Right angle partial projection theorem

If the plane of the right angle is neither perpendicular nor parallel to the plane of projections, and if at least one of its sides is parallel to this plane, then the right angle is projected onto it without distortion.

Let the corner ABC- straight (Fig. 65) and side sun|| H, hence the projection bc|| BC. side AB continue until the intersection with the plane H and through the dot TO draw a straight line KN|| bc. Hence, KN || BC.

It follows from this that the angle BKN- straight. According to the three perpendiculars theorem, the angle bKN- right, therefore, an angle kbc= 90°.

Rice. 65. Spatial model of projection of a right angle

Note. This right angle projection theorem has two converse theorems (no proofs given).

1. If the projection of a flat angle is a right angle, then the projected angle will be right only if at least one of the sides of this angle is parallel to the projection plane.

2. If the projection of some angle, in which one side is parallel to the plane of projections, is a right angle, then the projected angle is also a right one.

Based on these theorems, it can be established that the angles shown in Fig. 66, in space - straight lines.

Rice. 66. Projection of a right angle on the Monge diagram:

A- one of the sides of the angle is horizontal; b- one of the sides of the corner - frontal

Consider the angle IN(Fig. 66 A).

In space angle IN straight line, because the diagram shows that the straight line AB is horizontal ( h'|| X) and ∠ a= 90° (according to the first inverse theorem).

Consider the angle IN(Fig. 66 b).

In space angle IN straight, because one of its sides is a frontal ( AB|| V;ab|| X) and frontal projection ∠ b′ = 90°.

A simple conclusion follows from this theorem - a perpendicular can be drawn to a straight line where the straight line is projected in full size.

When solving positional and metric problems of descriptive geometry, based on these theorems, it is possible to build two mutually perpendicular lines, which, ultimately, allows you to determine distances, build mutually perpendicular planes.

Let's consider a few tasks on the topic of this material.

Task 1. Through the dot A draw a line perpendicular to a line M(Fig. 67).

Analyzing the graphic condition of the problem, we note that m|| X, which means that the line M is a frontal ( M|| V).

Therefore, the construction of the desired straight line must begin with a frontal projection, drawing it perpendicular to the projection m׳, since on the frontal projection plane the line M projected without distortion and on the frontal projection plane V the right angle between the given and newly constructed lines will be projected without distortion.

1. We build a frontal projection of the desired segment a'b'⊥ m′.

2. Determine the position of the point b׳ on the projection m׳ and by the projection connection we determine the horizontal projection b on the projection m.

3. We build a horizontal projection of the desired segment ab.

Rice. 67. Construction of a perpendicular to a line M Rice. 68. Building height in ∆ ABC

Task 2. Through the top WITH draw the height of the triangle ABC(Fig. 68).

Solution. We analyze the diagram and note that the side of the triangle AB|| H, while its horizontal projection is displayed in full size.

Therefore, the construction of the height must begin with a horizontal projection.

The order of execution of the graphical part of the task:

1. From a point With draw a line perpendicular to the side ab.

2. Point d- the base of the height, cd is the horizontal projection of the height.

3. Projecting a point d on the frontal projection of the side a'b' and get the frontal projection of the point d' and build a frontal height projection c'd'.

Task 3. Determine distance from point TO to straight N(Fig. 69).

Solution. It should be noted that when solving problems for determining distances, it is necessary to build not only projections of the distance, but also to determine its natural value.

The shortest distance from a point to a line is the value of the perpendicular dropped from that point to the line. Analyzing the diagrams, we note that the straight line N is frontal and is displayed on the frontal projection without distortion.

Therefore, the construction of the projection of the perpendicular must begin with its frontal projection.

The order of execution of the graphical part of the task:

1. From a point k′ drop the perpendicular onto the projection of the line n', we get a point e′. Frontal projection of the perpendicular - k′ e′.

2. We project the resulting point onto the horizontal projection of the straight line n, get a point e and the horizontal projection of the perpendicular ke.

3. Judging by the projections, a straight line KE general position. By the method of a right-angled triangle we determine its actual size | KE |.

Distance from point TO to straight N equal to the length of the segment - TO O e′.

KE, N = K o e′= 30 mm.

3.5. Special plane lines

Straight lines occupying a special position in the plane:

1. Plane level lines.

2. Lines of the greatest inclination of the plane to the projection planes.

Plane level lines

Plane level lines- straight lines lying in a given plane and parallel to the projection planes: horizontals, fronts, profile lines.

Horizontal plane - straight line lying in a given plane and parallel to the plane of projections N. It should be remembered that all horizontal lines of the same plane are parallel to each other.

The horizontal projection of the horizontal is parallel to the horizontal trace of the plane, the horizontal trace of the plane is the zero horizontal of the plane. To build a horizontal plane R, given by traces, it is necessary on the frontal projection R V mark a point d"- frontal projection of the horizontal trace (Fig. 67 A). Through it we draw a frontal projection of the horizontal parallel to the axis X. on axle X find the horizontal projection d. A straight line drawn from a point d parallel to the track R N plane, represents the horizontal projection of the horizontal.

On fig. 70 b horizontal projections are drawn through point projections D and points 1 straight EU plane defined by a triangle CDE. Building a horizontal always starts with a frontal projection d"1", which is parallel to the axis X. By the property of belonging, find the horizontal projection of a point 1 and carry out a horizontal projection of the horizontal.

Rice. 70. Horizontal plane:

A- in the plane R, given by traces; b– in the plane given by ∆ CDE

Plane front- a straight line lying in a plane and parallel to the plane of projections V(Fig. 71).

The construction of the frontal and profile lines is carried out similarly to the construction of the horizontal, based on the known properties of the projections of the level lines and the belonging property, and it starts from the projection that is parallel to the corresponding projection axis. All frontals of the same plane are parallel to each other. The same can be said about the profile straight lines of the plane level.

Plane level profile line- this is a straight line lying in a given plane and parallel to the profile plane of projections (Fig. 72).

Rice. 71. Plane front:

A- in the plane R, given by traces; b– in the plane given by ∆ CDE

Rice. 72. Profile line level BE plane ∆ ABC

Direct projections.

Drawing reversibility

Drawing reversibility. By projecting onto one projection plane, an image is obtained that does not allow one to unambiguously determine the shape and dimensions of the depicted object. The projection A 1 (see Fig. 1.4.) does not determine the position of the point itself in space, since it is not known how far it is removed from the projection plane P 1. In such cases, one speaks of irreversibility drawing , since it is impossible to reproduce the original according to such a drawing. To eliminate uncertainty, the images are supplemented with the necessary data. In practice, various methods of supplementing a single-projection drawing are used.

CHAPTER 2

A straight line can be considered as the result of the intersection of two planes (Fig. 2.1, 2.2.).

The line in space is unlimited. The bounded part of a straight line is called a segment.

The projection of a straight line is reduced to the construction of projections of its two arbitrary points, since two points completely determine the position of the straight line in space. Lowering from point A and B (Fig. 2.2.) Perpendiculars to the intersection with the plane P 1, determine their horizontal projections A 1 and B 1. Segment A 1 B 1 is a horizontal projection of the straight line AB. A similar result is obtained by drawing perpendiculars to P 1 from arbitrary points of the line AB. The combination of these perpendiculars (projecting rays) forms a horizontally projecting plane a, which intersects with the plane P 1 along the straight line A 1 B 1 - the horizontal projection of the straight line AB. Based on the same considerations, a frontal projection of A 2 B 2 straight line AB is obtained (Fig. 2.2).

One projection of a straight line does not determine its position in space. Indeed, the segment A 1 B 1 (Fig. 2.1.) Can be a projection of an arbitrary segment lying in the projecting plane a. The position of a straight line in space is uniquely determined by the combination of its two projections. Restoring from the point horizontal A 1 B 1 and frontal P 1 and P 2, two projecting planes a and b are obtained, intersecting along a single straight line AB.

The complex drawing (Figure 2.3) shows a segment AB of a straight line in general position, where A 1 B 1 is horizontal, A 2 B 2 is frontal and A 3 B 3 is the profile projection of the segment. To build the third projection of the segment. To construct the third projection of a straight line segment using two data, you can use the same methods as for constructing the third projection of a point: projection (Fig. 2.4.), coordinate (Fig. 2.5.) and using a constant straight line drawing (Fig. 2.6.).

2.2. The position of the straight line relative to the projection plane.

In Figure 1.5. a parallelepiped with a truncated top and an arbitrary triangular pyramid are shown. The edges of the parallelepiped and the pyramid occupy different positions in space relative to the projection planes. To build and read drawings, you need to be able to analyze the position of a straight line. According to their position in space, lines are divided into lines of particular and lines of general position.

Direct private provision can be projecting and direct level.

Projecting lines are called straight lines perpendicular to one of the projection planes, i.e. parallel to the other two planes P 1, is called a horizontally projecting line; its horizontal projection A 1 B 1 is a point, and the frontal and profile projections are straight lines parallel to the O z axis. The straight line CD (Fig. 2.7.) Perpendicular to the plane of projections P 2 is called the frontally projecting straight line; its frontal projection C 2 D 2 is a point, and the horizontal and profile projections are straight lines parallel to the Oy axis. The straight line MN (Fig. 2.8.) Perpendicular to the plane of projections P 3 is called the profile projecting line; its profile projection M 3 N 3 is a point, and the horizontal and frontal projections are straight lines parallel to the Ox axis.

Therefore, on one of the projection planes, the projecting line is depicted as a point, and on the other two, as segments occupying a horizontal or vertical position, the magnitude of which is horizontal or vertical, the magnitude of which is equal to the natural value of the straight line segment itself.

Level lines are straight lines parallel to one of the projection planes. The straight line AB (Fig. 2.9.), Parallel to the horizontal plane of the projections P 1, is called the horizontal line, or, in short, the horizontal. Its frontal projection A 2 B 2 is parallel to the projection axis Ox, and the horizontal A 1 B 1 is equal to the natural value of the straight line segment (A 1 B 1 \u003d AB). The angle b between the horizontal projection A 1 B 1 and the Ox axis is equal to the natural value of the angle of inclination of the straight line AB to the projection plane P 2.

The straight line CD (Fig. 2.10.) Parallel to the frontal plane of the projections P 2 is called the frontal straight line, or, in short, the frontal. Its horizontal projection C 1 D 1 is parallel to the Ox axis, and the frontal C 2 D 2 is equal to the natural value of the straight line segment (C 2 D 2 = CD). The angle a between the frontal projection C 2 D 2 and the Ox axis is equal to the actual value of the angle of inclination of the straight line to the projection plane P 1 .

The straight line MN (Fig. 2.11.) Parallel to the profile plane of the projections P 3 is called the profile straight line. Its frontal M 2 N 2 and horizontal M 1 N 1 projections are perpendicular to the Ox axis, and the profile projection is equal to the natural size of the segment (M 3 N 3 = MN). The angles a and b between the profile projection and the axes Oy 3 and Oz are equal to the actual value of the angles of inclination of the straight line to the plane of projections P 1 and P 2.

Consequently, the level lines on one of the projection planes are projected in full size, and on the other two - in the form of segments of reduced size, occupying a vertical or horizontal position in the drawing. According to the drawing, you can determine the magnitude of the angles of inclination of these lines to the projection planes.

If a straight line lies in the projection plane, then one of its projections (of the same name) coincides with the straight line itself, and the other two coincide with the projection axes. For example, the straight line AB (Fig. 2.12) lies in the P 1 plane. Its horizontal projection A 1 B 1 merges with the straight line AB, and the frontal A 2 B 2 merges with the Ox axis. Such a straight line is called a zero horizontal, since the height of its points (z-coordinate) is equal to zero.

Direct general position called a straight line inclined to all projection planes. Its projections form sharp or obtuse angles with the axes Ox, Oy and Oz, i.e. none of its projections are parallel or perpendicular to the axes. The value of the projections of a line in general position is always less than the natural value of the segment itself. Directly from the drawing, without additional constructions, it is impossible to determine the actual value of the straight line and its angle of inclination to the projection planes.

If the point lies on a straight line, then the projections of the point are on the same projections of the straight line and on a common communication line.

On fig. 2.13. point C lies on the line AB, since its projections C 1 and C 2 are respectively on the horizontal A 1 B 1 and on the frontal A 2 B 2 projections of the line. The points M and N do not belong to the line, since one of the projections of each point is not on the projection of the line with the same name.

The projections of a point divide the projections of a straight line in the same ratio in which the point itself divides a line segment, i.e. Using this rule, divide the given segment by a straight line in the desired ratio. For example, in fig. 2.14. the straight line EF is divided by the point K in the ratio 3:5. The division is made in a manner known from geometric drawing.

The position of a straight line in space is determined by the position of its two points. Therefore, to construct projections of a straight line, it is sufficient to construct projections of two points belonging to it and connect them to each other.

Depending on the position relative to the projection planes, straight lines of general and particular positions are distinguished.

straight lines private position parallel to one or two projection planes.

straight level lines- straight lines, parallel to one plane of projections and inclined to two others. There are three types of such lines.

sch, called horizontal straight and denoted by the letter To(Fig. 3.1). Its segment is projected onto the plane sch without distortion. The angle between its horizontal projection To " and axis OH equal to the angle of inclination f 2 of the horizontal line to the plane n 2, and the angle between its projection To"and axis OU - the angle of inclination f 3 to the plane ts 3 . All points of the same horizontal straight line have the same coordinate b

A straight line parallel to a plane n 2, called frontal straight and denoted by the letter / (Fig. 3.2). Its segment is projected without distortion onto the plane 7Г 2 - The angles of inclination of the frontal line TO THE PLANE are projected onto the same plane in true value L(angle φ[) and plane l 3 (angle φ 3). All points of the same frontal straight line have the same coordinate y.

A straight line parallel to the plane l 3 is called profile straight R(Fig. 3.3). Its segment without distortion is projected onto the plane l 3 . For the same

the plane is projected to the true value of the angles of inclination of the profile line TO THE PLANE 7Гі (angle (pі) and the plane l 2 (angle (p 2). All points of the profile straight line have the same coordinate X.

Projecting straight lines- straight lines perpendicular to one projection plane and parallel to the other two.

A straight line perpendicular to the plane l is called horizontally projecting(Fig. 3.4). It is projected onto the l i plane as a point, and its frontal and profile projections are parallel to the axis 01. A segment of a horizontally projecting straight line is projected without distortion on the plane l 2 and Lz. Therefore, the horizontally projecting line is both frontal / and profile R straight line.

A straight line perpendicular to the plane l 2 is called front projecting(Fig. 3.5). It is projected onto the l 2 plane as a point, and its horizontal and profile projections are parallel to the axis OU. A segment of a frontally projecting straight line is projected without distortion on the plane Li and l 3. The front projecting line is also horizontal To and profile R straight.

A straight line perpendicular to the plane l 3 is called profile projecting(Fig. 3.6). Its profile projection is a point, and the horizontal and frontal

on i parallel to the axis OH. A segment of such a straight line is projected to the true value on the plane A] and 712, therefore it is also horizontal AND, and frontal / straight.

A straight line that is not parallel to any of the main projection planes is called a straight line. general position(Fig. 3.7). On surface P, P 2 and 7D 3 its segment is projected with distortion, since it is inclined towards them and the angles of inclination in the drawing are also distorted. Thus, according to the drawing of a straight line in general position, it is impossible to measure the length of its segment or angles of inclination to the projection planes. To determine these quantities, additional constructions are required.

The position of a straight line in space is completely determined by any two of its points. In the general case, the projection of a straight line is a straight line, in a particular case, a point if the straight line is perpendicular to the plane of projections. To construct projections of a straight line, it is sufficient to have either the projections of two of its points, or the projection of one point of the straight line and the direction of the straight line in space.

According to their location in space relative to the projection planes, straight lines are divided into straight lines of general position, level and projecting .

2.2.1. Lines of general position. These are straight lines, not parallel and not perpendicular to the projection planes. projections A 1 B 1, A 2 B 2 And A 3 B 3 segment AB straight AB general position (Fig. 2.18, A) are inclined at acute angles to the axes x 12 , y 13 And z 23. The lengths of the projections of the segments of this line are always less than the segment itself. Three-picture complex drawing of a line segment in general position, built on two points A And IN, shown in Figure 2.18, b.

2.2.2. Direct level. These are straight lines parallel to one of the projection planes - P 1,P 2 or P 3. Therefore, we have three types of level lines:

1) horizontal level a (horizontal ), parallel P 1(straight a with a segment AB on it in Fig. 2.19 A, b);

2) frontal level (frontal ), parallel P 2(straight b with a segment CD on it in Fig. 2.20, A);

3) profile level , parallel P 3(straight With with a segment EF on it in Fig. 2.20, b). On fig. 2.20 visual representations of straight lines b And c relative to the projection planes are not shown.

The same-named projections of line segments of the level are projected in full size, and the opposite-named ones are parallel to the axes separating them from the same-named ones. At the same time, for the horizontal, the projection of the same name is horizontal, and the projection of opposite names is frontal and profile, etc.

Angles of inclination of straight lines a,b And c to projection planes P 1, P 2 And P 3 are designated accordingly α , β And γ (in Fig. 2.19 the angles α , β And γ not shown).

2.2.3. Projecting lines. These are straight lines perpendicular to one of the projection planes and parallel to the other two. Therefore, we have three types of projecting lines:

1) horizontally projecting straight, perpendicular P 1(straight A with a segment AB on it in Fig. 2.21, A);

2) front-projecting straight, perpendicular P 2(straight b with a segment CD on it in Fig. 2.21, b);

3)profile-projecting straight, perpendicular P 3(straight c with a segment EF on it in Fig. 2.21, V).

On fig. 2.21 projections of invisible points are enclosed in brackets. The issue of determining the visibility of points on projections will be considered in more detail below in the section "Crossing lines".

For projecting lines, the projections of the same name are points, which follows from the essence of the projecting line along which the projection is carried out.

Each opposite projection of the projecting line is perpendicular to the axis separating it from the projection of the same name, and the opposite projection of a segment located on the level line is the natural value of this segment.

2.2.4. Determining the natural size of a line segment in general position. The natural value of the line of particular position can be immediately determined on the complex drawing of this line.

To determine the natural size of a segment of a straight line in general position, you can apply the one considered earlier (see Section 2.1.2) method of replacing projection planes . Figure 2.22 shows the definition of the natural value ( N.V.) segment AB line of general position and determination of its angles of inclination to Π 1(corner α ) and to Π 2(corner β ) in this manner.

Additional plane Π 4 carried out in parallel AB (x 14 ||A 1 B 1). Straight AB converted to frontal position, therefore A 4 B 4- natural size AB.

Drawing an additional plane Π 5 ||AB(x 25 ||A 2 B 2), it is also possible to determine the actual size AB. A 5 B 5- natural size AB. Straight AB in system Π 2-Π 5 became horizontal.

Figure 2.23 shows the definition of the natural value AB triangle method. The natural value of the segment is equal to the hypotenuse of a right triangle, one leg of which is one of the projections of the segment, and the other is the algebraic difference of the distances of its ends from the plane Π 1(ΔZ).

2.2.5. Mutual position of lines. Straight lines in space can be parallel, intersect and cross.

Parallel lines. It follows from the properties of parallel projections that if lines in space are parallel, then all three pairs of their projections of the same name are parallel. The opposite is also obvious: if the same-name projections of lines are parallel, then the lines in space are parallel.

To determine the parallelism of straight lines, in the general case, it is sufficient that two pairs of projections of the same name are parallel. If the parallelism of the level lines is determined, then one of the two pairs of parallel projections must be a projection onto the plane of the same name.

On fig. 2.24 projections of parallel lines are shown a And b general position, where a 1 ║ b 1 And a 2 ║ b 2. On fig. 2.25 two horizontal lines are shown c And d. For horizontals, the frontal and profile projections are always parallel to the axes separating them from the horizontal projections of the same name, i.e. c 2║d2║x 12 And c 3║d3║y 3. But their horizontal projections are not parallel, i.e. c 1╫d1. Therefore, direct c And d are not parallel.

Intersecting lines. Two intersecting lines lie in the same plane and have one common point. From the properties of parallel projections, it is known that if a point lies on a line, then its projections lie on the projections of the line. If a point lies on both lines, i.e., at the point of intersection of the lines, then its projection must immediately lie on two projections of the lines of the same name, and therefore, at the point of intersection of the projections of the lines.

So, if the segments AB And CD two lines intersect at a point K, then the projections of the segments A 1 B 1 And C 1 D 1 intersect at a point K1, which is the projection of the point K(Fig. 2.26, A). That's why, if the same-name projections of lines intersect at points lying on the same line of the projection connection, then the lines intersect in space (Fig. 2.26, b).

To determine whether the lines intersect or not, it is sufficient that this condition is satisfied for any two projections. The exception is the case when one of the intersecting lines is the profile level. In this case, to check the intersection of lines, it is necessary to build a profile projection.

Let through the dot A need to be horizontal b that intersects the line a(Fig. 2.27, A). To do this, through the dot A2 carry out b 2 ║ x 12(stage 1) to the intersection with a 2 at the point K2(fig.2.27, b). Further, using the projection connection line on a 1 find a point K1(step 2) and by connecting the dots A 1 And K1(stage 3), we get b 1.

Crossing straight lines. Crossed lines a And b do not lie in the same plane and, therefore, are not parallel and do not have common points (Fig. 2.28, A). Therefore, if the lines are skew, then at least one pair of their like projections is not parallel, and the intersection points of like projections do not lie on the same projection connection line (Fig. 2.28, b).

Each such point of intersection is a projection of two points belonging to lines; these two points lie on the same projecting ray and are called competing .