points in all views.

Teacher: Well done! This is the purpose of our lesson, and the topic: Construction of projections of points on the surface of an object. Write the topic of the lesson in your notebook.
You and I know that any point or segment on the image of an object is a projection of a vertex, edge, face, i.e. each view is an image not from one side (ch. view, top view, left view), but the whole object.
In order to correctly find the projections of individual points lying on the faces, you must first find the projections of this face, and then use the connection lines to find the projections of the points.

(We look at the drawing on the board, we work in a notebook where 3 projections of the same part are made at home).

- Opened a notebook with a completed drawing (An explanation of the construction of points on the surface of an object with leading questions on the board, and students fix it in a notebook.)

Teacher: Consider a point AT. What plane is the face with this point parallel to?

Answer: The face is parallel to the frontal plane.

Teacher: We set the projection of a point b' in frontal projection. Draw down from the point b' vertical line of communication to the horizontal projection. Where will the horizontal projection of the point be? AT?

Answer: At the intersection with the horizontal projection of the face that was projected into the edge. And is at the bottom of the projection (view).

Teacher: Point profile projection b'' where will it be located? How will we find it?

Answer: At the intersection of the horizontal line of communication from b' with a vertical edge on the right. This edge is the projection of the face with a point AT.

THOSE WANTING TO CONSTRUCT THE NEXT PROJECTION OF THE POINT ARE CALLED TO THE BOARD.

Teacher: Point projections BUT are also located using communication lines. Which plane is parallel to the edge with a point BUT?

Answer: The face is parallel to the profile plane. We set a point on the profile projection a'' .

Teacher: On what projection is the face projected into the edge?

Answer: On the front and horizontal. Let's draw a horizontal connection line to the intersection with a vertical edge on the left on the frontal projection, we get a point a' .

Teacher: How to find the projection of a point BUT on a horizontal projection? After all, communication lines from the projection of points a' and a'' do not intersect the projection of the face (edge) on the horizontal projection on the left. What can help us?

Answer: You can use a constant straight line (it determines the position of the view on the left) from a'' draw a vertical line of communication until it intersects with a constant straight line. From the intersection point, a horizontal line of communication is drawn, until it intersects with a vertical edge on the left. (This is the face with point A) and denotes the projection with a point a .

2) Teacher: Everyone has a task card on the table, with a tracing paper attached. Consider the drawing, now try on your own, without redrawing the projections, to find the given projections of points on the drawing.

– Find in the textbook p. 76 fig. 93. Test yourself. Who performed correctly - score "5" "; one mistake - ''4''; two - ''3''.

(The grades are set by the students themselves in the self-control sheet).

- Collect cards for testing.

3)Group work: Time limited: 4min. + 2 min. checks. (Two desks with students are combined, and a leader is selected within the group).

For each group, tasks are distributed in 3 levels. Students choose tasks by levels, (as they wish). Solve problems on the construction of points. Discuss the construction under the supervision of the leader. Then the correct answer is displayed on the board with the help of a codoscope. Everyone checks that the points are projected correctly. With the help of the group leader, grades are given on assignments and in self-control sheets (see. Appendix 2 and Annex 3 ).

HEALTH-SAVING PAUSE. REFLECTION

"Pharaoh's Pose"- sit on the edge of a chair, straighten your back, bend your arms at the elbows, cross your legs and put on your toes. Inhale, tighten all the muscles of the body while holding the breath, exhale. Do 2-3 times. Close your eyes tightly, to the stars, open. Mark your mood.

STAGE III. PRACTICAL PART. (Individual tasks)

There are task cards to choose from with different levels. Students choose their own option. Find projections of points on the surface of an object. Works are handed over and evaluated for the next lesson. (Cm. Appendix 4 , Annex 5 , Appendix 6 ).

STAGE IV. FINAL

1) Homework assignment. (Instruction). Performed by levels:

B - understanding, on "3". Exercise 1 fig. 94a p. 77 - according to the assignment in the textbook: complete the missing projections of points on these projections.

B - application, on "4". Exercise 1 Fig. 94 a, b. complete the missing projections and mark the vertices on the visual image in 94a and 94b.

A - analysis, on "5". (Increased difficulty.) Ex. 4 fig.97 - construct missing projections of points and designate them with letters. There is no visual image.

2)Reflective analysis.

1. Determine the mood at the end of the lesson, mark it on the self-control sheet with any sign.
2. What new did you learn at the lesson today?
3. What form of work is most effective for you: group, individual and would you like it to be repeated in the next lesson?
4. Collect checklists.

3)"Wrong Teacher"

Teacher: You have learned how to build projections of vertices, edges, faces and points on the surface of an object, following all the construction rules. But here you were given a drawing, where there are errors. Now try yourself as a teacher. Find the mistakes yourself, if you find all 8–6 mistakes, then the score is “5”, respectively; 5–4 errors - “4”, 3 errors - “3”.

Answers:

Consider the profile plane of projections. Projections on two perpendicular planes usually determine the position of the figure and make it possible to find out its real dimensions and shape. But there are times when two projections are not enough. Then apply the construction of the third projection.

The third projection plane is carried out so that it is perpendicular to both projection planes at the same time (Fig. 15). The third plane is called profile.

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

Figure 15a shows the point BUT 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 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 near the axis z in the direction indicated by the arrow in Figure 15.

Figure 16 shows the position of the projections a, a and a points BUT, obtained as a result of combining all three planes with the drawing plane.

As a result of the cut, the y-axis occurs on the diagram in two different places. 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).

Figure 16 shows three projections a, a and a points A have a strictly defined position on the diagram and are subject to unambiguous conditions:

a and a must always be located on one vertical straight line perpendicular to the axis X;

a and a must always be located on the same horizontal line perpendicular to the axis z;

3) when drawn through a horizontal projection and a horizontal line, but through a profile projection a- a vertical straight line, the constructed lines will necessarily intersect on the bisector of the angle between the projection axes, since the figure Oa at a 0 a n is a square.

When constructing three projections of a point, it is necessary to check the fulfillment of all three conditions for each point.

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.

Point distance BUT to the profile plane is the coordinate X, wherein X = a˝A(Fig. 15), the distance to the frontal plane - by the coordinate y, and y = aa, 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 box, and the measurements of this box correspond to the coordinates of this point, i.e., each of the coordinates is presented in Figure 15 four times, i.e.:

x = a˝A = Oa x = a y a = a z á;

y = а́А = Оа y = a x a = a z a˝;

z = aA = Oa z = a x a′ = a y a˝.

On the diagram (Fig. 16), the x and z coordinates occur three times:

x \u003d a z a ́ \u003d Oa x \u003d a y a,

z = a x á = 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 the vertical axis:

y \u003d Oa y \u003d a x a

and twice - located horizontally:

y \u003d Oa y \u003d a z a˝.

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

It should be noted 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 build projections of a point on the diagram.

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

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

1) horizontal and frontal projections a and a X X;

2) frontal and profile projections a and a should be located on 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 the profile projection a away from axis z, since the projections a′ and a˝ have a common coordinate at.

If the 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, its two coordinates are zero.

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

Projection of a straight line

Two points are needed to define a line. A point is defined by two projections on the horizontal and frontal planes, i.e., a straight line is determined using the projections of its two points on the horizontal and frontal planes.

Figure 17 shows projections ( a and a, b and b) two points BUT and B. With their help, the position of some straight line AB. When connecting the same-name projections of these points (i.e. a and b, a and b) you can get projections ab and ab direct AB.

Figure 18 shows the projections of both points, and Figure 19 shows the projections of a straight line passing through them.

If the projections of a straight line are determined by the projections of its two points, then they are denoted by two adjacent Latin letters corresponding to the designations of the projections of points taken on the straight line: with strokes to indicate the frontal projection of the straight line or without strokes - for the horizontal projection.

If we consider not individual points of a straight line, but its projections as a whole, then these projections are indicated by numbers.

If some point With lies on a straight line AB, its projections с and с́ are on the projections of the same line ab and ab. Figure 19 illustrates this situation.

Straight traces

trace straight- this is the point of its intersection with some plane or surface (Fig. 20).

Horizontal track straight some point is called H where the line meets the horizontal plane, and frontal- dot V, in which this straight line meets the frontal plane (Fig. 20).

Figure 21a shows the horizontal trace of a straight line, and its frontal trace, in Figure 21b.

Sometimes the profile trace of a straight line is also considered, W- the point of intersection of a straight line with a profile plane.

The horizontal trace is in the horizontal plane, i.e. its horizontal projection h coincides with this trace, and the frontal h lies on the x-axis. The frontal trace lies in the frontal plane, so its frontal projection ν́ coincides with it, and the horizontal v lies on the x-axis.

So, H = h, and V= v. Therefore, to denote traces of a straight line, letters can be used h and v.

Various positions of the line

The straight line is called direct general position, if it is neither parallel nor perpendicular to any of the projection planes. The projections of a line in general position are also neither parallel nor perpendicular to the projection axes.

Straight lines that are parallel to one of the projection planes (perpendicular to one of the axes). Figure 22 shows a straight line that is parallel to the horizontal plane (perpendicular to the z-axis), is a horizontal straight line; figure 23 shows a straight line that is parallel to the frontal plane (perpendicular to the axis at), is the frontal straight line; figure 24 shows a straight line that is parallel to the profile plane (perpendicular to the axis X), is a profile straight line. Despite the fact that each of these lines forms a right angle with one of the axes, they do not intersect it, but only intersect with it.

Due to the fact that the horizontal line (Fig. 22) is parallel to the horizontal plane, its frontal and profile projections will be parallel to the axes that define the horizontal plane, i.e., the axes X and at. Therefore projections ab|| X and a˝b˝|| at z. The horizontal projection ab can take any position on the plot.

At the frontal line (Fig. 23) projection ab|| x and a˝b˝ || z, i.e. they are perpendicular to the axis at, and therefore in this case the frontal projection ab the line can take any position.

At the profile line (Fig. 24) ab|| y, ab|| z, and both are perpendicular to the x-axis. Projection a˝b˝ can be placed on the diagram in any way.

When considering the plane that projects the horizontal line onto the frontal plane (Fig. 22), you can see that it projects this line onto the profile plane as well, i.e. it is a plane that projects the line onto two projection planes at once - the frontal and profile. For this reason it is called doubly projecting plane. In the same way, for the frontal line (Fig. 23), the doubly projecting plane projects it onto the planes of the horizontal and profile projections, and for the profile (Fig. 23) - onto the planes of the horizontal and frontal projections.

Two projections cannot define a straight line. Two projections 1 and one profile straight line (Fig. 25) without specifying the projections of two points of this straight line on them will not determine the position of this straight line in space.

In a plane that is perpendicular to two given planes of symmetry, there may be an infinite number of lines for which the data on the diagram 1 and one are their projections.

If a point is on a line, then its projections in all cases lie on the projections of the same name on this line. The opposite situation is not always true for the profile line. On its projections, you can arbitrarily indicate the projections of a certain point and not be sure that this point lies on a given line.

In all three special cases (Fig. 22, 23 and 24), the position of the straight line with respect to the plane of projections is its arbitrary segment AB, taken on each of the straight lines, is projected onto one of the projection planes without distortion, that is, onto the plane to which it is parallel. Line segment AB horizontal straight line (Fig. 22) gives a life-size projection onto a horizontal plane ( ab = AB); line segment AB frontal straight line (Fig. 23) - in full size on the plane of the frontal plane V ( ab = AB) and the segment AB profile straight line (Fig. 24) - in full size on the profile plane W (a˝b˝\u003d AB), i.e. it is possible to measure the actual size of the segment on the drawing.

In other words, with the help of diagrams, one can determine the natural dimensions of the angles that the line under consideration forms with the projection planes.

The angle that a straight line makes with a horizontal plane H, it is customary to denote the letter α, with the frontal plane - the letter β, with the profile plane - the letter γ.

Any of the straight lines under consideration has no trace on a plane parallel to it, i.e., the horizontal straight line has no horizontal trace (Fig. 22), the frontal straight line has no frontal trace (Fig. 23), and the profile straight line has no profile trace (Fig. 24 ).