The method of replacing planes is to find the actual size of the segment. Methods for converting a complex drawing. Determining the distance between parallel planes

L E C T I O N 10

METHOD FOR REPLACING PROJECTION PLANES

1. The essence of the method of replacing planes

2. Applying the method of replacing planes to a line segment

3. Applying the method of replacing planes to a flat figure

1. The essence of the method of replacing planes

This method consists in the fact that the given system of projection planes is replaced by a new system so that the object (straight line or plane), without changing its position in space, is in a particular position relative to the new system of projection planes. Projection planes form a new orthogonal system.

Depending on the conditions of the problem, it is necessary to replace either one of the given projection planes, or both, if by replacing one projection plane it is not possible to obtain the necessary location of the projected object relative to the projection plane.

Take in the system of projection planes H and V arbitrary point BUT and construct its rectangular projections a and a"(Fig. 60). Let us replace the frontal plane V with a new plane V 1 , perpendicular to the plane H, i.e. from the system of planescolor:black"> let's go to the system with new j-axis x 1 . Projection

making a point BUT to the planeV 1get a new projection a1". Horizontal projection a points BUT belongs to both systems of projection planes. It is clear from the constructions thata1"aXi= Aa =a"ax =zA ,i.e. when changing the plane V planeV1,perpendicular to the plane H, the coordinate of the projected point remains unchanged.

To get a drawing, we combine all three planes - H, V to V 1- in one plane (Fig. 60). In the new projection systema and a"are on a projection bond line perpendicular to the new axisx 1 .At the same time, the distanceaXia 1 "=axa"=zA .

Replacing the horizontal projection plane H new planeH1 , perpendicular to the plane V, from the system of projection planesfont-size:14.0pt;color:black"> are moving to the new system(Fig. 61).

By constructing the projections of the point BUT in both systems, we notice that the coordinate at remains unchanged. In the drawing, the segmentoxla 1 =axa =yA ,which allows us to build a new projection a1 given point BUT on a perpendicular from a" to the new axisx 1.

Sequential replacement of two projection planes is shown in fig. 62. Plane first V replaced by a planeV1 perpendicular to the planeH, and a new projection is constructed a1 points BUT. Then the plane H replaced by a plane H1 perpendicular to the planeV1 , and built a new projection a1. Thus, a successive transition from the system of projection planes to the system, and then to the system, is completed.

position:relative; z-index:-10">

In the system of planes, projections of a point BUT will a( and a1", the sequential construction of which is determined by the invariance of the coordinatez in the system of planes and coordinatesy1 in the system of planes

We will consider the solution of problems by this method using two examples.

2. Application of the plane replacement method

to a straight line

Example 1. Determine the length of the segment AB straight on its projectionsaband a "b"(Fig. 63).

The problem is solved by replacing one of the given projection planes with a new projection plane parallel to the segment AB. The segment is projected onto the new plane in its true value.

When changing the plane V planeV1,parallel to the segment AB, new axle o x1 carried out parallel to the horizontal projectionab(rice .63 a ). Dropping out of points a andbperpendiculars to the o-axis x1 and put aside for themaXla 1 "=axa" and bXib 1" =bxb",get a new projection a1" b "1,equal to the segment AB, and also the angle en, equal to the angle of inclination of a straight line to a plane N.

On fig. 63 b the solution of the same problem is given by replacing the plane H plane H1, parallel to the segment AB. In this case, the axis x1 positioned parallel to the frontal projectiona"b"and similarly to the previous one, we obtain the projection a1 b 1equal to a given segment, and the angle α v , at the angle of inclination of the straight line to the plane V.

3. Application of the plane replacement method

to a flat shape

Example 2. Determine the size and shape of a triangle ABC according to his projectionsabc and a" b"With"(Fig. 64).

The triangle is projected without distortion onto a projection plane parallel to it. In the general case, this cannot be achieved by one replacement of the projection planes, therefore, two projection planes are successively replaced.

First replace the plane V planeV 1perpendicular to the plane of the triangle. To do this, draw a horizontal line in the plane of the triangleADand a plane is placed perpendicular to itV1.In the drawing, the construction is reduced to drawing the axis x1, perpendicular planadhorizontal. HorizontalADprojected onto a planeV 1 exactly a 1" ≡ d1, and the triangle - into a segmentb 1c 1 .

Then change the plane H plane H1 parallel to the plane of the triangleABC.Axis o x2 will be parallel to the projectionb 1 "a1" c1", and the projection b 1 а1с1will display the true size of the triangle.

The essence of the method for replacing projection planes is that the given system of projection planes is replaced by a new system so that the geometric figures are in a particular position relative to the new system of projection planes.

Let's see how the projections of the point change B if the plane V replace with a new projection plane V 1(Fig. 5.1, a). Plane V 1 draw perpendicular to the plane H, whose position remains unchanged. planes H and V 1 intersect in a straight line 0x 1, defining a new projection axis. In the new system of projection planes, instead of projections b and b" get new projections b and b 1 ′. It is easy to verify that the distance from the new point projection b 1 ′ to new axle 0x 1(coordinate Z) is equal to the distance from the replaced projection b" to the axle to be replaced 0x. To go from a spatial drawing to a diagram, you need to combine the plane V 1 with plane H. On the diagram (Fig. 5.1, 6 ) to build a new projection b 1 ′ we use the invariance of the coordinate Z points B. For this, it is enough from the horizontal projection b draw a perpendicular to the new axis 0x 1 and from the point b X 1 set aside a coordinate Z, determined by the distance b "b x (Z B) in the old system.

Replacing the horizontal plane H new plane H 1(Fig. 5.1, in) is performed similarly, with the only difference that now the frontal projection of the point does not change b", to build a new horizontal projection b 1 required from stored frontal projection b" draw a communication line to a new axis 0x 1 and postpone from the new axis a distance equal to the distance from the replaced projection b to the axle to be replaced 0x.

Projection planes can only be replaced sequentially; both planes cannot be changed at once.

Let's look at examples of how the projection planes are replaced and new projections of figures are built.

Task 1. Determine the length of a line segment AB general position.

Replacing the plane V plane V1, parallel to the segment AB(Fig. 5.2, a). Making a new axle X 1 parallel ab and on the perpendiculars drawn to it from the points a and b, postpone a X 1 a 1 ′ = a x a " and b X 1 b 1 ′ = b x b". Getting a new projection a 1 ′ b 1 ′ = AB and at the same time the angle α inclination of a straight line to a plane N.

If the plane H plane H1 parallel to the segment AB(Fig. 5.2, b), then we get a 1 b 1 \u003d AB and angle β inclination of a straight line to a plane v.

Task 2. Determine the actual size and shape of the triangle ABC.

The problem is solved by sequential replacement of two projection planes.

Plane first V replace with a plane V 1, perpendicular to the plane of the triangle (Fig. 5.3). To do this, draw a horizontal line in the plane of the triangle AD (ad, a"d") and new axle X 1 positioned perpendicular to ad. On the new projection plane, the triangle is projected into a straight line b 1 ′a 1 ′c 1 . At the second stage, the plane H replace with a plane H 1, parallel to the plane of the triangle, positioning the axis X 2 parallel to a straight line b 1 ′a 1 ′c 1 ′. Built projection a 1 b 1 c 1 determines the actual size and shape of a triangle ABC.

The essence of the method is that the position of the depicted figure in space remains unchanged, and the original system of projection planes, relative to which the figure is set, is replaced by a new one.

When choosing a new projection plane, the basic principle of orthogonal projection (Monge's method) must be fulfilled - the mutual perpendicularity of the projection planes, i.e. a new projection plane must be placed perpendicular to one of the main initial projection planes.

Let the system of projection planes be given P 1 and P 2(hereinafter we will abbreviate ). Projecting any point BUT onto these planes and find its projections A 2 and A 1(Fig. 9.5).

Suppose that in solving some problem we found it expedient to replace the plane P 2 other frontal plane P 4, perpendicular to the plane P 1. Line of intersection of projection planes P 1 and P 4 is called the new projection axis and is denoted X 1 . Let's construct orthogonal projections of a point BUT in system . Since the plane P 1 remained the same, then the projection of the point BUT to this plane will not change its position.

To get a new frontal projection of a point onto a new plane P 4 drop the perpendicular from BUT to the plane P 4. Base A 4 this perpendicular determines the desired frontal projection of the point BUT.

Let's establish what connection exists between the projections A (A 1, A 2) and A (A 1 A 4) the same point in both systems.

They have a common horizontal projection, and since the distance of the point BUT from the plane P 1 has not changed, then /AA 1 /=/A 2 A x /=/A 4 A x1 ¹ /, i.e., the distance of the new frontal projection to the new axis is equal to the distance of the replaced projection to the previous axis.

To go to the plot, rotate the plane P 4 around the axis X 1 and compatible with the plane P 1. Then the new frontal projection A 4 aligned with the plane P 1 and at the same time will be on one perpendicular to the axis x 1 with projection A 1.

On fig. 9.6 shows those constructions that need to be made on the diagram, so that from projections (A 1, A 2) points BUT in the system go to projections A 1 A 4) the same point in the system , it is necessary: ​​to draw a new axis of projections X 1, which determines the position of the horizontally projecting plane P 4, then from the horizontal projection of the point A 1 X 1. On the constructed perpendicular, set aside (from the new axis) the segment A x A 4 \u003d A x A 2. The point thus obtained A 4 is the projection of the point BUT to the plane P 4.

Replacing the horizontal plane P 1 new plane P 4 and construction of new point projections BUT in the system is carried out similarly to the considered case, with the only difference that now the frontal projection of the point remains unchanged, and to find a new horizontal projection A 4 points BUT necessary from the frontal projection of the point A 2 lower the perpendicular to the new axis X 1 and set aside on it from the point of intersection with the axis X 1 line segment A 4 A x ¹, equal to the distance of the old horizontal projection from the old axis A 1 A x(Fig. 9.7).

The considered examples allow us to establish the following general rule: in order to construct a projection of a point in a new system of projection planes, it is necessary to lower the perpendicular from the constant projection of the point to the new axis of projections and set aside on it from the new axis to the new projection a distance equal to the distance from the replaced projection to previous axis.

The purpose of the drawing transformation methods is to place a general geometric figure in a particular position relative to the projection planes in order to use the properties of its projections. For example, the transformation of a plane of general position into a level plane will make it possible to determine its actual size from the corresponding projection.

Methods for converting a complex drawing are divided into two groups according to the feature that determines the position of the figure and projection planes relative to each other or the direction of projection:

1. Change the position of the projection planes or the direction of projection so that the figure fixed in space is in a particular position. This group includes:

method of replacing projection planes;

additional projection method.

2. Change the position of the geometric figure in space so that it is in a particular position relative to a fixed system of projection planes. This group includes:

method of plane-parallel movement;

rotation method.

The tasks solved using the methods of converting a complex drawing are reduced to the following main tasks in which it is necessary to convert:

a straight line (plane, cylindrical or prismatic surface) into a projecting figure;

a straight line (flat line or plane) into a level figure.

We will consider successively all methods of transformation, with the exception of the method of additional projection, which it is recommended to familiarize yourself with in the textbook.

How to replace projection planes

The essence of the method consists in replacing the original system of mutually perpendicular projection planes with a new system of mutually perpendicular projection planes with the position of the geometric figure unchanged in space.

To solve a specific problem, one or two successive transformations are performed by the replacement method, for example, Π 1 Π 2 →Π 1 Π 4 or Π 1 Π 2 →Π 1 Π 4 → Π 5 Π 4 . In the second case, the transformation is called the composition of transformations. At each step in this method, only one projection plane is replaced, while the other remains common to the two systems.

Consider the mechanism and features of the method for replacing projection planes using the example of transforming a complex drawing of a point (Fig. 28).

When replacing, for example, the frontal projection plane Π 2 new vertical plane Π 4 horizontal plane Π 1 in this case is common for two systems of projection planes, as a result of which the projection BUT 1 points BUT to this plane is also common to these systems. At the same time, the value of the distance remains unchanged ( AA 1 ) from a given point to this plane of projections and, as a consequence, the equality of its projections on the plane Π 2 and Π 4 , i.e. AA 1 =BUT 2 BUT 12 =BUT 4 BUT 14 , which allows you to build a new projection on a complex drawing BUT 4 given point (see Fig. 28).

Another feature of the method for replacing projection planes is that a complex drawing is formed by combining the projection planes with the plane that is common to the two systems. In the considered in Fig. 28, such a plane is the horizontal projection plane.

As an example, consider the problem of transforming a line in general position into a projecting line. To achieve the final result, it is necessary to replace two projection planes using a composition of transformations, i.e. two successive transformations (Fig. 29).

Replacing one projection plane, for example, Π 2 on the Π 4 allows you to convert a line of general position only to a line of level, since it is impossible to immediately position a new vertical projection plane Π 4 perpendicular to a given line. Further, successively replacing the second projection plane Π 1 on the Π 5 and placing it perpendicular to the line AB, we get the final result (see Fig. 29).

GENERAL PROVISIONS

METHODS FOR CONVERTING A COMPLEX DRAWING

Lecture 4

The solution of a number of problems in descriptive geometry is greatly simplified when geometric figures occupy a particular position relative to the projection planes. Tasks for determining the relative position of figures and metric problems (determining the natural values ​​of planes, segments, etc.). To do this, there are various ways to convert a multidrawing. Each of them is based on one of the following principles:

1. on a change in the position of projection planes relative to fixed geometric shapes;

2. on changing the position of given geometric figures relative to fixed projection planes;

Let's consider some of them.

The essence of the method is that the given geometric figures are fixed in a given system of projection planes ( P 1 , P 2). New projection planes ( P 4, P 5), with respect to which geometric figures will occupy a particular position. The new projection plane is chosen so that it is perpendicular to the non-replaceable projection plane.

Most problems are solved using one or two successive transformations of the original system of projection planes. Only one projection plane can be replaced at a time P 1(or P 2), another plane P 2(or P 1) should remain unchanged.
Figure 1 shows a visual representation of the method for replacing projection planes. Frontal plane P 2 replaced with a new frontal plane P 4. New point projections BUT (A 1 A 4), while, as can be seen from the figure, the height of point A remained the same.

It is necessary to remember the rule for constructing new projections of points with the replacement method:

1. communication lines are always perpendicular to new projection axes;
2. the distance from the new projection axis to the new point projection is always taken from the plane being replaced.

Figure 1. Visual representation of the method of replacing projection planes.

Figure 2. Image of the method of replacing projection planes on the diagram.

Most problems in descriptive geometry are solved on the basis of four problems:

1. Convert a general position line to a level line;
2. Transform a general position line into a projecting line;
3. Convert a generic plane to a projection plane;
4. Convert Generic Plane to Level Plane.

Task #1

Consider the solution tasks number 1 . Dana straight AB– in general position, we transform it into a level line (Fig. 3). To do this, we introduce a new frontal projection plane P 4, axis X 1.4 run in parallel A 1 B 1 AB – A 4 B 4. In the new system of projection planes, the straight line AB- frontal.

Figure 3

Converting a general position line to a level line (frontal)

Task #2

Dana straight AB- in general position, we transform it into a projecting line (Fig. 4). To solve this problem, it is necessary to perform two transformations in succession:

1. Transform the line of general position into the level line, that is, first solve problem No. 1;
2. Convert the level line to a projecting line.

Draw the condition of problem No. 1, solve it yourself, then proceed to perform the second transformation. Introducing a new horizontal projection plane P 5 X 4, 5 perpendicular to the projection A 4 B 4 and build a new projection of the line A 5 B 5. In the system of planes P 4, P 5, straight AB is a horizontally projecting line.

On the basis of tasks No. 1 and No. 2, the following tasks are solved:

1. determination of the distance from a point to a straight line;

2. determination of the distance between parallel and intersecting lines;

3. determination of the natural size of a straight line;

4. determination of the value of the dihedral angle.

Figure 4

Transformation of a line in general position into a projecting line.

Task number 3.

Given a plane ABC- in general position, we transform it into a projecting plane (Fig. 5). To solve this problem, it is necessary to draw a level line in the plane, if there is none. Draw a new projection axis perpendicular to the level line. In a triangle ABC draw a horizontal line h. Projection axis X 14 draw perpendicular h1, new plane projection A 4 B 4 C 4, we build according to the rules discussed in the previous tasks.

In the system of projection planes P 1, P 4, the plane of the triangle is the front-projecting plane.

Figure 5

Transformation of a generic plane into a projecting plane.

Task number 4.

Figure 6

Transformation of a generic plane into a level plane.

Given a plane ABC- general position, we transform it into a level plane (Fig. 6). To solve this problem, it is necessary to perform two transformations in succession:

1. Transform the plane of general position into a projecting plane, that is, first solve problem No. 3;
2. Convert projection plane to level plane.

Draw the condition of problem No. 3, solve it yourself, then proceed to perform the second transformation. Introducing a new horizontal projection plane P 5, for this we draw a new axis of projections X 4, 5 parallel to the projection A 4 B 4 C 4 and build a new projection of the triangle A 5 B 5 C 5. In the system of planes P 4, P 5, triangle ABC is the horizontal level plane.

On the basis of tasks No. 3 and No. 4, the following tasks are solved:

1. determination of the distance from a point to a plane;

2. determination of the distance between parallel planes;

3. determination of natural (true) values ​​of geometric shapes;

determination of the angles of inclination of the plane to the planes of projections

Plane-parallel movement method

All of the above problems can be solved using the method of plane-parallel displacement, in which the projection planes remain in place, and the projection of the figure moves (Fig. 7).

Figure 7. Determination of the natural size of the segment by the method of plane-parallel displacement.

Dana straight AB– in general position, we transform it into a level line (Fig. 7). To do this, move the projection A 1 B 1 parallel to axis X. We build a new projection of a straight line AB – A 2 ` B 2 ` , which will be - the natural size of the segment. This method is used to determine the natural values ​​of the edges of polyhedra when constructing a development.

Rotation method

A particular case of plane-parallel movement is the method of rotation around projecting lines and level lines.