Let's project a point A orthogonally on both projection planes:

AA 1 _|_ P 1 ;AA 1 ^P 1 =A 1 ;

AA 2 _|_ P 2 ;AA 2 ^P 2 =A 2 ;

Projection rays AA 1 and AA 2 mutually perpendicular and create a projecting plane in space AA 1 AA 2, perpendicular to both sides of the projections. This plane intersects the projection planes along lines passing through the projections of the point A.

To get a flat drawing, combine the horizontal plane of projections P 1 with the frontal plane P 2 rotating around the P 2 / P 1 axis (Fig. 61, a). Then both projections of the point will be on the same line perpendicular to the P 2 / P 1 axis. Straight A 1 A 2, connecting horizontal A 1 and frontal A 2 projection of a point is called vertical communication line.

The resulting flat drawing is called complex drawing. It is an image of an object on several combined planes. A complex drawing consisting of two orthogonal projections interconnected is called two-projection. In this drawing, the horizontal and frontal projections of the points always lie on the same vertical connection line.

Two interconnected orthogonal projections of a point uniquely determine its position relative to the projection planes. If we determine the position of the point A relative to these planes (Fig. 61, b) its height h (AA 1 =h) and depth f(AA 2 =f ), then these quantities in a complex drawing exist as segments of a vertical communication line. This circumstance makes it easy to reconstruct the drawing, that is, to determine from the drawing the position of the point relative to the projection planes. To do this, it is enough to restore a perpendicular to the drawing plane (considering it frontal) at point A 2 of the drawing with a length equal to the depth f. The end of this perpendicular will determine the position of the point A relative to the drawing plane.

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7. Self-test questions

SELF-TEST QUESTIONS

4. What is the name of the distance that determines the position of a point relative to the projection plane? P 1, P 2?

7. How to construct an additional projection of a point on a plane P 4 _|_ P 2 , P 4 _|_ P 1 , P 5 _|_ P 4 ?

9. How can you construct a complex drawing of a point using its coordinates?

33. Elements of a three-projection complex drawing of a point

§ 33. Elements of a three-projection complex drawing of a point

To determine the position of a geometric body in space and obtain additional information on their images, it may be necessary to construct a third projection. Then the third projection plane is located to the right of the observer, perpendicular to the horizontal projection plane at the same time P 1 and the frontal plane of projections P 2 (Fig. 62, a). As a result of the intersection of the frontal P 2 and profile P 3 projection planes we obtain a new axis P 2 / P 3 , which is located on the complex drawing parallel to the vertical communication line A 1 A 2(Fig. 62, b). Third point projection A- profile - appears to be associated with the frontal projection A 2 a new communication line called horizontal

Rice. 62

Noah. Frontal and profile projections of points always lie on the same horizontal connection line. Moreover A 1 A 2 _|_ A 2 A 1 And A 2 A 3 , _| _ P 2 / P 3 .

The position of a point in space in this case is characterized by its latitude- the distance from it to the profile plane of projections P 3, which we denote by the letter r.

The resulting complex drawing of a point is called three-projection.

In a three-projection drawing, the depth of a point AA 2 is projected without distortion on the planes P 1 and P 2 (Fig. 62, A). This circumstance allows us to construct a third - frontal projection of the point A along its horizontal A 1 and frontal A 2 projections (Fig. 62, V). To do this, you need to draw a horizontal communication line through the frontal projection of the point A 2 A 3 _|_A 2 A 1 . Then, anywhere in the drawing, draw the projection axis P 2 / P 3 _|_ A 2 A 3, measure the depth f of a point on the horizontal projection field and place it along the horizontal connection line from the projection axis P 2 / P 3. Let's get a profile projection A 3 points A.

Thus, in a complex drawing consisting of three orthogonal projections of a point, two projections are on the same connection line; communication lines are perpendicular to the corresponding projection axes; two projections of a point completely determine the position of its third projection.

It should be noted that in complex drawings, as a rule, the projection planes are not limited and their position is specified by axes (Fig. 62, c). In cases where the conditions of the problem do not require this,

It turns out that projections of points can be given without depicting axes (Fig. 63, a, b). Such a system is called baseless. Communication lines can also be drawn with a break (Fig. 63, b).

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34. Position of a point in three-dimensional angle space

§ 34. Position of a point in the space of a three-dimensional angle

The location of the projections of points in a complex drawing depends on the position of the point in the space of a three-dimensional angle. Let's look at some cases:

• the point is located in space (see Fig. 62). In this case it has depth, height and breadth;
• the point is located on the projection plane P 1- it has no height, P 2 - has no depth, Pz - has no breadth;
• the point is located on the axis of projections, P 2 / P 1 does not have depth and height, P 2 / P 3 does not have depth and latitude, and P 1 / P 3 does not have height and latitude.

35. Competing points

§ 35. Competing points

Two points in space can be located in different ways. In a separate case, they can be located so that their projections on some projection plane coincide. Such points are called competing. In Fig. 64, A a comprehensive drawing of the points is provided A And IN. They are located so that their projections coincide on the plane P 1 [A 1 == B 1]. Such points are called horizontally competing. If the projections of the points A and B coincide on the plane

P 2(Fig. 64, b), they are called frontally competing. And if the projections of the points A And IN coincide on the plane P 3 [A 3 == B 3 ] (Fig. 64, c), they are called profile competitors.

Visibility in the drawing is determined by competing points. For horizontally competing points, the one with greater height will be visible, for frontally competing points, the one with greater depth will be visible, and for profile competing points, the one with greater latitude will be visible.

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36. Replacing projection planes

§ 36. Replacement of projection planes

The properties of a three-projection drawing of a point allow using its horizontal and frontal projections to construct a third onto other projection planes entered to replace the given ones.

In Fig. 65, A showing point A and its projection is horizontal A 1 and frontal A 2. According to the conditions of the problem, it is necessary to replace the P 2 planes. Let's denote the new projection plane P 4 and place it perpendicular to P 1. At the intersection of planes P 1 and P 4 we get a new axis P 1 / P 4 . New point projection A 4 will be located on communication line passing through a point A 1 and perpendicular to the P 1 / P 4 axis .

Since the new plane P 4 replaces the frontal projection plane P 2, point height A is depicted equally in full size both on the P2 plane and on the P4 plane.

This circumstance allows us to determine the position of the projection A 4, in a system of planes P 1 _|_ P 4(Fig. 65, b) on a complex drawing. To do this, it is enough to measure the height of the point on the plane being replaced.

ity of the projection P 2, put it on a new connection line from the new axis of projections - and a new projection of the point A 4 will be built.

If a new projection plane is introduced instead of the horizontal projection plane, i.e. P 4 _|_ P 2 (Fig. 66, A), then in the new system of planes the new projection of the point will be on the same line of communication with the frontal projection, and A 2 A 4 _|_. In this case, the depth of the point is the same on the plane P 1, and on the plane P 4. On this basis they build A 4(Fig. 66, b) on the communication line A 2 A 4 at such a distance from the new axis P 1 / P 4 at what A 1 located from the P 2 / P 1 axis.

As already noted, the construction of new additional projections is always associated with specific tasks. In the future, a number of metric and positional problems will be considered that can be solved using the method of replacing projection planes. In problems where the introduction of one additional plane will not give the desired result, another additional plane is introduced, which is designated P 5. It is placed perpendicular to the already introduced plane P 4 (Fig. 67, a), i.e. P 5 P 4 and produce a construction similar to those previously discussed. Now the distances are measured on the replaced second of the main projection planes (in Fig. 67, b on the plane P 1) and postpone them on a new communication line A 4 A 5, from the new projection axis P 5 / P 4. In the new system of planes P 4 P 5, a new two-projection drawing is obtained, consisting of orthogonal projections A 4 and A 5 , connected by communication line

With rectangular projection, the system of projection planes consists of two mutually perpendicular projection planes (Fig. 2.1). They agreed to place one horizontally and the other vertically.

The projection plane located horizontally is called horizontal projection plane and denote sch, and the plane perpendicular to it is frontal plane of projectionsl 2. The system of projection planes itself is denoted p/p 2. Usually abbreviated expressions are used: plane L[, plane n 2. Line of intersection of planes sch And to 2 called projection axisOH. It divides each projection plane into two parts - floors. The horizontal projection plane has front and rear, and the frontal plane has upper and lower floors.

Planes sch And n 2 divide the space into four parts, called in quarters and designated by Roman numerals I, II, III and IV (see Fig. 2.1). The first quarter is called the part of space limited by the upper hollow frontal and anterior hollow horizontal planes of projections. For the remaining quarters of space, the definitions are similar to the previous one.

All mechanical engineering drawings are images built on the same plane. In Fig. 2.1 the system of projection planes is spatial. To move to images on the same plane, we agreed to combine the projection planes. Usually flat n 2 left motionless, and the plane P rotate in the direction indicated by the arrows (see Fig. 2.1) around the axis OH at an angle of 90° until it aligns with the plane n 2. With this rotation, the front floor of the horizontal plane goes down, and the back goes up. After combining the planes they look like

married to fig. 2.2. It is believed that the projection planes are opaque and the observer is always in the first quarter. In Fig. 2.2 the designation of planes that are invisible after combining the floors is taken in brackets, as is customary for highlighting invisible figures in drawings.

The projected point can be located in any quarter of space or on any projection plane. In all cases, to construct projections, projection lines are drawn through it and their meeting points with planes 711 and 712, which are projections, are found.

Consider the projection of a point located in the first quarter. The system of projection planes 711/712 and the point are specified A(Fig. 2.3). Two straight LINES are drawn through it, perpendicular to PLANES 71) AND 71 2. One of them will intersect plane 711 at point A ", called horizontal projection of point A, and the other is the plane 71 2 at the point A ", called frontal projection of point A.

Projecting straight lines AA" And AA" determine the projection plane a. It is perpendicular to the planes Kip 2, since it passes through the perpendiculars to them and intersects the projection planes along straight lines A "Ah and A "Ah. Projection axis OH perpendicular to the plane os, as the line of intersection of two planes 71| and 71 2, perpendicular to the third plane (a), and therefore to any straight line lying in it. In particular, 0X1A"A x And 0X1A "Ah.

When combining planes, a segment A "Ah, flat to 2, remains motionless, and the segment A "A x together with plane 71) will be rotated around the axis OH until aligned with plane 71 2. View of combined projection planes along with point projections A shown in Fig. 2.4, A. After combining the point A", Ax and A" will be located on one straight line, perpendicular to the axis OH. This implies that two projections of the same point

lie on a common perpendicular to the projection axis. This perpendicular connecting two projections of the same point is called projection communication line.

Drawing in Fig. 2.4, A can be greatly simplified. The designations of the combined projection planes are not marked in the drawings and the rectangles that conditionally limit the projection planes are not depicted, since the planes are unlimited. Simplified point drawing A(Fig. 2.4, b) also called diagram(from the French ?pure - drawing).

Shown in Fig. 2.3 quadrilateral AE4 "A H A" is a rectangle and its opposite sides are equal and parallel. Therefore, the distance from the point A to plane P, measured by a segment AA", in the drawing is determined by the segment A "Ah. The segment A "A x = AA" allows you to judge the distance from a point A to plane to 2. Thus, the drawing of a point gives a complete picture of its location relative to the projection planes. For example, according to the drawing (see Fig. 2.4, b) it can be argued that the point A located in the first quarter and removed from the plane n 2 at a smaller distance than from the plane since A "A x A "Ah.

Let's move on to projecting a point in the second, third and fourth quarters of space.

When projecting a point IN, located in the second quarter (Fig. 2.5), after combining the planes, both of its projections will be above the axis OH.

The horizontal projection of point C, specified in the third quarter (Fig. 2.6), is located above the axis OH, and the front one is lower.

Point D shown in Fig. 2.7, located in the fourth quarter. After combining the projection planes, both of its projections will be below the axis OH.

Comparing drawings of points located in different quarters of space (see Fig. 2.4-2.7), you can notice that each is characterized by its own location of projections relative to the axis of projections OH.

In special cases, the projected point may lie on the projection plane. Then one of its projections coincides with the point itself, and the other will be located on the axis of projections. For example, for a point E, lying on a plane sch(Fig. 2.8), the horizontal projection coincides with the point itself, and the frontal one is on the axis OH. At the point E, located on a plane to 2(Fig. 2.9), horizontal projection on the axis OH, and the front one coincides with the point itself.

A point in space is defined by any two of its projections. If it is necessary to construct a third projection based on two given ones, it is necessary to use the correspondence of segments of projection communication lines obtained when determining the distances from a point to the projection plane (see Fig. 2.27 and Fig. 2.28).

Examples of solving problems in the first octant

Given A 1; A 2	Build A 3
Given A 2; A 3	Build A 1
Given A 1; A 3	Build A 2

Let's consider the algorithm for constructing point A (Table 2.5)

Table 2.5

Algorithm for constructing point A
at given coordinates A ( x = 5, y = 20, z = -9)

In the following chapters we will consider images: straight lines and planes only in the first quarter. Although all the methods considered can be applied in any quarter.

Conclusions

Thus, based on the theory of G. Monge, it is possible to transform the spatial image of an image (point) into a planar one.

This theory is based on the following provisions:

1. The entire space is divided into 4 quarters using two mutually perpendicular planes p 1 and p 2, or into 8 octants by adding a third mutually perpendicular plane p 3.

2. The image of a spatial image on these planes is obtained using a rectangular (orthogonal) projection.

3. To convert a spatial image into a planar one, it is assumed that the plane p 2 is stationary, and the plane p 1 rotates around the axis x so that the positive half-plane p 1 is combined with the negative half-plane p 2, the negative part p 1 - with the positive part p 2.

4. Plane p 3 rotates around the axis z(line of intersection of planes) until aligned with plane p 2 (see Fig. 2.31).

The images obtained on the planes p 1, p 2 and p 3 by rectangular projection of images are called projections.

Planes p 1, p 2 and p 3, together with the projections depicted on them, form a planar complex drawing or diagram.

Lines connecting the projections of the image to the axes x, y, z, are called projection communication lines.

To more accurately determine images in space, a system of three mutually perpendicular planes p 1, p 2, p 3 can be used.

Depending on the conditions of the problem, you can choose either the p 1, p 2 or p 1, p 2, p 3 system for the image.

The system of planes p 1 , p 2 , p 3 can be connected to the Cartesian coordinate system, which makes it possible to define objects not only graphically or (verbally), but also analytically (using numbers).

This method of depicting images, in particular points, makes it possible to solve such positional problems as:

• location of the point relative to the projection planes (general position, belonging to the plane, axis);
• position of the point in the quarters (in which quarter the point is located);
• position of the points relative to each other (higher, lower, closer, further relative to the projection planes and the viewer);
• position of the point’s projections relative to the projection planes (equidistant, closer, further).

Metric tasks:

• equidistance of the projection from the projection planes;
• ratio of projection distance from projection planes (2–3 times, more, less);
• determining the distance of a point from the projection planes (when introducing a coordinate system).

Self-Reflection Questions

1. The intersection line of which planes is the axis z?

2. The intersection line of which planes is the axis y?

3. How is the line of projection connection between the frontal and profile projection of a point located? Show me.

4. What coordinates determine the position of the projection of a point: horizontal, frontal, profile?

5. In which quarter is point F (10; –40; –20) located? From which projection plane is point F farthest away?

6. The distance from which projection to which axis determines the distance of a point from the plane p 1? What coordinate of the point is this distance?

The position of a point in space can be specified by its two orthogonal projections, for example, horizontal and frontal, frontal and profile. The combination of any two orthogonal projections allows you to find out the value of all the coordinates of a point, construct a third projection, and determine the octant in which it is located. Let's look at several typical problems from the descriptive geometry course.

For a given complex drawing of points A and B, it is necessary:

Let us first determine the coordinates of point A, which can be written in the form A (x, y, z). Horizontal projection of point A - point A", having coordinates x, y. Let us draw perpendiculars from point A" to the x, y axes and find A x, A y, respectively. The x coordinate for point A is equal to the length of the segment A x O with a plus sign, since A x lies in the region of positive values ​​of the x axis. Taking into account the scale of the drawing, we find x = 10. The y coordinate is equal to the length of the segment A y O with a minus sign, since t. A y lies in the region of negative values ​​of the y axis. Taking into account the scale of the drawing, y = –30. The frontal projection of point A - point A"" has coordinates x and z. Let's drop the perpendicular from A"" to the z axis and find A z. The z coordinate of point A is equal to the length of the segment A z O with a minus sign, since A z lies in the region of negative values ​​of the z axis. Taking into account the drawing scale z = –10. Thus, the coordinates of point A are (10, –30, –10).

The coordinates of point B can be written as B (x, y, z). Consider the horizontal projection of point B - point B". Since it lies on the x axis, then B x = B" and the coordinate B y = 0. The abscissa x of point B is equal to the length of the segment B x O with a plus sign. Taking into account the drawing scale x = 30. The frontal projection of point B is t. B˝ has coordinates x, z. Let's draw a perpendicular from B"" to the z axis, thus finding B z. The applicate z of point B is equal to the length of the segment B z O with a minus sign, since B z lies in the region of negative values ​​of the z axis. Taking into account the scale of the drawing, we determine the value z = –20. So the coordinates of B are (30, 0, -20). All necessary constructions are presented in the figure below.

Construction of projections of points

Points A and B in plane P 3 have the following coordinates: A""" (y, z); B""" (y, z). In this case, A"" and A""" lie on the same perpendicular to the z axis, since they have a common z coordinate. Similarly, B"" and B""" lie on a common perpendicular to the z axis. To find the profile projection of point A, we plot along the y-axis the value of the corresponding coordinate found earlier. In the figure, this is done using a circular arc of radius A y O. After this, draw a perpendicular from A y until it intersects with the perpendicular restored from point A"" to the z axis. The intersection point of these two perpendiculars determines the position of A""".

Point B""" lies on the z axis, since the y ordinate of this point is zero. To find the profile projection of point B in this problem, you only need to draw a perpendicular from B"" to the z axis. The intersection point of this perpendicular with the z axis is B """.

Determining the position of points in space

Visually imagining the spatial layout, composed of projection planes P 1, P 2 and P 3, the location of the octants, as well as the order of transforming the layout into diagrams, you can directly determine that point A is located in the III octant, and point B lies in the plane P 2.

Another option for solving this problem is the method of exceptions. For example, the coordinates of point A are (10, -30, -10). A positive abscissa x allows us to judge that the point is located in the first four octants. A negative y-ordinate indicates that the point is in the second or third octant. Finally, the negative applicate z indicates that point A is located in the third octant. The following table clearly illustrates the above reasoning.

Coordinates of point B (30, 0, -20). Since the ordinate of point B is zero, this point is located in the projection plane P 2. The positive abscissa and negative applicate of t. B indicate that it is located on the border of the third and fourth octants.

Construction of a visual image of points in the system of planes P 1, P 2, P 3

Using a frontal isometric projection, we built a spatial layout of the III octant. It is a rectangular trihedron whose faces are the planes P 1, P 2, P 3, and the angle (-y0x) is 45 º. In this system, segments along the x, y, z axes will be plotted in natural size without distortion.

Let's start constructing a visual image of point A (10, -30, -10) with its horizontal projection A. Having plotted the corresponding coordinates along the abscissa and ordinate axis, we find the points A x and A y. The intersection of perpendiculars reconstructed from A x and A y respectively to the x and y axes determines the position of point A". Laying off from A" parallel to the z axis towards its negative values ​​the segment AA", the length of which is 10, we find the position of point A.

The visual image of point B (30, 0, -20) is constructed in a similar way - in the P2 plane along the x and z axes, you need to plot the corresponding coordinates. The intersection of the perpendiculars reconstructed from B x and B z will determine the position of point B.