The position of a plane in space is determined by three points that do not lie on one line, a line and a point taken outside the line, two intersecting lines and two parallel lines. Accordingly, the plane in the drawing (Fig. 3.1) can be set by the projections of three points that do not lie on the same straight line (a), a straight line and a point taken outside the straight line (b), two intersecting lines (V), two parallel lines (d). The projections of any flat figure can also serve as a reference to the plane in the drawing; for example, see fig. 3.10 image of a plane by projections of a triangle.
Position of the plane relative to projection planes
The plane relative to the planes of projections can take the following positions: 1) not perpendicular to the planes of projections; 2) perpendicular to one projection plane; 3) perpendicular to two projection planes.
A plane that is not perpendicular to any of the projection planes is called a plane. general position(see figure 3.1).
The second and third positions of the planes are special cases. The planes in these positions are called projecting planes.
A plane perpendicular to one projection plane. A visual representation of the plane a given by a triangle ABC and perpendicular to the plane ∏!, is shown in Fig. 3.2, its drawing is in fig. 3.3. This plane is called horizontally projecting.
Visual representation of the plane β defined by a parallelogram ABCD, perpendicular to the frontal projection plane, is shown in fig. 3.4, its drawing is in fig. 3.5. This plane is called front projecting.
Plane drawing in the form of a triangle with projections A "B" C "A" B "C", A "" B tn C"", perpendicular to the profile plane of projections, is shown in fig. 3.6. Such a plane is called a profile-projecting plane.
Plane traces. The line of intersection of the plane with the projection plane is called next. The line of intersection of some plane
sti a given by a triangle ABC, with the π plane, denoted a", a with the π2 plane - a" (see Fig. 3.2).
The line of intersection of the plane with the π plane is called the horizontal trace, with the π2 plane - the frontal trace, with the π plane - the profile trace.
For a plane a perpendicular to the π plane, the horizontal trace a "(see Fig. 3.2,3.3) is located at an angle to the x-axis, corresponding to the angle of inclination of this plane to the frontal projection plane, and the frontal trace a" is perpendicular to the x-axis.
Similarly, for some plane β perpendicular to the plane π2 (see Fig. 3.4,3.5), the frontal wake β" is located at an angle to the axis X, corresponding angle of inclination of this plane to the plane ∏), and the horizontal track β" is perpendicular to the axis X.
In the drawings, the trace that is perpendicular to the projection axis, usually, when it does not participate in the constructions, is not depicted.
Property of projections of geometric elements lying in projecting planes(see § 1.1, ∏. 1, V). The projecting plane is depicted as a straight line
line on the projection plane to which it is perpendicular. Therefore, any closed geometric figure, lying in the projecting plane, is projected onto this projection plane into a straight line segment.
Planes perpendicular to two projection planes. If a plane is perpendicular to two projection planes, then it is parallel to the third projection plane. Such a plane is called horizontal (parallel to the π plane), frontal (parallel to the π2 plane) and profile (parallel to the π3 plane).
Examples of their visual images and drawings are shown in fig. 3.7, a, b(frontal plane at and its own point A), in fig. 3.8, a, b (horizontal plane β and a point belonging to it IN), in fig. 3.9 a, b(profile plane a and the point Q belonging to it.
5.1 Defining a plane
The plane is defined by three arbitrary points that do not belong to the same straight line. The plane in space can be set:
three points that do not lie on one straight line (Figure 5.1, A);
a straight line and a point not belonging to it (Figure 5.1, b);
Two intersecting straight lines (Figure 5.1, V);
Two parallel straight lines (Figure 5.1, G);
any flat figure (figure 5.1, d).
Figure 5.1
Each of the listed ways of specifying a plane allows a transition to any other, since the position of a line in a plane is determined by two of its points or by one point and the direction of this line.
Often used is the method of specifying a plane using straight lines (mutually intersecting or parallel), along which this plane intersects with the projection planes P 1 P 2, P 3. Besides - this is a task of the plane with traces, while maintaining the clarity of the image (Figure 5.2).
Figure 5.2
5.2 Traces of the plane.
The line of intersection of the considered plane with the plane of projections (P 1 , P 2, P 3 ) is called the trace of the plane. In other words, the trace of a plane is a straight line lying in the plane of projections. The trace is assigned the name of the projection plane to which it belongs. For example, a horizontal trace is obtained at the intersection of a given plane with the P 1 plane and is denoted, the frontal trace is with the P 2 plane (), profile - with a plane P 3 (). Two traces of the same plane intersect on the projection axis at a point called the vanishing point of the traces. Each of the traces of the plane coincides with its projection of the same name, the remaining projections turn out to lie on the axes. For example, the horizontal trace of the plane Σ (Figure 5.2) coincides with its horizontal projection, its frontal projection is on the axis X, and the profile on the axis y. By the location of the traces of the plane, one can judge the position of this plane in space relative to the projection planes P 1, P 2, P 3.
5.3 Position of the plane relative to the projection planes
Any plane, arbitrarily taken in space, can occupy a general or particular position. A generic plane is a plane that is not perpendicular to any of the projection planes (see Figure 5.2). All other planes (except for the projection planes) refer to the planes of partial position and are divided into projecting planes and level planes. | A projecting plane is a plane perpendicular to one
from projection planes. For example, the horizontally projecting plane is perpendicular to the horizontal projection plane P 1 (Figure 5.3).
Figure 5.3
Horizontal projections of all geometric images (points, lines, figures) lying in this plane coincide with the horizontal trace 1 . The angle that is formed between the planes and P 2 is projected onto P 1 without distortion. Front trace 2 is perpendicular to the x-axis.
The front-projecting plane () is perpendicular to the frontal plane P 2 is shown in Figure 5.4. Frontal projections of all geometric images (points, lines, figures) lying in this plane coincide with the frontal trace of the plane 2 . The angle that is formed between the given plane and P 1 is projected onto P 2 without distortion. The horizontal trace of plane 1 is perpendicular to the x-axis.
Figure 5.4
The profile-projecting plane T (T 1, T 2) is perpendicular to the profile projection plane P 3 (Figure 5.5).
Figure 5.5
The profile projections of all geometric images (points, lines, figures) lying in this plane coincide with the profile trace of the plane T 3 . Angles and , which are formed between a given plane and the projection planes P 1 and P 2 (= T ^ P 1 ; = T^P 2 ), are projected onto the P 3 plane without distortion. Horizontal and frontal traces of the plane are parallel to the axis X.
The profile-projecting plane can pass through the x-axis: (Figure 5.6).
Figure 5.6
The traces of this plane 1 = 2 coincide with each other and with the x-axis, so they do not determine the position of the plane. In addition to traces, it is necessary to set a point in the plane (Figure 5.6). In a particular case, this plane may be a bisector plane. Angle ° \u003d °, and point A is equidistant from the projection planes P 1 and P 2 . A level plane is a plane that is perpendicular to two projection planes and parallel to a third one. There are three types of such planes (Figure 5.7):
The horizontal plane of the level is perpendicular to P 2, P 3 and parallel to P 1 (Figure 5.7, A);
The frontal plane of the level is perpendicular to P 1, P 3 and parallel to P 2 (Figure 5.7, b);
The profile plane of the level is perpendicular to P 1, P 2 and parallel to P 3 (Figure 5.7 V).
Figure 5.7
It follows from the definition of level planes that one of the projections of a point, line, figure belonging to these planes will coincide with the trace of the level plane of the same name, and the other projection will be the natural size of these geometric images.
5.4 Signs of membership of a point and a straight plane
To determine whether a point belongs to a straight line located in space, one should be guided by the following provisions:
A point belongs to a plane if it is possible to draw a line through it that lies in the plane;
A line belongs to a plane if it has at least two points in common with the plane;
· A line belongs to a plane if it passes through a point of the given plane parallel to the line belonging to this plane.
An infinite number of lines can be drawn through a single point on a plane. These can be arbitrary lines and lines that occupy a special position in relation to the projection planes P 1 P 2, P 3 . A straight line belonging to the plane under consideration, drawn parallel to the horizontal plane of projections, is called r horizontal planes.
A straight line belonging to the plane under consideration, drawn parallel to the frontal plane of projections, is called frontal planes.
Horizontal and frontal are level lines.
The horizontal of the plane should begin to be built from the frontal projection, because it is parallel to the axis x, the horizontal projection of the horizontal is parallel to the horizontal trace of the plane.
And since all the horizontals of the plane are parallel to each other, we can consider the horizontal trace of the plane as the zero horizontal (Figure 5.8).
The front of the plane should begin to be built from a horizontal projection, because it is parallel to the x-axis, the frontal projection of the frontal is parallel to the frontal trace. The frontal trace of the plane is the zero frontal. All fronts of the plane are parallel to each other (Figure 5.9).
Figure 5.8
Figure 5.9
The level line also includes a profile straight line lying in a given plane and parallel to P 3 .
The main lines of a special position in the plane, in addition to the level line, include the lines of the greatest inclination of the plane to the plane of projections.
5.5 Determining the angle of inclination of the plane to the projection planes
The plane of general position, located arbitrarily in space, is inclined to the planes of projections. To determine the value of the dihedral angle of inclination of a given plane to any projection plane, the lines of the greatest inclination of the plane to the projection plane are used: to P 1 - the slope line, to P 2 - the line of the greatest inclination of the plane to the plane P 2 .
The lines of the greatest inclination of the plane are straight lines that form the largest angle with the projection plane, are drawn in the plane perpendicular to the corresponding level line. The lines of greatest slope and its corresponding projection form linear angle, which measures the value of the dihedral angle formed by a given plane and the plane of projections (Figure 5.10).
Any geometric figure immersed in space consists of a certain set of points in space. A plane, as one of the geometric figures, is a collection of many points. From this definition of a plane, it is possible to establish ways of specifying its position in space. To do this, it is enough to recall the axiom of combination - through three points that do not lie on one straight line, you can draw a plane and, moreover, only one.
On fig. 21 shows ways to set the position of the plane in space:
a - three points that do not lie on one straight line;
b - a straight line and a point taken outside the straight line;
c - two intersecting straight lines;
d - two parallel lines.
On the complex drawing (Fig. 22), the plane can be set:
a - projections of three points that do not lie on one straight line;
b - projections of a straight line and a point taken outside the straight line;
c - projections of two intersecting lines;
d - projections of two parallel lines.
Each of those shown in Fig. 22 ways to define a plane in a drawing can be converted from one to another. So, for example, drawing a straight line through points A and B (Fig. 22, a), they get the assignment of the plane shown in Fig. 22b. From there, you can proceed to the method shown in Fig. 22, d, if through point C we draw a line parallel to line AB. If points A, B and C are connected in pairs by straight lines, then we get triangle ABC- a flat figure (Fig. 23), the projections of which can be set to a plane in the drawing.
At the same time, one should always remember that the plane, as a geometric figure, is unlimited and therefore one cannot limit oneself to constructions only within the area of \u200b\u200bthis triangle, since in the general case the projections of the plane occupy all of each of the projection planes: horizontal P I, frontal P 2 and profile P 3.
More clearly, a plane can be defined using straight lines along which it intersects the projection planes (Fig. 24, a).
These lines are called traces of the plane. In the general case, both traces must intersect at a point on the projection axis, which is called the “vanishing point of traces”.
From the whole variety of positions of the plane relative to a given system of projection planes, such are usually distinguished when.
Here, from the axioms of stereometry adopted by us, we obtain important theorems and corollaries about lines and planes. By themselves, they are fairly obvious. Consider their proofs, which show how any statement can be strictly deduced from the axioms with all the necessary references.
2.1 Defining a straight line with two points
Proof. In Section 1.1 it has already been proved that through every two points A and B there passes a line a.
Let us prove that this line is only one. The line a lies in some plane a. Let us assume that, in addition to the line a, the line b also passes through the points A, B (Fig. 31). According to axiom 3, a line that has two points in common with a plane lies in this plane. Since the line b has common points A and B with a, then b lies in the plane α.
Rice. 31
But in the plane a, planimetry is performed, and, consequently, only one straight line passes through the two points A and B. So lines a and b coincide. Thus, only one straight line passes through points A and B.
Consequence. In space (as well as on a plane), two distinct lines cannot have more than one common point.
Two lines that have a single common point are called intersecting.
Comment. Not always a proposition that is true in planimetry is also true in solid geometry. So, for example, in a plane through two given points N, S, only one circle with a diameter NS passes, and in the space of such circles there is an infinite set - in each plane passing through the points N, S, such a circle lies (Fig. 32, a) .
Rice. 32
But the straight line passing through the points N, S in space is only one. This is a common line of all planes passing through the points N, S (Fig. 32, b).
Having proved that there is only one line in space through every two points, we can define a line in space by any pair of its points, without worrying about the plane in which this line lies. A straight line passing through points A, B is denoted (AB).
The same is true for segments: every two points in space serve as the ends of a single segment.
2.2 Defining a plane with three points
Proof. Let the points A, B, C do not lie on one straight line. According to the axiom of the plane, some plane a passes through these points (see Fig. 6). We prove that there is only one.
Let us assume that another plane (3) other than a passes through the points A, B, C. The planes a and p have common points (for example, the point A). According to axiom 2, the intersection of the planes α and β is their common line. On this all common points of the planes α and β lie on the straight line, and hence the points A, B, C. But this contradicts the condition of the theorem, since, according to it, A, B, C do not lie on one straight line. Thus, through the points A, B, C passes only one plane α.
The plane passing through three points A, B, C, which do not lie on one straight line, is denoted (ABC).
It is easy to illustrate Theorem 2. For example, the position of a door is fixed by two door hinges and a lock.
2.3 Specifying a plane with a line and a point and two lines
Proof. Let a line a and a point A not lying on it be given. Take two points B and C on the line a (Fig. 33). The point A does not lie with them on the same line, since only one line passes through the points B and C - this is the line a, and the point A does not lie on it by the condition of the theorem.
Rice. 33
Through the points A, B, C, which do not lie on one straight line, passes (by Theorem 2) the only plane ABC. The line a has two common points B and C with it and, therefore, by axiom 3 lies in it. Thus, the plane ABC is the plane passing through the line a and the point A.
We prove the uniqueness of such a plane by contradiction.
Let there be another plane β containing the line a and the point A. Then it contains the points B and C. By Theorem 2, it must coincide with the plane ABC. The resulting contradiction proves the uniqueness.
Here is an illustration of this theorem: turning the cover of a book, you fix its position with your fingers at every moment.
Proof. Let lines a and b intersect at point A. Take another point B on line b (Fig. 34). By Theorem 3, the plane a passes through the line a and the point B. According to axiom 3, the line b lies in this plane, since it has two common points A and B with it. Hence, the plane a passes through the lines a and b. Prove the uniqueness of such a plane yourself by contradiction.
Rice. 34
Now we know three ways to define a plane:
- three points that do not lie on one straight line;
- a straight line and a point not lying on it;
- two intersecting lines.
Questions for self-control
- What do you know about the ways of setting a straight line in space?
- What do you know how to define a plane?
The plane is one of the most important figures in planimetry, so you need to understand well what it is. Within the framework of this material, we will formulate the very concept of a plane, show how it is denoted in writing, and introduce the necessary notation. Then we will consider this concept in comparison with a point, line or other plane and analyze their options. relative position. All definitions will be illustrated graphically, and the necessary axioms will be formulated separately. In the last paragraph, we will indicate how to correctly define a plane in space in several ways.
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A plane is one of the simplest figures in geometry along with a line and a point. We have previously explained that a point and a line are placed on a plane. If this plane is placed in three-dimensional space, then we get points and lines in space.
In life, an idea of what a plane is can be given to us by objects such as the surface of a floor, table, or wall. But it must be borne in mind that in life their dimensions are limited, but here the concept of a plane is associated with infinity.
We will denote straight lines and points located in space similarly to those placed on a plane - using lowercase and uppercase Latin letters (B, A, d, q, etc.) If in the conditions of the problem we have two points that are located on a straight line, then you can choose such designations that will correspond to each other, for example, the line D B and the points D and B .
To represent plane in writing, small Greek letters are traditionally used, such as α, γ, or π.
If we need a graphical representation of a plane, then usually a closed space of arbitrary shape or a parallelogram is used for this.
The plane is usually considered together with straight lines, points, other planes. Problems with this concept usually contain some variants of their location relative to each other. Let's consider individual cases.
The first way of mutual arrangement is that the point is located on a plane, i.e. belongs to her. We can formulate an axiom:
Definition 1
Every plane has points.
This arrangement is also called passing a plane through a point. To denote this in writing, the symbol ∈ is used. So, if we need to write in literal form that a certain plane π passes through the point A, then we write: A ∈ π.
If a certain plane is given in space, then the number of points belonging to it is infinite. And what is the minimum number of points that will be enough to determine the plane? The answer to this question is the following axiom.
Definition 2
Through three points that are not located on the same straight line, there is only one plane.
Knowing this rule, you can enter a new designation of the plane. Instead of a small Greek letter, we can use the names of the points that lie in it, for example, the plane A B C.
Another way of mutual arrangement of a point and a plane can be expressed using the third axiom:
Definition 3
You can select at least 4 points that will not be in the same plane.
We have already noted above that three points will be enough to designate a plane in space, and the fourth can be located both in it and outside it. If you need to indicate the absence of a point belonging to a given plane in writing, then the sign ∉ is used. An entry of the form A ∉ π is correctly read as "point A does not belong to the plane π"
Graphically, the last axiom can be represented as follows:
The simplest option is that the line is in a plane. Then at least two points of this line will be located in it. Let's formulate an axiom:
Definition 4
If at least two points of a given line are in a certain plane, this means that all points of this line are located in this plane.
To record the belonging of a straight line to a certain plane, we use the same symbol as for a point. If we write “ a ∈ π ”, then this will mean that we have a line a , which is located in the plane π . Let's depict this in the figure:
The second variant of the relative position is when the straight line intersects the plane. In this case, they will have only one common point - the point of intersection. To write such an arrangement in literal form, we use the symbol ∩ . For example, the expression a ∩ π = M reads as "the line a intersects the plane π at some point M". If we have an intersection point, then we also have an angle at which the line intersects the plane.
Graphically, this arrangement looks like this:
If we have two lines, one of which lies in a plane and the other intersects it, then they are perpendicular each friend. In writing, this is indicated by the symbol ⊥ . We will consider the features of this position in a separate article. In the figure, this location will look like this:
If we are solving a problem that has a plane, we need to know what the normal vector of the plane is.
Definition 5
A normal vector of a plane is a vector that lies on a perpendicular line with respect to the plane and is not equal to zero.
Examples of normal plane vectors are shown in the figure:
The third case of the relative position of a straight line and a plane is their parallelism. In this case, they do not have a single common point. To indicate such relationships in writing, the symbol ∥ is used. If we have a record of the form a ∥ π, then it should be read like this: "the line a is parallel to the plane ∥". We will analyze this case in more detail in the article about parallel planes and lines.
If a straight line is located inside a plane, then it divides it into two equal or unequal parts (half-planes). Then such a straight line will be called the boundary of half-planes.
Any 2 points located in the same half-plane lie on the same side of the boundary, and two points belonging to different half-planes lie on opposite sides of the boundary.
1. The simplest option - two planes coincide with each other. Then they will have at least three common points.
2. One plane can intersect another. This creates a straight line. We derive an axiom:
Definition 6
If two planes intersect, then a common straight line is formed between them, on which all possible points of intersection lie.
On a chart it will look like this:
In this case, an angle is formed between the planes. If it is equal to 90 degrees, then the planes will be perpendicular to each other.
3. Two planes can be parallel to each other, that is, not have a single point of intersection.
If we have not two, but three or more intersecting planes, then such a combination is usually called a bundle or a bunch of planes. We will write more about this in a separate article.
In this paragraph, we will see what are the ways to define a plane in space.
1. The first method is based on one of the axioms: the only plane passes through 3 points that do not lie on one straight line. Therefore, we can define a plane by simply specifying three such points.
If we have a rectangular coordinate system in three-dimensional space, in which a plane is given using this method, then we can write an equation for this plane (for more details, see the corresponding article). Let's portray this method on the image:
2. The second way is to set a plane using a straight line and a point that does not lie on this straight line. This follows from the axiom about a plane passing through 3 points. See picture:
3. The third way is to set a plane that passes through two intersecting lines (as we remember, in this case there is also only one plane.) Let's illustrate the method as follows:
4. The fourth method is based on parallel lines. Recall which lines are called parallel: they must lie in the same plane and not have a single intersection point. It turns out that if we indicate two such lines in space, then we can thereby determine for them that very single plane. If we have a rectangular coordinate system in space in which a plane is already defined in this way, then we can derive the equation for such a plane.
In the figure, this method will look like this:
If we remember what the sign of parallelism is, we can derive another way to define a plane:
Definition 7
If we have two intersecting lines that lie in a certain plane that are parallel to two lines in another plane, then these planes themselves will be parallel.
Thus, if we specify a point, then we can specify a plane that passes through it, and the plane to which it will be parallel. In this case, we can also derive the equation of the plane (we have a separate material on this).
Recall one theorem studied in the course on geometry:
Definition 8
Through a certain point in space, only one plane can pass, which will be parallel to a given line.
This means that you can define a plane by specifying a specific point through which it will pass, and a line that will be perpendicular to it. If a plane is defined in this way in a rectangular coordinate system, then we can write a plane equation for it.
Also, we can specify not a straight line, but a normal vector of the plane. Then it will be possible to formulate the general equation.
We have considered the main ways in which you can set a plane in space.
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