A system of three mutually perpendicular planes. Projection onto three mutually perpendicular projection planes Three mutually perpendicular planes

Point position	Visual image	Complex drawing	Characteristic signs
belongs to the plane  1			A 1 – below the X axis, A 2 – on the X axis
belongs to the plane  1			B 1 – above the X axis, B 2 – on the X axis
belongs to the plane  2			C 2 – above the X axis, C 1 – on the X axis
belongs to the plane  2			D 1 – on the X axis, D 2 – below the X axis
belongs to the X axis			E 1 coincides with E 2 and belongs to the X axis

Task No. 1.

Construct a complex drawing of point A if:

the point is located in the second quarter and is equidistant from the planes  1 and  2.

the point is located in the third quarter, and its distance to the  1 plane is twice as great as to the  2 plane.

the point is located in the IV quarter, and its distance to the  1 plane is greater than to the  2 plane.

Task No. 2.

Determine in which quarters the points are located (Fig. 2.21).

Task No. 3.

Construct a visual representation of the points in the quarters:

a) A – general position in the third quarter;

b) B – general position in the IV quarter;

c) C – in the second quarter, if its distance from  1 is 0;

d) D – in the first quarter, if its distance from  2 is 0.

Task No. 4.

Construct a complex drawing of points A, B, C, D (see task 3).

§ 5. System of three mutually perpendicular planes

In practice, research and imaging, a system of two mutually perpendicular planes does not always provide the possibility of an unambiguous solution. So, for example, if you move point A along the X axis, its image will not change.

The position of the point in space (Fig. 2.22) has changed (Fig. 2.24), but the images in the complex drawing remain unchanged (Fig. 2.23 and Fig. 2.25).

To solve this problem, a system of three mutually perpendicular planes is introduced, since when drawing up drawings, for example, machines and their parts, not two, but more images are required. On this basis, in some constructions when solving problems, it is necessary to introduce  1,  2 and other projection planes into the system.

Consider three mutually perpendicular planes 1 ,  2 ,  3 ( rice. 2.26). The vertical plane 3 is called the profile plane of projection. Intersecting each other, planes 1 ,  2 ,  3 form the projection axes, while the space is divided into 8 octants.

 1  2 = x; -x

 1  3 = y; -y

 2  3 = z; -z

0 – point of intersection of the projection axes.

These planes divide the entire space into VIII parts, which are called octants (from the Latin okto eight). The planes have no thickness, are opaque and infinite. The observer is located in the first quarter (for systems  1,  2) or the first octant (for systems  1,  2,  3) at an infinite distance from the projection planes.

Task No. 4.

Task No. 3.

Task No. 2.

Task No. 1.

Formation of a complex drawing (diagram)

For the convenience of using the resulting images from the spatial system of planes, let’s move on to the planar one.

For this:

1. Apply the method of rotating the plane p 1 around the X axis until it aligns with the plane p 2 (Fig. 2.7)

2. Combine planes p 1 and p 2 into one drawing plane (Fig. 2.8)


Rice. 2.7	Rice. 2.8

Projections A 1 and A 2 are located on the same connection line perpendicular to the X axis. This line is called the projection connection line (Fig. 2.9).

Since the projection plane is considered infinite in space, the boundaries of the plane p 1, p 2 need not be depicted (Fig. 2.10).

As a result of combining the planes p 1 and p 2, a complex drawing or diagram is obtained (from the French epure drawing), i.e. drawing in the system p 1 and p 2 or in the system of two projection planes. Having replaced the visual image with a diagram, we have lost the spatial picture of the location of projection planes and points. But the diagrams provide accuracy and easy-to-measure images with significant simplicity of construction. To imagine a spatial picture from a diagram requires the work of imagination: for example, according to Fig. 2.11 you need to imagine the picture shown in Fig. 2.12.

If there is a projection axis in the complex drawing along projections A 1 and A 2, you can establish the position of point A relative to p 1 and p 2 (see Fig. 2.5 and 2.6). Comparing Fig. 2.11 and 2.12 it is easy to establish that the segment A 2 A X is the distance from point A to the plane p 1, and the segment A 1 A X is the distance from point A to p 2. The location of A 2 above the projection axis means that point A is located above the plane p 1. If A 1 on the diagram is located below the projection axis, then point A is in front of the plane p 2. Thus, the horizontal projection of the geometric image determines its position relative to the frontal plane of projections p 2 , and the frontal projection of the geometric image - relative to the horizontal plane of projections p 1 .


Rice. 2.11	Rice. 2.12

§ 4. Characteristics of the position of a point in the system p 1 and p 2

A point defined in space can have different positions relative to the projection planes (Fig. 2.13).

Let's consider possible options for the location of a point in the space of the first quarter:

1. A point is located in the space of the first quarter at any distance from the X axis and planes p 1 p 2, for example, points A, B (such points are called points of general position) (Fig. 2.14 and Fig. 2.15).

3. Point K belongs simultaneously to both the plane p 1 and p 2, that is, it belongs to the X axis (Fig. 2.18):

Based on the above, we can draw the following conclusion:

1. If a point is located in the space of the first quarter, then its projection A 2 is located above the X axis, and A 1 is below the X axis; A 2 A 1 – lie on the same perpendicular (connection line) to the X axis (Fig. 2.14).

2. If a point belongs to the plane p 2, then its projection C 2 C (coincides with the point C itself) and the projection C 1 X (belongs to the X axis) and coincides with C X: C 1 C X.

3. If a point belongs to the plane p 1, then its projection D 1 onto this plane coincides with the point D D 1 itself, and the projection D 2 belongs to the X axis and coincides with D X: D 2 D X.

4. If a point belongs to the X axis, then all its projections coincide and belong to the X axis: K K 1 K 2 K X.

Exercise:

1. Characterize the position of points in the space of the first quarter (Fig. 2.19).

2. Construct a visual image and a comprehensive drawing of the point according to the description:

a) point C is located in the first quarter, and is equidistant from the planes p 1 and p 2.

b) point M belongs to the plane p 2.

c) point K is located in the first quarter, and its distance to p 1 is twice as large as to the plane p 2.

d) point L belongs to the X axis.

3. Construct a complex drawing of a point according to the description:

a) point P is located in the first quarter, and its distance from the plane p 2 is greater than from the plane p 1.

b) point A is located in the first quarter and its distance to the plane p 1 is 3 times greater than to the plane p 2.

c) point B is located in the first quarter, and its distance to the plane is p 1 =0.

4. Compare the position of the points relative to the projection planes p 1 and p 2 and with each other. The comparison is made based on characteristics or features. For points, these characteristics are the distance to the planes p 1; p 2 (Fig. 2.20).

The application of the above theory when constructing images of a point can be carried out in various ways:

words (verbal);
graphically (drawings);
visual image (volumetric);
planar (complex drawing).

The ability to translate information from one method to another contributes to the development of spatial thinking, i.e. from verbal to visual (volumetric), and then to planar, and vice versa.

Let's look at this with examples (Table 2.1 and Table 2.2).

Table 2.1

Example of dot image
in a system of two projection planes

Quarter space	Visual image	Complex drawing	Characteristic signs
I			Frontal projection of point A above the X axis, horizontal projection of point A below the X axis
II			Frontal and horizontal projections of point B above the X axis
III			Frontal projection of point C below the X axis, horizontal projection of point C above the X axis
IV			Frontal and horizontal projections of point D below the X axis

Table 2.2

An example of an image of points belonging to the planes p 1 and p 2

Point position	Visual image	Complex drawing	Characteristic signs
Point A belongs to the plane p 1			A 1 – below the X axis, A 2 – on the X axis
Point B belongs to plane p 1			B 1 – above the X axis, B 2 – on the X axis
Point C belongs to the plane p 2			C 2 – above the X axis, C 1 – on the X axis
Point D belongs to the plane p 2			D 1 – on the X axis, D 2 – below the X axis
Point E belongs to the X axis			E 1 coincides with E 2 and belongs to the X axis

Construct a complex drawing of point A if:

1. The point is located in the II quarter and is equidistant from the planes p 1 and p 2.

2. The point is located in the third quarter, and its distance to the plane p 1 is twice as large as to the plane p 2.

3. The point is located in the IV quarter, and its distance to the p1 plane is greater than to the p2 plane.

Determine in which quarters the points are located (Fig. 2.21).

1. Construct a visual image of the points in the quarters:

a) A – general position in the third quarter;

b) B – general position in the IV quarter;

c) C – in the second quarter, if its distance from p 1 is 0;

d) D – in the first quarter, if its distance from p 2 is 0.

Construct a complex drawing of points A, B, C, D (see task 3).

The position of the point in space (Fig. 2.22) has changed (Fig. 2.24), but the images in the complex drawing remain unchanged (Fig. 2.23 and Fig. 2.25).


Rice. 2.22	Rice. 2.23

Rice. 2.24	Rice. 2.25

To solve this problem, a system of three mutually perpendicular planes is introduced, since when drawing up drawings, for example, machines and their parts, not two, but more images are required. On this basis, in some constructions when solving problems, it is necessary to introduce p 1, p 2 and other projection planes into the system.

These planes divide the entire space into VIII parts, which are called octants (from the Latin okto eight). The planes have no thickness, are opaque and infinite. The observer is located in the first quarter (for systems p 1, p 2) or the first octant (for systems p 1, p 2, p 3) at an infinite distance from the projection planes.

§ 6. Point in the system p 1, p 2, p 3

The construction of projections of a certain point A, located in the first octant, onto three mutually perpendicular planes p 1, p 2, p 3 is shown in Fig. 2.27. Using the combination of projection planes with the p 2 plane and using the method of rotating the planes, we obtain a complex drawing of point A (Fig. 2.28):

AA 1 ^ p 1 ; AA 2 ^ p 2 ; AA 3 ^ p 3,

where A 3 – profile projection of point A; А Х, А y, А Z – axial projections of point A.

Projections A 1, A 2, A 3 are called, respectively, the frontal, horizontal and profile projection of point A.


Rice. 2.27	Rice. 2.28

The projection planes, intersecting in pairs, define three axes x, y, z, which can be considered as a system of Cartesian coordinates: axis X called the abscissa axis, axis y– ordinate axis, axis Z– applicate axis, the point of intersection of the axes, denoted by the letter ABOUT, is the origin of coordinates.

Thus, the viewer looking at the object is in the first octant.

To obtain a complex drawing, we apply the method of rotating the planes p 1 and p 3 (as shown in Fig. 2.27) until aligned with the plane p 2. The final view of all planes in the first octant is shown in Fig. 2.29.

Here are the axes Oh And Oz, lying in the fixed plane p 2, are depicted only once, the axis Oh shown twice. This is explained by the fact that, rotating with the plane p 1, the axis y on the diagram it is combined with the axis Oz, and rotating with the plane p 3, this same axis coincides with the axis Oh.

Let's look at Fig. 2.30, where is the point in space A, given by coordinates (5,4,6). These coordinates are positive, and she herself is in the first octant. The construction of an image of the point itself and its projections on a spatial model is carried out using a coordinate rectangular parallelogram. To do this, we plot segments on the coordinate axes, corresponding to the length segments: Oah = 5, OAy = 4, OAz= 6. On these segments ( ОАx, ОАy, ОАz), as on the edges, we build a rectangular parallelepiped. One of its vertices will define a given point A.

Speaking about the system of three projection planes in a complex drawing (Fig. 2.30), it is necessary to note the following.

When solving problems, two projections are sometimes not enough. Therefore, a third plane is introduced perpendicular to the planes P 1 and P 2. They call her profile plane (P 3 ) .

Three planes divide space into 8 parts - octants (Fig. 6). As before, we will assume that the viewer looking at the object is in the first octant. To obtain a diagram (Fig. 7), any geometric image of the plane P 1 and P 3 is rotated, as shown in Fig. 6.

The projection planes, intersecting in pairs, define three axes x, y And z, which can be considered as a system of Cartesian coordinates in space with the origin at the point ABOUT.

To obtain a diagram, points in the system of three projection planes, planes P 1 and P 3, are rotated until aligned with plane P 2 (Fig. 8). When designating axes on a diagram, negative semi-axes are usually not indicated.

To find the profile projection of the points proceed as follows: from the frontal projection A 2 points A draw a straight line perpendicular to the axis Z and on this straight line from the axis z plot a segment equal to the coordinate at points A(Fig. 9).

Fig.8 Fig. 9
Coordinates are numbers that are assigned to a point to determine its position in space or on a surface. In three-dimensional space, the position of a point is determined using rectangular Cartesian coordinates x, y And z(abscissa, ordinate and applicate):

A
?
bscissa X = ………..= …..…..= ….….. = ……….. – distance from the point to the plane P 3;

ordinate at = ……….= ………= …...... = ………… – distance from the point to the plane P 2;

applicate z= …….. = ………= ……..= ………… – distance from the point to the plane P 1
A 1 A 2 – vertical connection line perpendicular to the x axis;

A 2 A 3 – horizontal connection line perpendicular to the axisz.
A
?
1 (….,….) Projection position of each point

A 2 (….,….) is defined by two coordinates

A 3 (….,….)
If a point belongs to at least one projection plane, it occupies private position relative to projection planes. If a point does not belong to any of the projection planes, it occupies general position.

Lecture No. 2
STRAIGHT

1. Direct. 2. Position of the line relative to the projection planes. 3. The point belongs to a straight line. 4. Traces are straight. 5. Division of a straight line segment in a given ratio. 6. Determination of the length of a straight line segment and the angles of inclination of the straight line to the projection planes. 7. Mutual position of lines.
1STRAIGHT
The projection of a line in the general case is a straight line, except for the case when the line is perpendicular to the plane (Fig. 10).

To construct a diagram of a straight line, determine the coordinates x, y, z two points on a straight line and transfer these values to the drawing.

2 POSITION OF THE LINE RELATIVE TO THE PROJECTION PLANES
IN

Depending on the position of the line in relation to the projection planes, it can occupy both general and particular positions.

P the projection of a generic line is less than the straight line itself.

There is an ascending straight line - this is a straight line, which rises as it moves away from the observer (Fig. 11) and a descending straight line, which decreases.

h   P 1 ; Z = const

h 2  0x sign

h 3  0at horizontal

h 1 =  h – property

horizontal

 – angle of inclination of the straight line to

plane P 1

 – angle of inclination of the straight line to

plane P 2

 – angle of inclination of the straight line to

plane P 3


?
= 0

 = (h 1  P 2) designate


Rice. 12. Horizontal
= (h 1  P 3) in the drawing

f   P 2 ; y = const

f 1  0x sign

f 3  0z frontal

f 2 = f – frontal property

?
= 0

 = (f 2  P 1) designate

 = (f 2  P 3) in the drawing

Rice. 13. Front

R   P 3 ; x = const

R 1  0at sign

R 2  0z profile straight

R 3 =  R – profile property

straight
 = 0


?
= (R 3  P 1) designate

 = (R 3  P 2) in the drawing

Rice. 14. Profile straight

A P 1

A 2  0X sign

A 3  0at

?
=

b P 2

b 1  0X sign

b 3  0z

?
=

c P 3

c 1  0at sign

With 2  0z

?
=

3 BELONGING OF A STRAIGHT POINT
T theorem: If a point in space belongs to a line, then on the diagram the projections of this point are on the same projections of the line (Fig. 18):

M  AB,

E AB.
Fair converse theorem :

M 1  A 1 B 1 ;

M 2  A 2 B 2  M AB.

4 TRACES DIRECT
WITH
?
ice – this is the point intersected by a straight line with the projection plane (Fig. 19). Since the trace belongs to one of the projection planes, one of its coordinates must be equal to zero.

mark on H = k ∩ P 1 – horizontal trace

drawing (Fig. 19) F = k ∩ P 2 – frontal trace

?
P =k ∩ P 3 – profile trace

Rule for constructing traces:

To construct a horizontal trace of a straight line..... it is necessary to carry out a frontal projection..... straight line..... continue until it intersects with the axis X, then from the point of intersection with the axis X restore a perpendicular to it, and continue the horizontal..... projection of the straight line...... until it intersects with this perpendicular.

The frontal trace is constructed in a similar way.

5 DIVISION OF A LINE SEGMENT IN A GIVEN RELATIONSHIP
From the properties of parallel projection it is known that if a point divides a line segment in a given ratio, then the projections of this point divide the same projections of the line in the same ratio.

Therefore, in order to divide a certain segment on a diagram in a given ratio, it is necessary to divide its projections in the same ratio.

Knowing this condition, you can determine whether a point belongs to TO straight AB : A 2 TO 2 : TO 2 IN 2 ¹ A 1 TO 1 : TO 1 IN 1 Þ TO Ï AB

Example: To split a line AB in a 2:3 ratio from a point A 1 let's draw an arbitrary segment A 1 IN 0 1 divided into five equal parts (Fig. 20): A 1 K 0 1 = 2 parts, K 0 1 B 0 1 = 3 parts, A 1 TO 0 1 :TO 0 1 IN 0 1 =2: 3

Connect the dot IN 0 1 with dot IN 1 and drawing from the point TO 0 1 straight parallel ( IN 1 IN 0 1) we obtain the projection of the point TO 1 . According to Thales' theorem (If equal segments are laid out on one side of an angle and parallel lines are drawn through their ends, intersecting the other side, then equal segments are laid down on the other side) A 1 TO 1: TO 1 IN 1 = = 2: 3, then we find TO 2. Thus the projections of the point TO divide the same projections of a segment AB in this regard, hence the point TO divides a segment AB in a ratio of 2:3.

6 DETERMINING THE LENGTH OF A STRAIGHT SEGMENT AND ANGLES

TILTING STRAIGHT TO PROJECTION PLANES
Length of the segment AB can be determined from a right triangle ABC ,where A WITH  = A 1 B 1 ,  CB  = DZ, corner a- angle of inclination of the segment to the plane P 1 . To do this, on the diagram (Fig. 21) from the point B 1 draw a segment at an angle of 90  B 1 B 1 0 = DZ, the resulting segment A 1 B 1 0 and will be the natural value of the segment AB , and the angle B 1 A 1 B 1 0 = α . The considered method is called the method right triangle . However, all constructions can be explained as the rotation of a triangle ABC around the side AC until it becomes parallel to the plane P 1 , in this case the triangle is projected onto the projection plane without distortion. For determining b- the angle of inclination of the segment to the plane P 2 the constructions are similar (Fig. 22). Only in a triangle ABC side  Sun  = DU and the triangle is aligned with the plane P 2 .

? Designate the projections of the line and

determine the angle α.

Designate the projections of the line and

determine the angle α.

Designate the projections of the line and

determine the angle β.

7 MUTUAL POSITION OF STRAIGHTS
Lines in space can intersect, cross and be parallel.

1. Intersecting lines - these are lines that lie in the same plane and have a common point (a ∩ b = K).

Theorem: If straight lines intersect in space, then their projections of the same name intersect in the drawing (Fig. 23).

T the point of intersection of projections of the same name is located on the same perpendicular to the axis X (TO 1 TO 2  O X).

TO = a ∩ b  TO  a; TO  b  TO 1 = a 1 ∩ b 1 ;

TO 2 = a 2 ∩ b 2 .
The converse theorem is also true:

If TO 1  A 1 ; TO 2  b 2, then

TO 1 = A 1 ∩ b 1 ;

TO 2 = A 2 ∩ b 2  TO = A ∩ b.
2. Crossing lines - these are straight lines that do not lie in the same plane and do not have a common point (Fig. 24).

Pairs of points 1 And 2 , lying on the horizontally projecting line are called horizontally competing, and points 3 And 4 – frontally competitive. Visibility on the diagram is determined from them.

P about horizontally competing points 1 And 2 Visibility relative to P 1 is determined. Dot 1 closer to the observer's eye, it will be visible on the P 1 plane. Since point 1 m, then straight m will be higher than the straight line n.

Which line will be visible in relation to the plane P 2 ?
3. Parallel lines - these are lines that lie in the same plane and have an improper common point.

Theorem:

E If the lines are parallel in space, then their projections of the same name are parallel in the drawing (Fig. 25).

If k  m  k 1 m 1 , k 2 m 2 , k 3 m 3
The converse theorem is true:

If k 1 m 1 ; k 2 m 2  k  m
Lecture No. 3
PLANE

1. Methods for defining a plane in a drawing. Traces of a plane. 2. Position of the plane relative to the projection planes. 3. Belonging of a point and a straight plane. 4. Main (special) lines of the plane.
1 WAYS TO SET THE PLANE IN THE DRAWING.

TRACE PLANE

Plane- an infinite ruled surface in all directions, which throughout its entire length has no curvature or refraction.

The plane in the drawing can be specified:

Three points that do not lie on the same line - P (A, B, C) , rice. 26.
A straight line and a point not lying on this line – P (m, A; A m) , rice. 27.
Rice. 29 Fig. thirty
Specifying a plane using traces
Trace plane – line of intersection of the plane with the projection plane (Fig. 31).
Horizontal track is obtained by the intersection of the plane P with the horizontal plane of projections (P P1 = P ∩ P 1).
P P2 = P ∩ P 2 – frontal trace ;
R P3 = P ∩ P 3 – profile trace ;
R x, R y, R z – vanishing points .

10.1 Dihedral angle. Angle between planes

Two intersecting lines form two pairs of vertical angles. Just as two intersecting lines on a plane form a pair of vertical angles (Fig. 89, a), so two intersecting planes in space form two pairs of vertical dihedral angles (Fig. 89, b).

Rice. 89

A dihedral angle is a figure that consists of two half-planes that have a common boundary straight line and do not lie in the same plane (Fig. 90). The half-planes themselves are called the faces of a dihedral angle, and their common boundary straight line is called its edge.

Rice. 90

Dihedral angles are measured as follows.

Let us take point O on edge p of a dihedral angle with faces α and β. Draw rays a and b from point O at its faces, perpendicular to edge p: a - in face α and b - in face β (Fig. 91, a).

Rice. 91

An angle with sides a, b is called a linear dihedral angle.

The magnitude of the linear angle does not depend on the choice of its vertex on the edge of the dihedral angle.

Indeed, let’s take another point O 1 of the edge p and draw the rays a 1 ⊥ p and b 1 ⊥ p in the faces α and β (Fig. 91, b).

Let us plot on ray a the segment OA, on ray a 1 the segment O 1 A 1, equal to the segment OA, on ray b the segment OB, and on ray b 1 the segment O 1 B 1, equal to the segment OB (Fig. 91, c).

In rectangles OAA 1 O 1 and 0BB 1 0 1, the sides AA 1 and BB 1 are equal to their common side OO 1 and parallel to it. Therefore AA 1 = BB 1 and AA 1 || BB 1.

Consequently, the quadrilateral ABV 1 A 1 is a parallelogram (Fig. 91, d), which means AB = A 1 B 1. Therefore, triangles ABO and A 1 B 1 O 1 are equal (on three sides) and angle ab is equal to angle a 1 b 1.

Now we can give the following definition: the magnitude of a dihedral angle is the magnitude of its linear angle.

The angle between intersecting planes is the size of the smaller of the dihedral angles formed by them. If this angle is 90°, then the planes are called mutually perpendicular. The angle between parallel planes is assumed to be 0°.

The angle between the planes α and β, as well as the value of the dihedral angle with faces α and β, is denoted ∠αβ.

The angle between the faces of a polyhedron that have a common edge is the value of the dihedral angle corresponding to these faces.

10.2 Properties of mutually perpendicular planes

Property 1. A straight line lying in one of two mutually perpendicular planes and perpendicular to their common straight line is perpendicular to the other plane.

Proof. Let the planes α and β be mutually perpendicular and intersect along a straight line c. Let straight line a lie in the plane α and a ⊥ с (Fig. 92). Line a intersects c at some point O. Let us draw a line b in the plane β through point O, perpendicular to line c. Since α ⊥ β, then a ⊥ b. Since a ⊥ b and a ⊥ c, then α ⊥ β based on the perpendicularity of the line and the plane.

Rice. 92

The second property is the converse of the first property.

Property 2. A straight line that has a common point with one of two mutually perpendicular planes and is perpendicular to the other plane lies in the first of them.

Proof. Let the planes α and β be mutually perpendicular and intersect along a straight line c, the straight line a ⊥ β and a have a common point A with a (Fig. 93). Through point A we draw a straight line p in the plane α, perpendicular to the straight line c. According to property 1 p ⊥ β. Lines a and p pass through point A and are perpendicular to the plane β. Therefore, they coincide, since only one straight line passes through a point, perpendicular to a certain plane. Since the straight line p lies in the α plane, then the straight line a lies in the α plane.

Rice. 93

A consequence of property 2 is the following sign of perpendicularity of a line and a plane: if two planes perpendicular to a third plane intersect, then the line of their intersection is perpendicular to the third plane.

Proof. Let two planes α and β, intersecting along a straight line a, be perpendicular to the plane γ (Fig. 94). Then through any point of line a we draw a line perpendicular to the plane γ. According to property 2, this line lies both in the plane α and in the plane β, i.e., it coincides with line a. So, a ⊥ γ.

Rice. 94

10.3 Sign of perpendicularity of planes

Let's start with practical examples. The plane of a door hung on a jamb perpendicular to the floor is perpendicular to the plane of the floor in any position of the door (Fig. 95). When they want to check whether a flat surface (wall, fence, etc.) is installed vertically, they do this using a plumb line - a rope with a load. The plumb line is always directed vertically, and the wall stands vertically if the plumb line, located along it, does not deviate. These examples tell us the following simple sign of the perpendicularity of planes: if a plane passes through a perpendicular to another plane, then these planes are mutually perpendicular.

Rice. 95

Proof. Let the plane α contain a line a perpendicular to the plane β (see Fig. 92). Then straight line a intersects plane β at some point O. Point O lies on line c along which planes α and β intersect. Let us draw a line b in the β plane through point O, perpendicular to line c. Since a ⊥ β, then a ⊥ b and a ⊥ c. This means that the linear angles of the dihedral angles formed by intersecting planes α and β are straight. Therefore, planes α and β are mutually perpendicular.

Note that each two of the three straight lines a, b and c, considered now (see Fig. 92), are mutually perpendicular. If we build another line passing through point O and perpendicular to two of these three lines, then it will coincide with the third line. This fact speaks about the three-dimensionality of the space around us: there is no fourth line perpendicular to each of the lines a, b and c.

Questions for self-control

How is the dihedral angle calculated?
How to calculate the angle between planes?
What planes are called mutually perpendicular?
What properties of mutually perpendicular planes do you know?
What sign of perpendicularity of planes do you know?

There are many parts whose shape information cannot be conveyed by two drawing projections. In order for information about the complex shape of a part to be presented sufficiently fully, projection is used on three mutually perpendicular projection planes: frontal - V, horizontal – H and profile - W .

The system of projection planes is a trihedral angle with its vertex at the point ABOUT. The intersections of the planes of a trihedral angle form straight lines - the axes of projections ( OX, OY, OZ) (Fig. 23).

An object is placed in a trihedral corner so that its formative edge and base are parallel to the frontal and horizontal projection planes, respectively. Then, projection rays are passed through all points of the object, perpendicular to all three projection planes, on which frontal, horizontal and profile projections of the object are obtained. After projection, the object is removed from the trihedral angle, and then the horizontal and profile projection planes are rotated by 90°, respectively, around the axes OH And OZ until aligned with the frontal projection plane and a drawing of the part containing three projections is obtained.

Rice. 23. Projection onto three mutually perpendicular

projection planes

The three projections of the drawing are interconnected with each other. Frontal and horizontal projections preserve the projection connection of images, i.e. projection connections are established between frontal and horizontal, frontal and profile, as well as horizontal and profile projections (see Fig. 23). Projection lines define the location of each projection on the drawing field.

In many countries of the world, another system of rectangular projection onto three mutually perpendicular projection planes has been adopted, which is conventionally called “American”. Its main difference is that the trihedral angle is located in space differently, relative to the projected object, and the planes unfold in other directions projections. Therefore, the horizontal projection appears above the frontal one, and the profile projection appears to the right of the frontal one.

The shape of most objects is a combination of various geometric bodies or their parts. Therefore, to read and complete drawings, you need to know how geometric bodies are depicted in a system of three projections.

Concept of view

You know that frontal, horizontal and profile projections are images of a projection drawing. Projection images of the external visible surface of an object are called views.

View- This is an image of the visible surface of an object facing the observer.

Main types. The standard establishes six main views that are obtained when projecting an object placed inside a cube, the six faces of which are taken as projection planes (Fig. 24). Having projected an object onto these faces, they are turned until they are aligned with the frontal plane of projections (Fig. 25).

Rice. 24. Getting Basic Views

Front view(main view) is placed at the site of the frontal projection. View from above placed on the horizontal projection site (under the main view). Left view located at the site of the profile projection (to the right of the main view). View on right located to the left of the main view. The bottom view is above the main view. The rear view is placed to the right of the left view.

Rice. 25. Main types

The main views, as well as the projections, are located in a projection relationship. The number of views in the drawing is chosen to be minimal, but sufficient to accurately represent the shape of the depicted object. In views, if necessary, it is allowed to show invisible parts of the surface of an object using dashed lines (Fig. 26).

The main view should contain the most information about the item. Therefore, the part must be positioned in relation to the frontal plane of projections so that its visible surface can be projected with the greatest number of form elements. In addition, the main view should give a clear idea of the features of the form, showing its silhouette, surface curves, ledges, recesses, holes, which ensures quick recognition of the shape of the depicted product.