The projection of a point on three planes of projections of the coordinate angle begins with obtaining its image on the plane H - the horizontal plane of projections. To do this, through point A (Fig. 4.12, a) a projecting beam is drawn perpendicular to the plane H.

In the figure, the perpendicular to the H plane is parallel to the Oz axis. The point of intersection of the beam with the plane H (point a) is chosen arbitrarily. The segment Aa determines how far point A is from the plane H, thus indicating unambiguously the position of point A in the figure with respect to the projection planes. Point a is a rectangular projection of point A onto the plane H and is called the horizontal projection of point A (Fig. 4.12, a).

To obtain an image of point A on the plane V (Fig. 4.12, b), a projecting beam is drawn through point A perpendicular to the frontal projection plane V. In the figure, the perpendicular to the plane V is parallel to the Oy axis. On the H plane, the distance from point A to plane V will be represented by a segment aa x, parallel to the Oy axis and perpendicular to the Ox axis. If we imagine that the projecting beam and its image are carried out simultaneously in the direction of the plane V, then when the image of the beam intersects the Ox axis at the point a x, the beam intersects the plane V at the point a. Drawing from the point a x in the V plane perpendicular to the Ox axis , which is the image of the projecting beam Aa on the plane V, the point a is obtained at the intersection with the projecting beam. Point a "is the frontal projection of point A, i.e. its image on the plane V.

The image of point A on the profile plane of projections (Fig. 4.12, c) is built using a projecting beam perpendicular to the W plane. In the figure, the perpendicular to the W plane is parallel to the Ox axis. The projecting beam from point A to plane W on the plane H will be represented by a segment aa y, parallel to the Ox axis and perpendicular to the Oy axis. From the point Oy parallel to the Oz axis and perpendicular to the Oy axis, an image of the projecting beam aA is built and, at the intersection with the projecting beam, the point a is obtained. Point a is the profile projection of the point A, i.e., the image of the point A on the plane W.

The point a "can be constructed by drawing from the point a" the segment a "a z (the image of the projecting beam Aa" on the plane V) parallel to the Ox axis, and from the point a z - the segment a "a z parallel to the Oy axis until it intersects with the projecting beam.

Having received three projections of point A on the projection planes, the coordinate angle is deployed into one plane, as shown in Fig. 4.11, b, together with the projections of the point A and the projecting rays, and the point A and the projecting rays Aa, Aa "and Aa" are removed. The edges of the combined projection planes are not carried out, but only the projection axes Oz, Oy and Ox, Oy 1 (Fig. 4.13) are carried out.

An analysis of the orthogonal drawing of a point shows that three distances - Aa", Aa and Aa" (Fig. 4.12, c), characterizing the position of point A in space, can be determined by discarding the projection object itself - point A, on a coordinate angle deployed in one plane (Fig. 4.13). The segments a "a z, aa y and Oa x are equal to Aa" as opposite sides of the corresponding rectangles (Fig. 4.12, c and 4.13). They determine the distance at which point A is located from the profile plane of projections. Segments a "a x, a" a y1 and Oa y are equal to segment Aa, determine the distance from point A to the horizontal plane of projections, segments aa x, a "a z and Oa y 1 are equal to segment Aa", which determines the distance from point A to frontal projection plane.

The segments Oa x, Oa y and Oa z located on the projection axes are a graphic expression of the sizes of the X, Y and Z coordinates of point A. The point coordinates are denoted with the index of the corresponding letter. By measuring the size of these segments, you can determine the position of the point in space, i.e., set the coordinates of the point.

On the diagram, the segments a "a x and aa x are arranged as one line perpendicular to the Ox axis, and the segments a" a z and a "a z - to the Oz axis. These lines are called projection connection lines. They intersect the projection axes at points a x and and z, respectively.The line of the projection connection connecting the horizontal projection of point A with the profile one turned out to be “cut” at the point a y.

Two projections of the same point are always located on the same projection connection line perpendicular to the projection axis.

To represent the position of a point in space, two of its projections and a given origin (point O) are sufficient. 4.14, b, two projections of a point completely determine its position in space. Using these two projections, you can build a profile projection of point A. Therefore, in the future, if there is no need for a profile projection, diagrams will be built on two projection planes: V and H.

Rice. 4.14. Rice. 4.15.

Let's consider several examples of building and reading a drawing of a point.

Example 1 Determination of the coordinates of the point J given on the diagram by two projections (Fig. 4.14). Three segments are measured: segment Ov X (X coordinate), segment b X b (Y coordinate) and segment b X b "(Z coordinate). Coordinates are written in the following order: X, Y and Z, after the letter designation of the point, for example , B20; 30; 15.

Example 2. Construction of a point according to the given coordinates. Point C is given by coordinates C30; ten; 40. On the Ox axis (Fig. 4.15) find a point with x, at which the line of the projection connection intersects the projection axis. To do this, the X coordinate (size 30) is plotted along the Ox axis from the origin (point O) and a point with x is obtained. Through this point, perpendicular to the Ox axis, a projection connection line is drawn and the Y coordinate is laid down from the point (size 10), the point c is obtained - the horizontal projection of the point C. The coordinate Z (size 40) is plotted upwards from the point c x along the projection connection line (size 40), the point is obtained c" - frontal projection of point C.

Example 3. Construction of a profile projection of a point according to the given projections. The projections of the point D - d and d are set. Through the point O, the projection axes Oz, Oy and Oy 1 are drawn (Fig. 4.16, a). it to the right behind the Oz axis. The profile projection of the point D will be located on this line. It will be located at the same distance from the Oz axis as the horizontal projection of the point d is located: from the Ox axis, i.e. at a distance dd x. The segments d z d "and dd x are the same, since they determine the same distance - the distance from point D to the frontal projection plane. This distance is the Y coordinate of point D.

Graphically, the segment d z d "is built by transferring the segment dd x from the horizontal plane of projections to the profile one. To do this, draw a line of projection connection parallel to the Ox axis, get a point d y on the Oy axis (Fig. 4.16, b). Then transfer the size of the segment Od y to the Oy 1 axis , drawing from point O an arc with a radius equal to the segment Od y, until it intersects with the axis Oy 1 (Fig. 4.16, b), get the point dy 1. This point can be constructed and, as shown in Fig. 4.16, c, drawing a straight line at an angle 45 ° to the Oy axis from the point d y. From the point d y1 draw a projection connection line parallel to the Oz axis and lay a segment on it equal to the segment d "d x, get the point d".

Transferring the value of the segment d x d to the profile plane of the projections can be done using a constant straight line drawing (Fig. 4.16, d). In this case, the projection connection line dd y is drawn through the horizontal projection of the point parallel to the Oy 1 axis until it intersects with a constant straight line, and then parallel to the Oy axis until it intersects with the continuation of the projection connection line d "d z.

Particular cases of the location of points relative to projection planes

The position of the point relative to the projection plane is determined by the corresponding coordinate, i.e., the value of the segment of the projection connection line from the Ox axis to the corresponding projection. On fig. 4.17 the Y coordinate of point A is determined by the segment aa x - the distance from point A to plane V. The Z coordinate of point A is determined by the segment a "a x - the distance from point A to plane H. If one of the coordinates is zero, then the point is located on the projection plane Fig. 4.17 shows examples of different locations of points relative to the projection planes.The Z coordinate of point B is zero, the point is in plane H. Its frontal projection is on the Ox axis and coincides with point b x. The Y coordinate of point C is zero, the point is located on the plane V, its horizontal projection c is on the x-axis and coincides with the point c x.

Therefore, if a point is on the projection plane, then one of the projections of this point lies on the projection axis.

On fig. 4.17, the Z and Y coordinates of point D are zero, therefore, point D is on the projection axis Ox and its two projections coincide.

In this article, we will find answers to questions about how to create a projection of a point onto a plane and how to determine the coordinates of this projection. In the theoretical part, we will rely on the concept of projection. We will give definitions of terms, accompany the information with illustrations. Let's consolidate the acquired knowledge by solving examples.

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Projection, types of projection

For convenience of consideration of spatial figures, drawings depicting these figures are used.

Definition 1

Projection of a figure onto a plane- a drawing of a spatial figure.

Obviously, there are a number of rules used to construct a projection.

Definition 2

projection- the process of constructing a drawing of a spatial figure on a plane using construction rules.

Projection plane is the plane in which the image is built.

The use of certain rules determines the type of projection: central or parallel.

A special case of parallel projection is perpendicular projection or orthogonal projection: in geometry, it is mainly used. For this reason, the adjective “perpendicular” itself is often omitted in speech: in geometry they simply say “projection of a figure” and mean by this the construction of a projection by the method of perpendicular projection. In special cases, of course, otherwise can be stipulated.

We note the fact that the projection of a figure onto a plane is, in fact, the projection of all points of this figure. Therefore, in order to be able to study a spatial figure in a drawing, it is necessary to acquire the basic skill of projecting a point onto a plane. What we will talk about below.

Recall that most often in geometry, speaking of projection onto a plane, they mean the use of perpendicular projection.

We will make constructions that will enable us to obtain the definition of the projection of a point onto a plane.

Suppose a three-dimensional space is given, and in it - a plane α and a point M 1 that does not belong to the plane α. Draw a straight line through a given point M 1 a perpendicular to the given plane α. The point of intersection of the line a and the plane α will be denoted as H 1 , by construction it will serve as the base of the perpendicular dropped from the point M 1 to the plane α .

If a point M 2 is given, belonging to a given plane α, then M 2 will serve as a projection of itself onto the plane α.

Definition 3

is either the point itself (if it belongs to a given plane), or the base of the perpendicular dropped from given point to a given plane.

Finding the coordinates of the projection of a point on a plane, examples

Let in three-dimensional space given: rectangular coordinate system O x y z, plane α, point M 1 (x 1, y 1, z 1) . It is necessary to find the coordinates of the projection of the point M 1 onto a given plane.

The solution obviously follows from the above definition of the projection of a point onto a plane.

We denote the projection of the point M 1 onto the plane α as H 1 . According to the definition, H 1 is the point of intersection of the given plane α and the line a through the point M 1 (perpendicular to the plane). Those. the coordinates of the projection of the point M 1 we need are the coordinates of the point of intersection of the line a and the plane α.

Thus, to find the coordinates of the projection of a point onto a plane, it is necessary:

Get the equation of the plane α (in case it is not set). An article about the types of plane equations will help you here;

Determine the equation of the line a passing through the point M 1 and perpendicular to the plane α (study the topic of the equation of the straight line passing through a given point perpendicular to a given plane);

Find the coordinates of the point of intersection of the line a and the plane α (article - finding the coordinates of the point of intersection of the plane and the line). The data obtained will be the coordinates of the projection of the point M 1 onto the plane α that we need.

Let's consider the theory on practical examples.

Example 1

Determine the coordinates of the projection of the point M 1 (- 2, 4, 4) onto the plane 2 x - 3 y + z - 2 \u003d 0.

Decision

As we can see, the equation of the plane is given to us, i.e. there is no need to compose it.

Let's write the canonical equations of the straight line a passing through the point M 1 and perpendicular to the given plane. For these purposes, we determine the coordinates of the directing vector of the straight line a. Since the line a is perpendicular to the given plane, then the directing vector of the line a is the normal vector of the plane 2 x - 3 y + z - 2 = 0. Thus, a → = (2 , - 3 , 1) – direction vector of the line a .

Now we compose the canonical equations of a straight line in space passing through the point M 1 (- 2, 4, 4) and having a direction vector a → = (2 , - 3 , 1) :

x + 2 2 = y - 4 - 3 = z - 4 1

To find the desired coordinates, the next step is to determine the coordinates of the point of intersection of the line x + 2 2 = y - 4 - 3 = z - 4 1 and the plane 2 x - 3 y + z - 2 = 0 . To this end, we pass from the canonical equations to the equations of two intersecting planes:

x + 2 2 = y - 4 - 3 = z - 4 1 ⇔ - 3 (x + 2) = 2 (y - 4) 1 (x + 2) = 2 (z - 4) 1 ( y - 4) = - 3 (z + 4) ⇔ 3 x + 2 y - 2 = 0 x - 2 z + 10 = 0

Let's make a system of equations:

3 x + 2 y - 2 = 0 x - 2 z + 10 = 0 2 x - 3 y + z - 2 = 0 ⇔ 3 x + 2 y = 2 x - 2 z = - 10 2 x - 3 y + z = 2

And solve it using Cramer's method:

∆ = 3 2 0 1 0 - 2 2 - 3 1 = - 28 ∆ x = 2 2 0 - 10 0 - 2 2 - 3 1 = 0 ⇒ x = ∆ x ∆ = 0 - 28 = 0 ∆ y = 3 2 0 1 - 10 - 2 2 2 1 = - 28 ⇒ y = ∆ y ∆ = - 28 - 28 = 1 ∆ z = 3 2 2 1 0 - 10 2 - 3 2 = - 140 ⇒ z = ∆ z ∆ = - 140 - 28 = 5

Thus, the desired coordinates of a given point M 1 on a given plane α will be: (0, 1, 5) .

Answer: (0 , 1 , 5) .

Example 2

Points А (0 , 0 , 2) are given in a rectangular coordinate system O x y z of three-dimensional space; In (2, - 1, 0) ; C (4, 1, 1) and M 1 (-1, -2, 5). It is necessary to find the coordinates of the projection M 1 onto the plane A B C

Decision

First of all, we write the equation of a plane passing through three given points:

x - 0 y - 0 z - 0 2 - 0 - 1 - 0 0 - 2 4 - 0 1 - 0 1 - 2 = 0 ⇔ x y z - 2 2 - 1 - 2 4 1 - 1 = 0 ⇔ ⇔ 3 x - 6y + 6z - 12 = 0 ⇔ x - 2y + 2z - 4 = 0

Let's write down parametric equations straight line a, which will pass through the point M 1 perpendicular to the plane A B C. The plane x - 2 y + 2 z - 4 \u003d 0 has a normal vector with coordinates (1, - 2, 2), i.e. vector a → = (1 , - 2 , 2) – direction vector of the line a .

Now, having the coordinates of the point of the line M 1 and the coordinates of the directing vector of this line, we write the parametric equations of the line in space:

Then we determine the coordinates of the point of intersection of the plane x - 2 y + 2 z - 4 = 0 and the line

x = - 1 + λ y = - 2 - 2 λ z = 5 + 2 λ

To do this, we substitute into the equation of the plane:

x = - 1 + λ , y = - 2 - 2 λ , z = 5 + 2 λ

Now, using the parametric equations x = - 1 + λ y = - 2 - 2 λ z = 5 + 2 λ, we find the values ​​of the variables x, y and z at λ = - 1: x = - 1 + (- 1) y = - 2 - 2 (- 1) z = 5 + 2 (- 1) ⇔ x = - 2 y = 0 z = 3

Thus, the projection of the point M 1 onto the plane A B C will have coordinates (- 2, 0, 3) .

Answer: (- 2 , 0 , 3) .

Let us dwell separately on the question of finding the coordinates of the projection of a point on the coordinate planes and planes that are parallel to the coordinate planes.

Let points M 1 (x 1, y 1, z 1) and coordinate planes O x y , O x z and O y z be given. The projection coordinates of this point on these planes will be respectively: (x 1 , y 1 , 0) , (x 1 , 0 , z 1) and (0 , y 1 , z 1) . Consider also the planes parallel to the given coordinate planes:

C z + D = 0 ⇔ z = - D C , B y + D = 0 ⇔ y = - D B

And the projections of the given point M 1 on these planes will be points with coordinates x 1 , y 1 , - D C , x 1 , - D B , z 1 and - D A , y 1 , z 1 .

Let us demonstrate how this result was obtained.

As an example, let's define the projection of the point M 1 (x 1, y 1, z 1) onto the plane A x + D = 0. The rest of the cases are similar.

The given plane is parallel to the coordinate plane O y z and i → = (1 , 0 , 0) is its normal vector. The same vector serves as the directing vector of the straight line perpendicular to the plane O y z . Then the parametric equations of a straight line drawn through the point M 1 and perpendicular to a given plane will look like:

x = x 1 + λ y = y 1 z = z 1

Find the coordinates of the point of intersection of this line and the given plane. We first substitute into the equation A x + D = 0 equalities: x = x 1 + λ, y = y 1, z = z 1 and get: A (x 1 + λ) + D = 0 ⇒ λ = - D A - x one

Then we calculate the desired coordinates using the parametric equations of the straight line for λ = - D A - x 1:

x = x 1 + - D A - x 1 y = y 1 z = z 1 ⇔ x = - D A y = y 1 z = z 1

That is, the projection of the point M 1 (x 1, y 1, z 1) onto the plane will be a point with coordinates - D A , y 1 , z 1 .

Example 2

It is necessary to determine the coordinates of the projection of the point M 1 (- 6 , 0 , 1 2) on coordinate plane O x y and onto the plane 2 y - 3 = 0 .

Decision

The coordinate plane O x y will correspond to an incomplete general equation plane z = 0 . The projection of the point M 1 onto the plane z \u003d 0 will have coordinates (- 6, 0, 0) .

The plane equation 2 y - 3 = 0 can be written as y = 3 2 2 . Now just write the coordinates of the projection of the point M 1 (- 6 , 0 , 1 2) onto the plane y = 3 2 2:

6 , 3 2 2 , 1 2

Answer:(- 6 , 0 , 0) and - 6 , 3 2 2 , 1 2

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Consider the projections of points onto two planes, for which we take two perpendicular planes(Fig. 4), which we will call the horizontal frontal and planes. The line of intersection of these planes is called the projection axis. We project one point A onto the considered planes using a flat projection. To do this, it is necessary to lower the perpendiculars Aa and A from the given point onto the considered planes.

Projection onto a horizontal plane is called plan view points BUT, and the projection a? on the frontal plane is called front projection.

The points to be projected into descriptive geometry usually denoted using capital Latin letters A, B, C. Small letters are used to designate horizontal projections of points. a, b, c... Frontal projections are indicated in small letters with a stroke at the top a?, b?, c?…

The designation of points with Roman numerals I, II, ... is also used, and for their projections - with Arabic numerals 1, 2 ... and 1?, 2? ...

When the horizontal plane is rotated by 90°, a drawing can be obtained in which both planes are in the same plane (Fig. 5). This picture is called point plot.

Through perpendicular lines Ah and ah? draw a plane (Fig. 4). The resulting plane is perpendicular to the frontal and horizontal planes because it contains perpendiculars to these planes. Therefore, this plane is perpendicular to the line of intersection of the planes. The resulting straight line intersects the horizontal plane in a straight line aa x, and the frontal plane - in a straight line huh? X. Straight aah and huh? x are perpendicular to the axis of intersection of the planes. I.e Aaah? is a rectangle.

When combining the horizontal and frontal projection planes a and a? will lie on one perpendicular to the axis of intersection of the planes, since when the horizontal plane rotates, the perpendicularity of the segments aa x and huh? x is not broken.

We get that on the projection diagram a and a? some point BUT always lie on the same perpendicular to the axis of intersection of the planes.

Two projections a and a? of some point A can uniquely determine its position in space (Fig. 4). This is confirmed by the fact that when constructing a perpendicular from the projection a to the horizontal plane, it will pass through point A. Similarly, the perpendicular from the projection a? to the frontal plane will pass through the point BUT, i.e. point BUT lies on two definite lines at the same time. Point A is their intersection point, i.e. it is definite.

Consider a rectangle Aaa X a?(Fig. 5), for which the following statements are true:

1) Point distance BUT from the frontal plane is equal to the distance of its horizontal projection a from the axis of intersection of the planes, i.e.

ah? = aa X;

2) point distance BUT from the horizontal plane of projections is equal to the distance of its frontal projection a? from the axis of intersection of the planes, i.e.

Ah = huh? X.

In other words, even without the point itself on the plot, using only its two projections, you can find out at what distance from each of the projection planes this point is located.

The intersection of two projection planes divides the space into four parts, which are called quarters(Fig. 6).

The axis of intersection of the planes divides the horizontal plane into two quarters - the front and back, and the frontal plane - into the upper and lower quarters. The upper part of the frontal plane and the anterior part of the horizontal plane are considered as the boundaries of the first quarter.

Upon receipt of the diagram, the horizontal plane rotates and coincides with the frontal plane (Fig. 7). In this case, the front of the horizontal plane will coincide with the bottom of the frontal plane, and the back of the horizontal plane with the top of the frontal plane.

Figures 8-11 show points A, B, C, D, located in different quarters of space. Point A is in the first quarter, point B is in the second, point C is in the third, and point D is in the fourth.

When the points are located in the first or fourth quarters of their horizontal projections located on the front of the horizontal plane, and on the diagram they will lie below the axis of intersection of the planes. When a point is located in the second or third quarter, its horizontal projection will lie on the back of the horizontal plane, and on the diagram it will be above the axis of intersection of the planes.

Front projections points that are located in the first or second quarters will lie on the upper part of the frontal plane, and on the diagram they will be located above the axis of intersection of the planes. When a point is located in the third or fourth quarter, its frontal projection is below the axis of intersection of the planes.

Most often, in real constructions, the figure is placed in the first quarter of the space.

In some particular cases, the point ( E) may lie on a horizontal plane (Fig. 12). In this case, its horizontal projection e and the point itself will coincide. The frontal projection of such a point will be on the axis of the intersection of the planes.

In the case where the point To lies on the frontal plane (Fig. 13), its horizontal projection k lies on the axis of intersection of the planes, and the frontal k? shows the actual location of that point.

For such points, the sign that it lies on one of the projection planes is that one of its projections is on the axis of intersection of the planes.

If a point lies on the intersection axis of the projection planes, it and both of its projections coincide.

When a point does not lie on the projection planes, it is called dot general position . In what follows, if there are no special marks, the point under consideration is a point in general position.

2. Lack of projection axis

To explain how to obtain on the model projections of a point onto perpendicular projection planes (Fig. 4), it is necessary to take a piece of thick paper in the form of an elongated rectangle. It needs to be bent between projections. The fold line will depict the axis of the intersection of the planes. If after that the bent piece of paper is straightened again, we get a diagram similar to the one shown in the figure.

Combining two projection planes with the drawing plane, you can not show the fold line, i.e., do not draw the axis of intersection of the planes on the diagram.

When constructing on a diagram, you should always place projections a and a? point A on one vertical line (Fig. 14), which is perpendicular to the axis of intersection of the planes. Therefore, even if the position of the axis of the intersection of the planes remains undefined, but its direction is determined, the axis of the intersection of the planes can only be perpendicular to the straight line on the diagram ah?.

If there is no projection axis on the point diagram, as in the first figure 14 a, you can imagine the position of this point in space. To do this, draw in any place perpendicular to the line ah? projection axis, as in the second figure (Fig. 14) and bend the drawing along this axis. If we restore the perpendiculars at the points a and a? before they intersect, you can get a point BUT. When changing the position of the projection axis, different positions of the point relative to the projection planes are obtained, but the uncertainty in the position of the projection axis does not affect mutual arrangement several points or figures in space.

3. Projections of a point onto three projection planes

Consider the profile plane of projections. Projections on two perpendicular planes usually determine the position of the figure and make it possible to find out its real dimensions and shape. But there are times when two projections are not enough. Then apply the construction of the third projection.

The third projection plane is carried out so that it is perpendicular to both projection planes at the same time (Fig. 15). The third plane is called profile.

In such constructions, the common line of the horizontal and frontal planes is called axis X , the common line of the horizontal and profile planes - axis at , and the common straight line of the frontal and profile planes - axis z . Dot O, which belongs to all three planes, is called the point of origin.

Figure 15a shows the point BUT and three of its projections. Projection onto the profile plane ( a??) are called profile projection and denote a??.

To obtain a diagram of point A, which consists of three projections a, a a, it is necessary to cut the trihedron formed by all planes along the y axis (Fig. 15b) and combine all these planes with the plane of the frontal projection. The horizontal plane must be rotated about the axis X, and the profile plane is near the axis z in the direction indicated by the arrow in Figure 15.

Figure 16 shows the position of the projections ah, huh? and a?? points BUT, obtained as a result of combining all three planes with the drawing plane.

As a result of the cut, the y-axis occurs on the diagram in two different places. On a horizontal plane (Fig. 16), it takes a vertical position (perpendicular to the axis X), and on the profile plane - horizontal (perpendicular to the axis z).

Figure 16 shows three projections ah, huh? and a?? points A have a strictly defined position on the diagram and are subject to unambiguous conditions:

a and a? must always be located on one vertical straight line perpendicular to the axis X;

a? and a?? must always be located on the same horizontal line perpendicular to the axis z;

3) when drawn through a horizontal projection and a horizontal line, but through a profile projection a??- a vertical straight line, the constructed lines will necessarily intersect on the bisector of the angle between the projection axes, since the figure Oa at a 0 a n is a square.

When constructing three projections of a point, it is necessary to check the fulfillment of all three conditions for each point.

4. Point coordinates

The position of a point in space can be determined using three numbers called its coordinates. Each coordinate corresponds to the distance of a point from some projection plane.

Point distance BUT to the profile plane is the coordinate X, wherein X = huh?(Fig. 15), the distance to the frontal plane - by the coordinate y, and y = huh?, and the distance to the horizontal plane is the coordinate z, wherein z = aA.

In Figure 15, point A occupies the width cuboid, and the dimensions of this box correspond to the coordinates of this point, i.e., each of the coordinates is presented in Figure 15 four times, i.e.:

x \u003d a? A \u003d Oa x \u003d a y a \u003d a z a?;

y \u003d a? A \u003d Oa y \u003d a x a \u003d a z a?;

z = aA = Oa z = a x a? = a y a?.

On the diagram (Fig. 16), the x and z coordinates occur three times:

x \u003d a z a? \u003d Oa x \u003d a y a,

z = a x a? = Oa z = a y a?.

All segments that correspond to the coordinate X(or z) are parallel to each other. Coordinate at represented twice by the vertical axis:

y \u003d Oa y \u003d a x a

and twice - located horizontally:

y \u003d Oa y \u003d a z a?.

This difference appeared due to the fact that the y-axis is present on the diagram in two different positions.

It should be noted that the position of each projection is determined on the diagram by only two coordinates, namely:

1) horizontal - coordinates X and at,

2) frontal - coordinates x and z,

3) profile - coordinates at and z.

Using coordinates x, y and z, you can build projections of a point on the diagram.

If point A is given by coordinates, their record is defined as follows: A ( X; y; z).

When constructing point projections BUT the following conditions must be checked:

1) horizontal and frontal projections a and a? X X;

2) frontal and profile projections a? and a? should be located on the same perpendicular to the axis z, since they have a common coordinate z;

3) horizontal projection and also removed from the axis X, like the profile projection a away from axis z, since the projection ah? and huh? have a common coordinate at.

If the point lies in any of the projection planes, then one of its coordinates is equal to zero.

When a point lies on the projection axis, its two coordinates are zero.

If a point lies at the origin, all three of its coordinates are zero.

A point, as a mathematical concept, has no dimensions. Obviously, if the object of projection is a zero-dimensional object, then it is meaningless to talk about its projection.

Fig.9 Fig.10

In geometry under a point, it is advisable to take a physical object that has linear dimensions. Conventionally, a ball with an infinitely small radius can be taken as a point. With this interpretation of the concept of a point, we can talk about its projections.

When constructing orthogonal projections of a point, one should be guided by the first invariant property of orthogonal projection: the orthogonal projection of a point is a point.

The position of a point in space is determined by three coordinates: X, Y, Z, showing the distances at which the point is removed from the projection planes. To determine these distances, it is enough to determine the meeting points of these lines with the projection planes and measure the corresponding values, which will indicate the values ​​of the abscissa, respectively. X, ordinates Y and appliques Z points (Fig. 10).

The projection of a point is the base of the perpendicular dropped from the point to the corresponding projection plane. Horizontal projection points a call the rectangular projection of a point on the horizontal plane of projections, frontal projection a /- respectively on the frontal plane of projections and profile a // – on the profile projection plane.

Direct Aa, Aa / and Aa // are called projecting lines. At the same time, direct Ah, projecting point BUT on the horizontal plane of projections, called horizontally projecting line, Аa / and Aa //- respectively: frontally and profile-projecting straight lines.

Two projecting lines passing through a point BUT define the plane, which is called projecting.

When converting the spatial layout, the frontal projection of the point A - a / remains in place as belonging to a plane that does not change its position under the considered transformation. Horizontal projection - a together with the horizontal projection plane will turn in the direction of clockwise movement and will be located on one perpendicular to the axis X with front projection. Profile projection - a // will rotate together with the profile plane and by the end of the transformation will take the position indicated in Figure 10. At the same time - a // will be perpendicular to the axis Z drawn from the point a / and will be removed from the axis Z the same distance as the horizontal projection a away from axis X. Therefore, the relationship between horizontal and profile projections points can be set using two orthogonal line segments aa y and a y a // and a conjugating arc of a circle centered at the point of intersection of the axes ( O- origin). The marked connection is used to find the missing projection (for two given ones). The position of the profile (horizontal) projection according to the given horizontal (profile) and frontal projections can be found using a straight line drawn at an angle of 45 0 from the origin to the axis Y(this bisector is called a straight line) k is the Monge constant). The first of these methods is preferable, as it is more accurate.

Therefore:

1. Point in space removed:

from the horizontal plane H Z,

from the frontal plane V by the value of the given coordinate Y,

from profile plane W by the value of the coordinate. x.

2. Two projections of any point belong to the same perpendicular (one connection line):

horizontal and frontal - perpendicular to the axis x,

horizontal and profile - perpendicular to the Y axis,

frontal and profile - perpendicular to the Z axis.

3. The position of a point in space is completely determined by the position of its two orthogonal projections. Therefore - for any two given orthogonal projections point, it is always possible to construct its missing third projection.

If a point has three definite coordinates, then such a point is called point in general position. If a point has one or two coordinates equal to zero, then such a point is called private position point.

Rice. 11 Fig. 12

Figure 11 shows a spatial drawing of points of particular position, Figure 12 shows a complex drawing (diagrams) of these points. Dot BUT belongs to the frontal projection plane, the point AT– horizontal plane of projections, point With– profile plane of projections and point D– abscissa axis ( X).

A short course in descriptive geometry

Lectures are intended for students of engineering and technical specialties

Monge method

If information about the distance of a point relative to the projection plane is given not with the help of a numerical mark, but with the help of the second projection of the point, built on the second projection plane, then the drawing is called two-picture or complex. The basic principles for constructing such drawings are set forth by G. Monge.
The method set forth by Monge - the method of orthogonal projection, and two projections are taken on two mutually perpendicular projection planes - providing expressiveness, accuracy and readability of images of objects on a plane, was and remains the main method for drawing up technical drawings

Figure 1.1 Point in the system of three projection planes

The model of three projection planes is shown in Figure 1.1. The third plane, perpendicular to both P1 and P2, is denoted by the letter P3 and is called the profile plane. The projections of points onto this plane are denoted by capital letters or numbers with the index 3. The projection planes, intersecting in pairs, define three axes 0x, 0y and 0z, which can be considered as a system Cartesian coordinates in space with origin at point 0. Three projection planes divide space into eight trihedral angles - octants. As before, we will assume that the viewer viewing the object is in the first octant. To obtain a diagram, the points in the system of three projection planes of the P1 and P3 planes are rotated until they coincide with the P2 plane. When designating axes on a diagram, negative semiaxes are usually not indicated. If only the image of the object itself is significant, and not its position relative to the projection planes, then the axes on the diagram are not shown. Coordinates are numbers that correspond to a point to determine its position in space or on a surface. In three-dimensional space, the position of a point is set using rectangular Cartesian coordinates x, y, and z (abscissa, ordinate, and applicate).

To determine the position of a straight line in space, there are the following methods: 1. Two points (A and B). Consider two points in space A and B (Fig. 2.1). Through these points we can draw a straight line, we get a segment. In order to find the projections of this segment on the projection plane, it is necessary to find the projections of points A and B and connect them with a straight line. Each of the segment projections on the projection plane is smaller than the segment itself:<; <; <.

Figure 2.1 Determining the position of a straight line from two points

2. Two planes (a; b). This method of setting is determined by the fact that two non-parallel planes intersect in space in a straight line (this method is discussed in detail in the course of elementary geometry).

3. Point and angles of inclination to the projection planes. Knowing the coordinates of a point belonging to the line and its angle of inclination to the projection planes, you can find the position of the line in space.

Depending on the position of the straight line in relation to the projection planes, it can occupy both general and particular positions. 1. A straight line that is not parallel to any projection plane is called a straight line in general position (Fig. 3.1).

2. Straight lines parallel to the projection planes occupy a particular position in space and are called level lines. Depending on which projection plane the given line is parallel to, there are:

2.1. Direct projections parallel to the horizontal plane are called horizontal or contour lines (Fig. 3.2).

Figure 3.2 Horizontal straight line

2.2. Direct projections parallel to the frontal plane are called frontal or frontals (Fig. 3.3).

Figure 3.3 Frontal straight

2.3. Direct projections parallel to the profile plane are called profile projections (Fig. 3.4).

Figure 3.4 Profile straight

3. Straight lines perpendicular to the projection planes are called projecting. A line perpendicular to one projection plane is parallel to the other two. Depending on which projection plane the investigated line is perpendicular to, there are:

3.1. Frontally projecting straight line - AB (Fig. 3.5).

Figure 3.5 Front projection line

3.2. Profile projecting straight line - AB (Fig. 3.6).

Figure 3.6 Profile-projecting line

3.3. Horizontally projecting straight line - AB (Fig. 3.7).

Figure 3.7 Horizontally projecting line

Plane is one of the basic concepts of geometry. In a systematic exposition of geometry, the concept of a plane is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry. Some characteristic properties of a plane: 1. A plane is a surface that completely contains every line connecting any of its points; 2. A plane is a set of points equidistant from two given points.

Ways of graphical definition of planes The position of a plane in space can be determined:

1. Three points that do not lie on one straight line (Fig. 4.1).

Figure 4.1 Plane defined by three points that do not lie on one straight line

2. A straight line and a point that does not belong to this straight line (Fig. 4.2).

Figure 4.2 Plane defined by a straight line and a point not belonging to this line

3. Two intersecting straight lines (Fig. 4.3).

Figure 4.3 Plane defined by two intersecting straight lines

4. Two parallel lines (Fig. 4.4).

Figure 4.4 Plane defined by two parallel straight lines

Different position of the plane relative to the projection planes

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

1. A plane not perpendicular to any projection plane is called a plane in general position. Such a plane intersects all projection planes (has three traces: - horizontal S 1; - frontal S 2; - profile S 3). The traces of the generic plane intersect in pairs on the axes at the points ax,ay,az. These points are called vanishing points, they can be considered as the vertices of the trihedral angles formed by the given plane with two of the three projection planes. Each of the traces of the plane coincides with its projection of the same name, and the other two projections of opposite names lie on the axes (Fig. 5.1).

2. Planes perpendicular to the planes of projections - occupy a particular position in space and are called projecting. Depending on which projection plane the given plane is perpendicular to, there are:

2.1. The plane perpendicular to the horizontal projection plane (S ^ П1) is called the horizontally projecting plane. The horizontal projection of such a plane is a straight line, which is also its horizontal track. Horizontal projections of all points of any figures in this plane coincide with the horizontal trace (Fig. 5.2).

Figure 5.2 Horizontal projection plane

2.2. The plane perpendicular to the frontal plane of projections (S ^ P2) is the front-projecting plane. The frontal projection of the plane S is a straight line coinciding with the trace S 2 (Fig. 5.3).

Figure 5.3 Front projection plane

2.3. The plane perpendicular to the profile plane (S ^ П3) is the profile-projecting plane. A special case of such a plane is the bisector plane (Fig. 5.4).

Figure 5.4 Profile-projecting plane

3. Planes parallel to the planes of projections - occupy a particular position in space and are called level planes. Depending on which plane the plane under study is parallel to, there are:

3.1. Horizontal plane - a plane parallel to the horizontal projection plane (S //P1) - (S ^P2, S ^P3). Any figure in this plane is projected onto the plane P1 without distortion, and on the plane P2 and P3 into straight lines - traces of the plane S 2 and S 3 (Fig. 5.5).

Figure 5.5 Horizontal plane

3.2. Frontal plane - a plane parallel to the frontal projection plane (S //P2), (S ^P1, S ^P3). Any figure in this plane is projected onto the plane P2 without distortion, and on the plane P1 and P3 into straight lines - traces of the plane S 1 and S 3 (Fig. 5.6).

Figure 5.6 Frontal plane

3.3. Profile plane - a plane parallel to the profile plane of projections (S //P3), (S ^P1, S ^P2). Any figure in this plane is projected onto the plane P3 without distortion, and on the plane P1 and P2 into straight lines - traces of the plane S 1 and S 2 (Fig. 5.7).

Figure 5.7 Profile plane

Plane traces

The trace of the plane is the line of intersection of the plane with the projection planes. Depending on which of the projection planes the given one intersects, they distinguish: horizontal, frontal and profile traces of the plane.

Each trace of the plane is a straight line, for the construction of which it is necessary to know two points, or one point and the direction of the straight line (as for the construction of any straight line). Figure 5.8 shows finding traces of the plane S (ABC). The frontal trace of the plane S 2 is constructed as a line connecting two points 12 and 22, which are the frontal traces of the corresponding lines belonging to the plane S . The horizontal trace S 1 is a straight line passing through the horizontal trace of the straight line AB and S x. Profile trace S 3 - a straight line connecting the points (S y and S z) of the intersection of the horizontal and frontal traces with the axes.

Figure 5.8 Construction of plane traces

Determining the relative position of a straight line and a plane is a positional problem, for the solution of which the method of auxiliary cutting planes is used. The essence of the method is as follows: draw an auxiliary secant plane Q through the line and set the relative position of two lines a and b, the last of which is the line of intersection of the auxiliary secant plane Q and this plane T (Fig. 6.1).

Figure 6.1 Auxiliary cutting plane method

Each of the three possible cases of the relative position of these lines corresponds to a similar case of mutual position of the line and the plane. So, if both lines coincide, then the line a lies in the plane T, the parallelism of the lines indicates the parallelism of the line and the plane, and, finally, the intersection of the lines corresponds to the case when the line a intersects the plane T. Thus, there are three cases of the relative position of the line and the plane: belongs to the plane; The line is parallel to the plane; A straight line intersects a plane, a special case - a straight line is perpendicular to the plane. Let's consider each case.

Straight line belonging to the plane

Axiom 1. A line belongs to a plane if two of its points belong to the same plane (fig.6.2).

Task. Given a plane (n,k) and one projection of the line m2. It is required to find the missing projections of the line m if it is known that it belongs to the plane given by the intersecting lines n and k. The projection of the line m2 intersects the lines n and k at points B2 and C2, to find the missing projections of the line, it is necessary to find the missing projections of the points B and C as points lying on the lines n and k, respectively. Thus, the points B and C belong to the plane given by the intersecting lines n and k, and the line m passes through these points, which means that, according to the axiom, the line belongs to this plane.

Axiom 2. A line belongs to a plane if it has one common point with the plane and is parallel to any line located in this plane (Fig. 6.3).

Task. Draw a line m through point B if it is known that it belongs to the plane given by intersecting lines n and k. Let B belong to the line n lying in the plane given by the intersecting lines n and k. Through the projection B2 we draw the projection of the line m2 parallel to the line k2, to find the missing projections of the line, it is necessary to construct the projection of the point B1 as a point lying on the projection of the line n1 and draw the projection of the line m1 through it parallel to the projection k1. Thus, the points B belong to the plane given by the intersecting lines n and k, and the line m passes through this point and is parallel to the line k, which means that, according to the axiom, the line belongs to this plane.

Figure 6.3 A straight line has one common point with a plane and is parallel to a straight line located in this plane

Main lines in the plane

Among the straight lines belonging to the plane, a special place is occupied by straight lines that occupy a particular position in space:

1. Horizontals h - straight lines lying in a given plane and parallel to the horizontal plane of projections (h / / P1) (Fig. 6.4).

Figure 6.4 Horizontal

2. Frontals f - straight lines located in the plane and parallel to the frontal plane of projections (f / / P2) (Fig. 6.5).

Figure 6.5 Frontal

3. Profile straight lines p - straight lines that are in a given plane and parallel to the profile plane of projections (p / / P3) (Fig. 6.6). It should be noted that traces of the plane can also be attributed to the main lines. The horizontal trace is the horizontal of the plane, the frontal is the front and the profile is the profile line of the plane.

Figure 6.6 Profile straight

4. The line of the largest slope and its horizontal projection form a linear angle j, which measures the dihedral angle made up by this plane and the horizontal plane of projections (Fig. 6.7). Obviously, if a line does not have two common points with a plane, then it is either parallel to the plane or intersects it.

Figure 6.7 The line of the largest slope

Mutual position of a point and a plane

There are two options for the mutual arrangement of a point and a plane: either the point belongs to the plane, or it does not. If the point belongs to the plane, then only one of the three projections that determine the position of the point in space can be arbitrarily set. Let's consider an example (fig.6.8): Construction of a projection of a point A belonging to a plane of general position given by two parallel straight lines a(a//b).

Task. Given: the plane T(a,b) and the projection of the point A2. It is required to construct the projection A1 if it is known that the point A lies in the plane c,a. Through the point A2 we draw the projection of the line m2, which intersects the projections of the lines a2 and b2 at the points C2 and B2. Having built the projections of points C1 and B1, which determine the position of m1, we find the horizontal projection of point A.

Figure 6.8. Point belonging to the plane

Two planes in space can either be mutually parallel, in a particular case coinciding with each other, or intersect. Mutually perpendicular planes are a special case of intersecting planes.

1. Parallel planes. Planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane. This definition is well illustrated by the task, through point B, to draw a plane parallel to the plane given by two intersecting straight lines ab (Fig. 7.1). Task. Given: a plane in general position given by two intersecting lines ab and point B. It is required to draw a plane through point B parallel to the plane ab and define it by two intersecting lines c and d. According to the definition, if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane, then these planes are parallel to each other. In order to draw parallel lines on the diagram, it is necessary to use the property of parallel projection - the projections of parallel lines are parallel to each other d||a, c||b; d1||a1,с1||b1; d2||a2 ,с2||b2; d3||a3,с3||b3.

Figure 7.1. Parallel planes

2. Intersecting planes, a special case - mutually perpendicular planes. The line of intersection of two planes is a straight line, for the construction of which it is enough to determine its two points common to both planes, or one point and the direction of the line of intersection of the planes. Consider the construction of the line of intersection of two planes, when one of them is projecting (Fig. 7.2).

Task. Given: a plane in general position is given by a triangle ABC, and the second plane is a horizontally projecting T. It is required to construct a line of intersection of the planes. The solution of the problem is to find two points common to these planes through which a straight line can be drawn. The plane defined by the triangle ABC can be represented as straight lines (AB), (AC), (BC). The point of intersection of the line (AB) with the plane T - point D, the line (AC) -F. The segment defines the line of intersection of the planes. Since T is a horizontally projecting plane, the projection D1F1 coincides with the trace of the plane T1, so it remains only to construct the missing projections on P2 and P3.

Figure 7.2. Intersection of a generic plane with a horizontally projecting plane

Let's move on to the general case. Let two generic planes a(m,n) and b (ABC) be given in space (Fig. 7.3).

Figure 7.3. Intersection of planes in general position

Consider the sequence of constructing the line of intersection of the planes a(m//n) and b(ABC). By analogy with the previous problem, to find the line of intersection of these planes, we draw auxiliary secant planes g and d. Let us find the lines of intersection of these planes with the planes under consideration. Plane g intersects plane a along a straight line (12), and plane b - along a straight line (34). Point K - the point of intersection of these lines simultaneously belongs to three planes a, b and g, being thus a point belonging to the line of intersection of planes a and b. Plane d intersects planes a and b along lines (56) and (7C), respectively, their intersection point M is located simultaneously in three planes a, b, d and belongs to the straight line of intersection of planes a and b. Thus, two points are found belonging to the line of intersection of planes a and b - a straight line (KM).

Some simplification in constructing the line of intersection of planes can be achieved if the auxiliary secant planes are drawn through the straight lines that define the plane.

Mutually perpendicular planes. It is known from stereometry that two planes are mutually perpendicular if one of them passes through a perpendicular to the other. Through the point A, you can draw a set of planes perpendicular to the given plane a (f, h). These planes form a bundle of planes in space, the axis of which is the perpendicular dropped from the point A to the plane a. In order to draw a plane perpendicular to the plane given by two intersecting lines hf from point A, it is necessary to draw a straight line n perpendicular to the plane hf from point A (the horizontal projection n is perpendicular to the horizontal projection of the horizontal h, the frontal projection n is perpendicular to the frontal projection of the frontal f). Any plane passing through the line n will be perpendicular to the plane hf, therefore, to set the plane through points A, we draw an arbitrary line m. The plane given by two intersecting straight lines mn will be perpendicular to the hf plane (Fig. 7.4).

Figure 7.4. Mutually perpendicular planes

Plane-parallel movement method

Changing the relative position of the projected object and the projection planes by the method of plane-parallel movement is carried out by changing the position of the geometric object so that the trajectory of its points is in parallel planes. The carrier planes of the trajectories of moving points are parallel to any projection plane (Fig. 8.1). The trajectory is an arbitrary line. With a parallel transfer of a geometric object relative to the projection planes, the projection of the figure, although it changes its position, remains congruent to the projection of the figure in its original position.

Figure 8.1 Determination of the natural size of the segment by the method of plane-parallel movement

Properties of plane-parallel movement:

1. With any movement of points in a plane parallel to the plane P1, its frontal projection moves along a straight line parallel to the x axis.

2. In the case of an arbitrary movement of a point in a plane parallel to P2, its horizontal projection moves along a straight line parallel to the x axis.

Method of rotation around an axis perpendicular to the projection plane

The carrier planes of the points movement trajectories are parallel to the projection plane. Trajectory - an arc of a circle, the center of which is located on the axis perpendicular to the plane of projections. To determine the natural size of a line segment in general position AB (Fig. 8.2), we choose the axis of rotation (i) perpendicular to the horizontal projection plane and passing through B1. Let's rotate the segment so that it becomes parallel to the frontal projection plane (the horizontal projection of the segment is parallel to the x-axis). In this case, point A1 will move to A "1, and point B will not change its position. The position of point A" 2 is at the intersection of the frontal projection of the trajectory of movement of point A (a straight line parallel to the x axis) and the communication line drawn from A "1. The resulting projection B2 A "2 determines the actual size of the segment itself.

Figure 8.2 Determining the natural size of a segment by rotating around an axis perpendicular to the horizontal plane of projections

Method of rotation around an axis parallel to the projection plane

Consider this method using the example of determining the angle between intersecting lines (Fig. 8.3). Consider two projections of intersecting lines a and in which intersect at point K. In order to determine the natural value of the angle between these lines, it is necessary to transform orthogonal projections so that the lines become parallel to the projection plane. Let's use the method of rotation around the level line - horizontal. Let us draw an arbitrary frontal projection of the horizontal h2 parallel to the Ox axis, which intersects the lines at points 12 and 22 . Having defined the projections 11 and 11, we construct a horizontal projection of the horizontal h1 . The trajectory of movement of all points during rotation around the horizontal is a circle that is projected onto the P1 plane in the form of a straight line perpendicular to the horizontal projection of the horizontal.

Figure 8.3 Determination of the angle between intersecting lines, rotation around an axis parallel to the horizontal projection plane

Thus, the trajectory of the point K1 is determined by the straight line K1O1, the point O is the center of the circle - the trajectories of the point K. To find the radius of this circle, we find the natural value of the segment KO by the triangle method. The point K "1 corresponds to the point K, when the lines a and b lie in a plane parallel to P1 and drawn through the horizontal - the axis of rotation. With this in mind, we draw straight lines through the point K "1 and points 11 and 21, which now lie in a plane parallel to P1, and therefore the angle phi is the natural value of the angle between the lines a and b.

Method for replacing projection planes

Changing the relative position of the projected figure and the projection planes by changing the projection planes is achieved by replacing the P1 and P2 planes with new P4 planes (Fig. 8.4). New planes are selected perpendicular to the old ones. Some projection transformations require a double replacement of projection planes (Figure 8.5). A successive transition from one system of projection planes to another must be carried out by following the following rule: the distance from the new point projection to the new axis must be equal to the distance from the replaced point projection to the replaced axis.

Task 1: Determine the actual size of the segment AB of a straight line in general position (Fig. 8.4). From the property of parallel projection, it is known that a segment is projected onto a plane in full size if it is parallel to this plane. We choose a new projection plane P4, parallel to the segment AB and perpendicular to the plane P1. By introducing a new plane, we pass from the system of planes P1P2 to the system P1P4, and in the new system of planes the projection of the segment A4B4 will be the natural value of the segment AB.

Figure 8.4. Determination of the natural size of a straight line segment by replacing projection planes

Task 2: Determine the distance from point C to a line in general position given by segment AB (Fig. 8.5).

Figure 8.5. Determination of the natural size of a straight line segment by replacing projection planes