A point in space is determined 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 p1 is combined with the negative half-plane p2, the negative part p1 - with the positive part p2.

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 (equidistance, 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.

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.

Octants	Coordinate signs
	x	y	z
1	+	+	+
2	+	–	+
3	+	–	–
4	+	+	–
5	–	+	+
6	–	–	+
7	–	–	–
8	–	+	–

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.

Projecting a point onto three planes of coordinate angle projections begins with obtaining its image on the H plane - the horizontal projection plane. To do this, a projection beam is passed through point A (Fig. 4.12, a) perpendicular to 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 H plane (point a) is chosen arbitrarily. The segment Aa determines at what distance point A is located from the plane H, thereby clearly indicating the position of point A in the figure in relation 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 plane V (Fig. 4.12,b), a projection beam is passed through point A perpendicular to the frontal plane of projections V. In the figure, the perpendicular to plane V is parallel to the Oy axis. On plane H, the distance from point A to plane V will be represented by the segment aa x, parallel to the Oy axis and perpendicular to the Ox axis. If we imagine that the projecting ray and its image are carried out simultaneously in the direction of the plane V, then when the image of the ray intersects the Ox axis at point a x, the ray will intersect the V plane at point a." Drawing from point a x in the V plane a perpendicular to the Ox axis , which is the image of the projecting ray Aa on the plane V, at the intersection with the projecting ray, point a is obtained." 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 projection plane (Fig. 4.12, c) is constructed 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 ray from point A to plane W on plane H will be represented by a segment aa y, parallel to the Ox axis and perpendicular to the Oy axis. From point Oy, parallel to the Oz axis and perpendicular to the Oy axis, an image of the projecting ray aA is constructed and at the intersection with the projecting ray, point a is obtained." Point a" is a profile projection of point A, i.e., an image of point A on the plane W.

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

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

Analysis of the orthogonal drawing of the 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 turned into 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. 4.12c and 4.13). They determine the distance at which point A is located from the profile projection plane. The segments a"a x, a"a y1 and Oa y are equal to the segment Aa, defining the distance from point A to the horizontal projection plane, the segments aa x, a"a z and Oa y 1 are equal to the segment Aa", defining the distance from point A to frontal plane of projections.

The segments Oa x, Oa y and Oa z, located on the projection axes, are a graphic expression of the dimensions of the X, Y and Z coordinates of point A. The coordinates of the point are indicated 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 located 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 ax and a z respectively. The projection connection line connecting the horizontal projection of point A with the profile one turned out to be “cut” at point a y.

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

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

Rice. 4.14. Rice. 4.15.

Let's look at several examples of constructing and reading a drawing of a point.

Example 1. Determination of the coordinates of point J specified on the diagram in two projections (Fig. 4.14). Three segments are measured: segment OB X (X coordinate), segment b X b (Y coordinate) and segment b X b" (Z coordinate). The 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. Constructing a point at given coordinates. Point C is given by coordinates C30; 10; 40. On the Ox axis (Fig. 4.15) find the point c x at which the projection connection line 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. A projection connection line is drawn through this point perpendicular to the Ox axis and the Y coordinate (size 10) is laid down from the point, a point c is obtained - a horizontal projection of point C. The Z coordinate (size 40) is laid up from the point c x along the projection connection line, the point is obtained c" - frontal projection of point C.

Example 3. Construction of a profile projection of a point using given projections. The projections of point D are given - d and d". Through point O, the projection axes Oz, Oy and Оу 1 are drawn (Fig. 4.16, a). To construct a profile projection of point D point d", a projection connection line is drawn perpendicular to the Oz axis and continues it to the right behind the Oz axis. The profile projection of 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 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 define the same distance - the distance from point D to the frontal plane of projections. This distance is the Y coordinate of point D.

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

Transferring the value of the segment d x d to the profile plane of projections can be done using the constant straight line of the 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.

Special cases of location of points relative to projection planes

The position of a point relative to the projection plane is determined by the corresponding coordinate, i.e., the size of the segment of the projection connection line from the Ox axis to the corresponding projection. In 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 Figure 4.17 shows examples of different locations of points relative to projection planes. The Z coordinate of point B is zero, the point is in the H plane. Its frontal projection is on the Ox axis and coincides with the point b x. The Y coordinate of point C is zero, the point is located on plane V, its horizontal projection c is on the Ox axis and coincides with 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.

In Fig. 4.17, the Z and Y coordinates of point D are equal to zero, therefore, point D is located on the Ox projection axis and its two projections coincide.

Projection apparatus

The projection apparatus (Fig. 1) includes three projection planes:

π 1 – horizontal projection plane;

π 2 – frontal plane of projections;

π 3– profile projection plane .

The projection planes are mutually perpendicular ( π 1^ π 2^ π 3), and their intersection lines form the axes:

Intersection of planes π 1 And π 2 form an axis 0X (π 1∩ π 2 = 0X);

Intersection of planes π 1 And π 3 form an axis 0Y (π 1∩ π 3 = 0Y);

Intersection of planes π 2 And π 3 form an axis 0Z (π 2∩ π 3 = 0Z).

The intersection point of the axes (OX∩OY∩OZ=0) is considered the starting point (point 0).

Since the planes and axes are mutually perpendicular, such an apparatus is similar to the Cartesian coordinate system.

The projection planes divide the entire space into eight octants (in Fig. 1 they are indicated by Roman numerals). Projection planes are considered opaque, and the viewer is always in I-th octant.

Orthogonal projection with projection centers S 1, S 2 And S 3 respectively for horizontal, frontal and profile projection planes.

A.

From projection centers S 1, S 2 And S 3 projecting rays come out l 1, l 2 And l 3 A

- A 1 A;

- A 2– frontal projection of a point A;

- A 3– profile projection of a point A.

A point in space is characterized by its coordinates A(x,y,z). Points A x, A y And A z respectively on the axes 0X, 0Y And 0Z show coordinates x, y And z points A. In Fig. 1 gives all the necessary notations and shows the connections between the point A space, its projections and coordinates.

Point diagram

To get a plot of a point A(Fig. 2), in the projection apparatus (Fig. 1) the plane π 1 A 1 0X π 2. Then the plane π 3 with point projection A 3, rotate counterclockwise around the axis 0Z, until it is aligned with the plane π 2. Direction of plane rotations π 2 And π 3 shown in Fig. 1 arrows. At the same time, straight A 1 A x And A 2 A x 0X perpendicular A 1 A 2, and the straight lines A 2 A x And A 3 A x will be located on a common axis 0Z perpendicular A 2 A 3. In what follows we will call these lines respectively vertical And horizontal communication lines.

It should be noted that when moving from the projection apparatus to the diagram, the projected object disappears, but all information about its shape, geometric dimensions and its location in space is preserved.

A(x A , y A , z Ax A , y A And z A in the following sequence (Fig. 2). This sequence is called the method of constructing a point diagram.

1. Axes are drawn orthogonally OX, OY And OZ.

2. On the axis OX xA points A and get the position of the point A x.

3. Through the point A x perpendicular to the axis OX

A x along the axis OY the numerical value of the coordinate is plotted y A points A A 1 on the diagram.

A x along the axis OZ the numerical value of the coordinate is plotted z A points A A 2 on the diagram.

6. Through the point A 2 parallel to the axis OX a horizontal communication line is drawn. The intersection of this line and the axis OZ will give the position of the point A z.

7. On a horizontal communication line from a point A z along the axis OY the numerical value of the coordinate is plotted y A points A and the position of the profile projection of the point is determined A 3 on the diagram.

Characteristics of points

All points in space are divided into points of particular and general positions.

Points of particular position. The points belonging to the projection apparatus are called points of particular position. These include points belonging to projection planes, axes, origins and projection centers. The characteristic features of particular position points are:

Metamathematical – one, two or all numerical coordinate values ​​are equal to zero and (or) infinity;

On a diagram, two or all projections of a point are located on the axes and (or) located at infinity.

Points of general position. Points of general position include points that do not belong to the projection apparatus. For example, dot A in Fig. 1 and 2.

In the general case, the numerical values ​​of the coordinates of a point characterize its distance from the projection plane: coordinate X from the plane π 3; coordinate y from the plane π 2; coordinate z from the plane π 1. It should be noted that the signs for the numerical values ​​of the coordinates indicate the direction in which the point moves away from the projection planes. Depending on the combination of signs with the numerical values ​​of the coordinates of a point, it depends on which octane it is located in.

Two Image Method

In practice, in addition to the full projection method, the two-image method is used. It differs in that this method eliminates the third projection of the object. To obtain the projection apparatus of the two-image method, the profile projection plane with its projection center is excluded from the full projection apparatus (Fig. 3). Moreover, on the axis 0X a reference point is assigned (point 0 ) and from it perpendicular to the axis 0X in projection planes π 1 And π 2 draw axes 0Y And 0Z respectively.

In this device, the entire space is divided into four quadrants. In Fig. 3 they are indicated by Roman numerals.

Projection planes are considered opaque, and the viewer is always in I-th quadrant.

Let's consider the operation of the device using the example of projecting a point A.

From projection centers S 1 And S 2 projecting rays come out l 1 And l 2. These rays pass through the point A and intersecting with the projection planes form its projections:

- A 1– horizontal projection of a point A;

- A 2– frontal projection of a point A.

To get a plot of a point A(Fig. 4), in the projection apparatus (Fig. 3) the plane π 1 with the resulting projection of the point A 1 rotate clockwise around an axis 0X, until it is aligned with the plane π 2. Direction of plane rotation π 1 shown in Fig. 3 arrows. In this case, on the diagram of a point obtained by the method of two images, only one remains vertical communication line A 1 A 2.

In practice, plotting a point A(x A , y A , z A) is carried out according to the numerical values ​​of its coordinates x A , y A And z A in the following sequence (Fig. 4).

1. The axis is drawn OX and a reference point is assigned (point 0 ).

2. On the axis OX the numerical value of the coordinate is plotted xA points A and get the position of the point A x.

3. Through the point A x perpendicular to the axis OX a vertical communication line is drawn.

4. On a vertical communication line from a point A x along the axis OY the numerical value of the coordinate is plotted y A points A and the position of the horizontal projection of the point is determined A 1 OY is not drawn, but it is assumed that its positive values ​​are located below the axis OX, and negative ones are higher.

5. On a vertical communication line from a point A x along the axis OZ the numerical value of the coordinate is plotted z A points A and the position of the frontal projection of the point is determined A 2 on the diagram. It should be noted that in the diagram the axis OZ is not drawn, but it is assumed that its positive values ​​are located above the axis OX, and negative ones are lower.

Competing points

Points on the same projecting beam are called competing points. In the direction of the projecting beam, they have a common projection for them, i.e. their projections are identical. A characteristic feature of competing points on the diagram is the identical coincidence of their projections of the same name. The competition lies in the visibility of these projections relative to the observer. In other words, in space for an observer one of the points is visible, the other is not. And, accordingly, in the drawing: one of the projections of the competing points is visible, and the projection of the other point is invisible.

On the spatial projection model (Fig. 5) from two competing points A And IN visible point A according to two mutually complementary characteristics. Judging by the chain S 1 →A→B dot A closer to the observer than the point IN. And, accordingly, further from the projection plane π 1(those. z A > z A).

Rice. 5 Fig.6

If the point itself is visible A, then its projection is also visible A 1. In relation to the projection coinciding with it B 1. For clarity and, if necessary, on the diagram, invisible projections of points are usually enclosed in brackets.

Let's remove the points on the model A And IN. Their coinciding projections on the plane will remain π 1 and separate projections – on π 2. Let us conditionally leave the frontal projection of the observer (⇩) located in the center of projection S 1. Then, along the chain of images ⇩ → A 2 → B 2 it will be possible to judge that z A > z B and that the point itself is visible A and its projection A 1.

Let us similarly consider competing points WITH And D in appearance relative to the π 2 plane. Since the common projecting beam of these points l 2 parallel to the axis 0Y, then a sign of the visibility of competing points WITH And D determined by inequality y C > y D. Therefore, that point D closed by a dot WITH and accordingly the projection of the point D 2 will be covered by the projection of the point C 2 on surface π 2.

Let's consider how the visibility of competing points in a complex drawing is determined (Fig. 6).

Judging by the coincident projections A 1≡IN 1 the points themselves A And IN are on one projecting beam parallel to the axis 0Z. This means that the coordinates can be compared z A And z B these points. To do this, we use the frontal projection plane with separate images of the points. In this case z A > z B. It follows from this that the projection is visible A 1.

Points C And D in the complex drawing under consideration (Fig. 6) are also on the same projecting beam, but only parallel to the axis 0Y. Therefore, from comparison y C > y D we conclude that projection C 2 is visible.

General rule. Visibility for matching projections of competing points is determined by comparing the coordinates of those points in the direction of a common projection ray. The projection of the point whose coordinate is greater is visible. In this case, the coordinates are compared on the projection plane with separate images of the points.

PROJECTIONS OF A POINT.

ORTHOGONAL SYSTEM OF TWO PLANES OF PROJECTIONS.

The essence of the orthogonal projection method is that an object is projected onto two mutually perpendicular planes by rays orthogonal (perpendicular) to these planes.

One of the projection planes H is placed horizontally, and the second V is placed vertically. Plane H is called the horizontal plane of projections, V is called the frontal plane. The H and V planes are infinite and opaque. The line of intersection of the projection planes is called the coordinate axis and is designated OX. Projection planes divide space into four dihedral angles - quarters.

When considering orthogonal projections, it is assumed that the observer is in the first quarter at an infinitely large distance from the projection planes. Since these planes are opaque, only those points, lines and figures that are located within the same first quarter will be visible to the observer.

When constructing projections, it is necessary to remember that orthogonal projection of a pointthe base of a perpendicular drawn from a given point is called the planeto this plane.

The figure shows a point A and its orthogonal projections a 1 And a 2.

Full stop a 1 called horizontal projection points A, point a 2- her frontal projection. Each of them is the base of a perpendicular drawn from a point A respectively on the plane H And V.

It can be proven that point projectionalways located on straight lines, perpendicularcular axisOH and intersecting this axisat the same point. Indeed, projecting rays Aa 1 And Aa 2 define a plane perpendicular to the projection planes and the line of their intersection - the axis OH. This plane intersects H And V in straight lines a 1 ax And a 1 ax, which form with the axis OX and with each other right angles with the vertex at the point Ax.

The opposite is also true, i.e. if points are given on projection planesa 1 And a 2 , located on straight lines intersecting axis OXat a given point at a right angle,then they are projections of somepoint A. This point is determined by the intersection of perpendiculars constructed from the points a 1 And a 2 to planes H And V.

Note that the position of the projection planes in space may be different. For example, both planes, being mutually perpendicular, can be vertical. But even in this case, the above-proven assumption about the orientation of opposite projections of points relative to the axis remains valid.

To obtain a flat drawing consisting of the above projections, the plane H combined by rotation around an axis OX with plane V, as shown by the arrows in the figure. As a result, the front half-plane H will be aligned with the lower half-plane V, and the back half-plane H- with upper half-plane V.

A projection drawing in which the projection planes with everything that is depicted on them are combined in a certain way with one another is called diagram(from the French epure - drawing). The figure shows a diagram of a point A.

With this method of combining planes H And V projections a 1 And a 2 will be located on the same perpendicular to the axis OX. In this case, the distance a 1 a x — from the horizontal projection of a point to the axis OX A to plane V, and the distance a 2 a x — from the frontal projection of a point to the axis OX equal to the distance from the point itself A to plane H.

Let us agree to call straight lines connecting opposite projections of a point on a diagram projection communication lines.

The position of the projections of points on the diagram depends on which quarter the given point is located in. So, if the point IN located in the second quarter, then after combining the planes both projections will appear to lie above the axis OX.

If the point WITH is in the third quarter, then its horizontal projection, after combining the planes, will be above the axis, and its frontal projection will be below the axis OX. Finally, if the point D is located in the fourth quarter, then both of its projections will be under the axis OX. The figure shows the points M And N, lying on the projection planes. In this position, the point coincides with one of its projections, while its other projection turns out to lie on the axis OX. This feature is also reflected in the designation: near the projection with which the point itself coincides, a capital letter is written without an index.

It should also be noted that the two projections of a point coincide. This will happen if the point is in the second or fourth quarter at the same distance from the projection planes. Both projections are combined with the point itself if the latter is located on the axis OX.

ORTHOGONAL SYSTEM OF THREE PLANES OF PROJECTIONS.

It was shown above that two projections of a point determine its position in space. Since each figure or body is a collection of points, it can be argued that two orthogonal projections of an object (in the presence of letter designations) completely determine its shape.

However, in the practice of depicting building structures, machines and various engineering structures, the need arises to create additional projections. They do this for the sole purpose of making the projection drawing clearer and more readable.

The model of three projection planes is shown in the figure. The third plane, perpendicular and H And V, denoted by the letter W and is called profile.

Projections of points onto this plane will also be called profile, and are designated by capital letters or numbers with index 3 (ah,bh,cz, ...1z, 2z, 3 3...).

The projection planes, intersecting in pairs, define three axes: ABOUTX, ABOUTY And ABOUTZ, which can be considered as a system of rectangular Cartesian coordinates in space with the beginning at point O. The system of signs indicated in the figure corresponds to the “right-handed system” of coordinates.

Three projection planes divide space into eight trihedral angles - these are the so-called octants. The numbering of octants is given in the figure.

To obtain a diagram of the plane H And W rotate as shown in the figure until aligned with the plane V. As a result of rotation, the front half-plane H turns out to be combined with the lower half-plane V, and the back half-plane H- with upper half-plane V. When rotated 90° around an axis ABOUTZ anterior half-plane W aligns with the right half-plane V, and the back half-plane W- with left half-plane V.

The final view of all combined projection planes is given in the figure. In this drawing the axes ABOUTX And ABOUTZ, lying in a fixed plane V, are depicted only once, and the axis ABOUTY shown twice. This is explained by the fact that, rotating with the plane H, axis ABOUTY on the diagram it is combined with the axis ABOUTZ, and rotating with the plane W, the same axis is aligned with the axis ABOUTX.

In the future, when designating axes on the diagram, negative semi-axes (— ABOUTX, — ABOUTY, — ABOUTZ) will not be indicated.

THREE COORDINATES AND THREE PROJECTIONS OF A POINT AND ITS RADIUS-VECTOR.

Coordinates are numbers thatmatch the point to determinechanging its position in space or onsurfaces.

In three-dimensional space, the position of a point is determined using rectangular Cartesian coordinates x, y And z.

Coordinate X called abscissa, at— ordinate And z — applicate. Abscissa X determines the distance from a given point to a plane W, ordinate y - to plane V and applicate z - to plane H. Having adopted the system shown in the figure to measure the coordinates of a point, we will compose a table of coordinate signs in all eight octants. Any point in space A, given by coordinates will be denoted as follows: A(x, y,z).

If x = 5, y = 4 and z = 6, then the entry will take the following form A(5, 4, 6). This point A, all coordinates of which are positive, is in the first octant

Point coordinates A are at the same time the coordinates of its radius vector

OA with respect to the origin. If i, j, k— unit vectors directed respectively along the coordinate axes x, y,z(picture), then

OA =ABOUTA x i+OAyj + OAzk , Where OA X, OA U, OA g - vector coordinates OA

It is recommended to construct an image of the point itself and its projections on a spatial model (figure) using a coordinate rectangular parallelepiped. First of all, on the coordinate axes from the point ABOUT lay down segments correspondingly equal 5, 4 and 6 units of length. On these segments (ABOUTa x , ABOUTa y , ABOUTa z ), as on the edges, a rectangular parallelepiped is built. Its vertex, opposite to the origin, will determine the given point A. It is easy to see that to determine a point A it is enough to construct only three edges of the parallelepiped, for example ABOUTa x , a x a 1 And a 1 A or ABOUTa y , a y a 1 And a 1 A etc. These edges form a coordinate polyline, the length of each link of which is determined by the corresponding coordinate of the point.

However, constructing a parallelepiped allows you to determine not only the point A, but also all three of its orthogonal projections.

Rays projecting a point onto a plane H, V, W are those three edges of the parallelepiped that intersect at the point A.

Each of the orthogonal projections of a point A, being located on a plane, it is determined by only two coordinates.

So, horizontal projection a 1 determined by coordinates X And y, frontal projection a 2 — coordinates x andz, profile projection a 3 — coordinates at And z. But any two projections are determined by three coordinates. That is why specifying a point with two projections is equivalent to specifying a point with three coordinates.

On the diagram (figure), where all projection planes are combined, the projections a 1 And a 2 will be on the same perpendicular to the axis ABOUTX, and projections a 2 And a 3 — on one perpendicular to the axis OZ.

Regarding projections a 1 And a 3 , then they are connected by straight lines a 1 a y And a 3 a y , perpendicular to the axis ABOUTY. But since this axis on the diagram occupies two positions, then the segment a 1 a y cannot be a continuation of a segment a 3 a y .

Constructing point projections A (5, 4, 6) on the diagram according to the given coordinates, perform in the following sequence: first of all, a segment is plotted on the abscissa axis from the origin of coordinates ABOUTa x = x(in our case x =5), then through the point a x draw perpendicular to the axis ABOUTX, on which, taking into account the signs, we plot the segments a x a 1 = y(we get a 1 ) And a x a 2 = z(we get a 2 ). It remains to construct a profile projection of the point a 3 . Since the profile and frontal projections of the point must be located on the same perpendicular to the axis OZ , then through a 3 carry out direct a 2 a z ^ OZ.

Finally, the last question arises: at what distance from the axis ABOUTZ should be a 3 ?

Considering the coordinate parallelepiped (see figure), the edges of which a z a 3 = O a y = a x a 1 = y we conclude that the required distance a z a 3 equals u. Line segment a z a 3 laid to the right of the OZ axis if y>0, and to the left if y

Let's see what changes will occur on the diagram when the point begins to change its position in space.

Let, for example, a point A (5, 4, 6) will move in a straight line perpendicular to the plane V. With such a movement, only one coordinate will change y, showing the distance from a point to a plane V. The coordinates will remain constant x andz , and the projection of the point determined by these coordinates, i.e. a 2 will not change its position.

Regarding projections a 1 And a 3 , then the first one will begin to approach the axis ABOUTX, the second - to the axis ABOUTZ. In the figures, the new position of the point corresponds to the designation a 1 (a 1 1 a 2 1 a 3 1 ). At the moment when the point is on the plane V(y = 0), two of the three projections ( a 1 2 And a 3 2 ) will lie on the axes.

Having moved from I octant in II, the point will begin to move away from the plane V, coordinate at will become negative, its absolute value will increase. Horizontal projection of this point, being located on the back half-plane H, on the diagram it will appear above the axis ABOUTX, and the profile projection, being on the rear half-plane W, on the diagram it will be to the left of the axis ABOUTZ. As always, a segment a za 3 3 = y.

In subsequent diagrams we will not indicate with letters the points of intersection of the coordinate axes with the projection connection lines. This will simplify the drawing to some extent.

In the future, there will be diagrams without coordinate axes. This is what is done in practice when depicting objects, when only the image itself is significanttion of the object, and not its relative positionspecifically projection planes.

The projection planes in this case are determined with an accuracy only up to parallel translation (figure). They are usually moved parallel to themselves in such a way that all points of the object are above the plane H and in front of the plane V. Since the position of the X 12 axis turns out to be uncertain, the formation of the diagram in this case does not need to be associated with the rotation of the planes around the coordinate axis. When moving to the plane diagram H And V are combined so that opposite projections of points are located on vertical lines.

Axis-free diagram of points A and B(drawing) Notdetermines their positions in space,but allows one to judge their relative orientation. Thus, the segment △x characterizes the displacement of the point A relative to point IN in a direction parallel to the H and V planes. In other words, △x indicates how far the point A located to the left of the point IN. The relative displacement of a point in the direction perpendicular to the plane V is determined by the segment △y, i.e. the point And in in our example closer to the observer than the point IN, to a distance equal to △y.

Finally, the segment △z shows the excess of the point A above the point IN.

Proponents of axle-free study of a course in descriptive geometry rightly point out that when solving many problems one can do without coordinate axes. However, complete abandonment of them cannot be considered advisable. Descriptive geometry is designed to prepare the future engineer not only for competent execution of drawings, but also for solving various technical problems, among which problems of spatial statics and mechanics occupy not the least place. And for this it is necessary to develop the ability to orient this or that object relative to the Cartesian coordinate axes. These skills will also be necessary when studying such sections of descriptive geometry as perspective and axonometry. Therefore, on a number of diagrams in this book we save images of coordinate axes. Such drawings determine not only the shape of the object, but also its location relative to the projection planes.