The distance of a point from the horizontal projection plane is called. Construction of orthogonal projections of points

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 a, a 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 a, a 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.

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 = a˝A(Fig. 15), the distance to the frontal plane - by the coordinate y, and y = aa, and the distance to the horizontal plane is the coordinate z, wherein z = aA.

In Figure 15, point A occupies the width of a rectangular box, and the measurements 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 = a˝A = Oa x = a y a = a z á;

y = а́А = Оа y = a x a = 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 á = 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 projections a′ and a˝ 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.

Projection of a straight line

Two points are needed to define a line. A point is defined by two projections on the horizontal and frontal planes, i.e., a straight line is determined using the projections of its two points on the horizontal and frontal planes.

Figure 17 shows projections ( a and a, b and b) two points BUT and B. With their help, the position of some straight line AB. When connecting the same-name projections of these points (i.e. a and b, a and b) you can get projections ab and ab direct AB.

Figure 18 shows the projections of both points, and Figure 19 shows the projections of a straight line passing through them.

If the projections of a straight line are determined by the projections of its two points, then they are denoted by two adjacent Latin letters corresponding to the designations of the projections of points taken on the straight line: with strokes to indicate the frontal projection of the straight line or without strokes - for the horizontal projection.

If we consider not individual points of a straight line, but its projections as a whole, then these projections are indicated by numbers.

If some point With lies on a straight line AB, its projections с and с́ are on the projections of the same line ab and ab. Figure 19 illustrates this situation.

Straight traces

trace straight- this is the point of its intersection with some plane or surface (Fig. 20).

Horizontal track straight some point is called H where the line meets the horizontal plane, and frontal- dot V, in which this straight line meets the frontal plane (Fig. 20).

Figure 21a shows the horizontal trace of a straight line, and its frontal trace, in Figure 21b.

Sometimes the profile trace of a straight line is also considered, W- the point of intersection of a straight line with a profile plane.

The horizontal trace is in the horizontal plane, i.e. its horizontal projection h coincides with this trace, and the frontal h lies on the x-axis. The frontal trace lies in the frontal plane, so its frontal projection ν́ coincides with it, and the horizontal v lies on the x-axis.

So, H = h, and V= v. Therefore, to denote traces of a straight line, letters can be used h and v.

Various positions of the line

The straight line is called direct general position, if it is neither parallel nor perpendicular to any of the projection planes. The projections of a line in general position are also neither parallel nor perpendicular to the projection axes.

Straight lines that are parallel to one of the projection planes (perpendicular to one of the axes). Figure 22 shows a straight line that is parallel to the horizontal plane (perpendicular to the z-axis), is a horizontal straight line; figure 23 shows a straight line that is parallel to the frontal plane (perpendicular to the axis at), is the frontal straight line; figure 24 shows a straight line that is parallel to the profile plane (perpendicular to the axis X), is a profile straight line. Despite the fact that each of these lines forms a right angle with one of the axes, they do not intersect it, but only intersect with it.

Due to the fact that the horizontal line (Fig. 22) is parallel to the horizontal plane, its frontal and profile projections will be parallel to the axes that define the horizontal plane, i.e., the axes X and at. Therefore projections ab|| X and a˝b˝|| at z. The horizontal projection ab can take any position on the plot.

At the frontal line (Fig. 23) projection ab|| x and a˝b˝ || z, i.e. they are perpendicular to the axis at, and therefore in this case the frontal projection ab the line can take any position.

At the profile line (Fig. 24) ab|| y, ab|| z, and both are perpendicular to the x-axis. Projection a˝b˝ can be placed on the diagram in any way.

When considering the plane that projects the horizontal line onto the frontal plane (Fig. 22), you can see that it projects this line onto the profile plane as well, i.e. it is a plane that projects the line onto two projection planes at once - the frontal and profile. For this reason it is called doubly projecting plane. In the same way, for the frontal line (Fig. 23), the doubly projecting plane projects it onto the planes of the horizontal and profile projections, and for the profile (Fig. 23) - onto the planes of the horizontal and frontal projections.

Two projections cannot define a straight line. Two projections 1 and one profile straight line (Fig. 25) without specifying the projections of two points of this straight line on them will not determine the position of this straight line in space.

In a plane that is perpendicular to two given planes of symmetry, there may be an infinite number of lines for which the data on the diagram 1 and one are their projections.

If a point is on a line, then its projections in all cases lie on the projections of the same name on this line. The opposite situation is not always true for the profile line. On its projections, you can arbitrarily indicate the projections of a certain point and not be sure that this point lies on a given line.

In all three special cases (Fig. 22, 23 and 24), the position of the straight line with respect to the plane of projections is its arbitrary segment AB, taken on each of the straight lines, is projected onto one of the projection planes without distortion, that is, onto the plane to which it is parallel. Line segment AB horizontal straight line (Fig. 22) gives a life-size projection onto a horizontal plane ( ab = AB); line segment AB frontal straight line (Fig. 23) - in full size on the plane of the frontal plane V ( ab = AB) and the segment AB profile straight line (Fig. 24) - in full size on the profile plane W (a˝b˝\u003d AB), i.e. it is possible to measure the actual size of the segment on the drawing.

In other words, with the help of diagrams, one can determine the natural dimensions of the angles that the line under consideration forms with the projection planes.

The angle that a straight line makes with a horizontal plane H, it is customary to denote the letter α, with the frontal plane - the letter β, with the profile plane - the letter γ.

Any of the straight lines under consideration has no trace on a plane parallel to it, i.e., the horizontal straight line has no horizontal trace (Fig. 22), the frontal straight line has no frontal trace (Fig. 23), and the profile straight line has no profile trace (Fig. 24 ).

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.

Points that are to be projected in descriptive geometry are 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 plot 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 point of 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 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 of the position of the projection axis does not affect the relative position of several points or figures in space.

3. Projections of a point onto three projection planes

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.

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

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.

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.

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.

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 coordinates of a point, build a third projection, determine the octant in which it is located. Let's consider some typical tasks from the course of descriptive geometry.

According to the 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). The horizontal projection of point A is point A ", having coordinates x, y. Draw from point A" perpendiculars to the x, y axes and find, respectively, A x, A y. 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 x-axis values. Taking into account the scale of the drawing, we find x \u003d 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 y-axis values. Given the scale of the drawing, y = -30. The frontal projection of point A - point A"" has x and z coordinates. 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. Given the scale of the drawing, 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 \u003d B" and the coordinate B y \u003d 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 scale of the drawing, x = 30. The frontal projection of the point B - point B˝ has the coordinates x, z. 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 B coordinates are (30, 0, -20). All necessary constructions are shown in the figure below.

Construction of projections of points

Points A and B in the P 3 plane 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. In the same way, B"" and B""" lie on a common perpendicular to the z-axis. To find the profile projection of t. A, we set aside along the y-axis the value of the corresponding coordinate found earlier. In the figure, this is done using an arc of a circle of radius A y O. After that, we draw a perpendicular from A y to the intersection with the perpendicular restored from the 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, it is only necessary to draw a perpendicular from B"" to the z-axis. The point of intersection of this perpendicular with the z-axis is B """.

Determining the position of points in space

Visually imagining a spatial layout composed of projection planes P 1, P 2 and P 3, the location of octants, as well as the order of transformation of the layout into diagrams, you can directly determine that t. A is located in octant III, and t. 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). The positive abscissa x makes it possible 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 of z indicates that point A is in the third octant. The given reasoning is clearly illustrated by the following table.

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

Point B coordinates (30, 0, -20). Since the ordinate of t. B is equal to zero, this point is located in the projection plane П 2 . The positive abscissa and the negative applicate of point 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 the frontal isometric projection, we built a spatial layout of the third 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 full size without distortion.

The construction of a visual image of point A (10, -30, -10) will begin with its horizontal projection A ". Having set aside the corresponding coordinates along the abscissa and ordinates, we find the points A x and A y. The intersection of perpendiculars restored from A x and A y respectively to the x and y axes determines the position of point A". Putting from A" parallel to the z axis towards its negative values the segment AA", whose length is equal to 10, we find the position of point A.

A visual image of point B (30, 0, -20) is constructed in a similar way - in the P 2 plane, the corresponding coordinates must be plotted along the x and z axes. The intersection of the perpendiculars reconstructed from B x and B z will determine the position of point B.

Auxiliary line of multidrawing

In the drawing shown in fig. 4.7, a, projection axes are drawn, and the images are interconnected by communication lines. Horizontal and profile projections are connected by communication lines using arcs centered at a point O axis intersections. However, in practice, another implementation of the integrated drawing is also used.

On axisless drawings, images are also placed in a projection relationship. However, the third projection can be placed closer or further away. For example, a profile projection can be placed to the right (Fig. 4.7, b, II) or to the left (Fig. 4.7, b, I). This is important for saving space and ease of sizing.

Rice. 4.7.

If in a drawing made according to an axisless system it is required to draw communication lines between the top view and the left view, then an auxiliary straight line of the complex drawing is used. To do this, approximately at the level of the top view and slightly to the right of it, a straight line is drawn at an angle of 45 ° to the drawing frame (Fig. 4.8, a). It is called the auxiliary line of the complex drawing. The procedure for constructing a drawing using this straight line is shown in fig. 4.8, b, c.

If three views have already been built (Fig. 4.8, d), then the position of the auxiliary line cannot be chosen arbitrarily. First you need to find the point through which it will pass. To do this, it is enough to continue until the mutual intersection of the axis of symmetry of the horizontal and profile projections and through the resulting point k draw a straight line segment at an angle of 45 ° (Fig. 4.8, d). If there are no axes of symmetry, then continue until the intersection at the point k 1 horizontal and profile projections of any face projected as a straight line (Fig. 4.8, d).

Rice. 4.8.

The need to draw communication lines, and, consequently, an auxiliary straight line, arises when constructing missing projections and when performing drawings on which it is necessary to determine the projections of points in order to clarify the projections of individual elements of the part.

Examples of the use of the auxiliary line are given in the next paragraph.

Projections of a point lying on the surface of an object

In order to correctly build projections of individual elements of a part when making drawings, it is necessary to be able to find projections of individual points on all drawing images. For example, it is difficult to draw a horizontal projection of the part shown in Fig. 4.9 without using the projections of individual points ( A, B, C, D, E and etc.). The ability to find all the projections of points, edges, faces is also necessary for recreating in the imagination the shape of an object according to its flat images in the drawing, as well as for checking the correctness of the completed drawing.

Rice. 4.9.

Let's consider ways of finding the second and third projections of a point given on the surface of an object.

If one projection of a point is given in the drawing of an object, then first it is necessary to find the projections of the surface on which this point is located. Then choose one of the two methods described below for solving the problem.

First way

This method is used when at least one of the projections shows the given surface as a line.

On fig. 4.10, a a cylinder is shown, on the frontal projection of which the projection is set a" points BUT, lying on the visible part of its surface (given projections are marked with double colored circles). To find the horizontal projection of a point BUT, they argue as follows: the point lies on the surface of the cylinder, the horizontal projection of which is a circle. This means that the projection of a point lying on this surface will also lie on the circle. Draw a line of communication and mark the desired point at its intersection with the circle a. third projection a"

Rice. 4.10.

If the point AT, lying on the upper base of the cylinder, given by its horizontal projection b, then the communication lines are drawn to the intersection with straight line segments depicting the frontal and profile projections of the upper base of the cylinder.

On fig. 4.10, b shows the detail - emphasis. To construct projections of a point BUT, given by its horizontal projection a, find two other projections of the upper face (on which lies the point BUT) and, drawing the connection lines to the intersection with the line segments depicting this face, determine the desired projections - points a" and a". Dot AT lies on the left side vertical face, which means that its projections will also lie on the projections of this face. So from a given point b" draw communication lines (as indicated by arrows) until they meet with line segments depicting this face. frontal projection with" points WITH, lying on an inclined (in space) face, are found on the line depicting this face, and the profile with"- at the intersection of the connection line, since the profile projection of this face is not a line, but a figure. Construction of point projections D shown by arrows.

Second way

This method is used when the first method cannot be used. Then you should do this:

draw through the given projection of the point the projection of the auxiliary line located on the given surface;
find the second projection of this line;
to the found projection of the line, transfer the given projection of the point (this will determine the second projection of the point);
find the third projection (if required) at the intersection of communication lines.

On fig. 4.10, frontal projection is given a" points BUT, lying on the visible part of the surface of the cone. To find the horizontal projection through a point a" carry out a frontal projection of an auxiliary straight line passing through the point BUT and the top of the cone. Get a point V is the projection of the meeting point of the drawn line with the base of the cone. Having frontal projections of points lying on a straight line, one can find their horizontal projections. Horizontal projection s the top of the cone is known. Dot b lies on the circumference of the base. A line segment is drawn through these points and a point is transferred to it (as shown by the arrow). a", getting a point a. Third projection a" points BUT located at the crossroads.

The same problem can be solved differently (Fig. 4.10, G).

As an auxiliary line passing through a point BUT, they take not a straight line, as in the first case, but a circle. This circle is formed if at the point BUT intersect the cone with a plane parallel to the base, as shown in the visual representation. The frontal projection of this circle will be depicted as a straight line segment, since the plane of the circle is perpendicular to the frontal projection plane. The horizontal projection of a circle has a diameter equal to the length of this segment. Describing a circle of the specified diameter, draw from a point a" connection line to the intersection with the auxiliary circle, since the horizontal projection a points BUT lies on the auxiliary line, i.e. on the constructed circle. third projection as" points BUT found at the intersection of communication lines.

In the same way, you can find the projections of a point lying on a surface, for example, a pyramid. The difference will be that when it is crossed by a horizontal plane, not a circle is formed, but a figure similar to the base.

With rectangular projection, the system of projection planes consists of two mutually perpendicular projection planes (Fig. 2.1). One agreed to be placed horizontally, and the other vertically.

The plane of projections, located horizontally, is called horizontal projection plane and denote sch, and the plane perpendicular to it frontal projection planel 2 . The system of projection planes itself is denoted p / p 2. Usually use abbreviated expressions: plane L[, plane n 2 . Line of intersection of planes sch and to 2 called projection axisOH. It divides each projection plane into two parts - floors. The horizontal plane of projections has an anterior and posterior floors, while the frontal plane has an upper and lower floor.

planes sch and p 2 divide the space into four parts called quarters and denoted by Roman numerals I, II, III and IV (see Fig. 2.1). The first quarter is called the part of space bounded by the upper hollow frontal and front hollow horizontal projection planes. For the remaining quarters of the space, the definitions are similar to the previous one.

All engineering drawings are images built on the same plane. On fig. 2.1 the system of projection planes is spatial. To move to images on the same plane, we agreed to combine the projection planes. Usually plane p 2 left motionless, and the plane P turn in the direction indicated by the arrows (see Fig. 2.1), around the axis OH at an angle of 90 ° until it is aligned with the plane n 2 . With such a turn, the front floor of the horizontal plane goes down, and the back one rises. After alignment, the planes have the form depicted

female in fig. 2.2. It is believed that the projection planes are opaque and the observer is always in the first quarter. On fig. 2.2, the designation of planes invisible after alignment is taken in brackets, as is customary for highlighting invisible figures in the drawings.

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

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

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

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

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

The drawing in fig. 2.4, a can be greatly simplified. The designations of the combined projection planes in the drawings are not marked and the rectangles conditionally limiting the projection planes are not depicted, since the planes are unlimited. Simplified point drawing BUT(Fig. 2.4, b) also called diagram(From French ?pure - drawing).

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

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

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

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

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

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

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

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