Consider the projection of points on 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. On the considered planes, we project one point A using a plane projection. To do this, it is necessary to lower the perpendiculars Aa and A from this point to the considered planes.

The projection onto the horizontal plane is called horizontal projection points A and the projection a? on the frontal plane is called frontal projection.

The points that are to be projected are usually denoted in descriptive geometry using large Latin letters. A, B, C... Small letters are used to denote horizontal projections of points. a, b, c... Frontal projections are indicated by 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 you turn the horizontal plane by 90 °, you can get a drawing in which both planes are in the same plane (Fig. 5). This picture is called point plot.

Through perpendicular lines Aa and Huh? 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 NS. Straight aah and huh x are perpendicular to the axis of intersection of the planes. That is Aaah? is a rectangle.

When combining the horizontal and frontal projection planes a and a? will lie on the same 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 will not be violated.

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

Two projections a and a? 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. In the same way, the perpendicular from the projection a? to the frontal plane will pass through the point A, i.e. point A is located simultaneously on two definite lines. Point A is their intersection point, that is, it is definite.

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

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

Huh? = aa NS;

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

Aa = huh NS.

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

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 - front and back, and the frontal plane - into upper and lower quarters. The upper part of the frontal plane and the front part of the horizontal plane are considered to be the boundaries of the first quarter.

When receiving the plot, the horizontal plane rotates and is aligned with the frontal plane (Fig. 7). In this case, the front part of the horizontal plane will coincide with the lower part of the frontal plane, and the back part of the horizontal plane - with the upper part 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 in the second, point C in the third and point D in the fourth.

When the points are located in the first or fourth quarters, their horizontal projections are on the front of the horizontal plane, and on the plot 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.

Frontal projections points that are located in the first or second quarters will lie on the upper part of the frontal plane, and on the plot will be 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 special cases, the point ( E) can 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 located on the axis of intersection of the planes.

In the case when 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 this point.

For such points, a 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 axis of intersection 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 clarify the receipt of projections of a point on the model perpendicular to the projection plane (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 represent the axis of intersection of the planes. If, after that, the folded piece of paper is straightened again, we get a diagram similar to the one shown in the figure.

By combining two projection planes with the drawing plane, you can not show the bend line, that is, do not draw the axis of intersection of the planes on the plot.

When constructing on a plot, 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 intersection of the planes can be on the plot only perpendicular to the straight line ah?.

If there is no projection axis on the plot of a point, as in the first figure 14 a, you can represent the position of this point in space. To do this, draw anywhere perpendicular to the straight line ah? the 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 A... When you change 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 multiple points or shapes in space.

3. Projections of a point onto three projection planes

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

The third projection plane is drawn so that it is perpendicular to both projection planes simultaneously (Fig. 15). The third plane is usually called profile.

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

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

To obtain a plot 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 frontal projection plane. The horizontal plane must be rotated about the axis NS, and the profile plane is about the axis z in the direction indicated by the arrow in Figure 15.

Figure 16 shows the position of the projections huh? and a?? points A, resulting from the alignment of all three planes with the plane of the drawing.

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

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

a and a? should always be located on the same vertical line perpendicular to the axis NS;

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

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

Defined point distance A to the profile plane is the coordinate NS, wherein NS = huh?(Fig. 15), the distance to the frontal plane is 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 of a rectangular parallelepiped, and the measurements of this parallelepiped correspond to the coordinates of this point, i.e., each of the coordinates is shown in Figure 15 four times, i.e.:

x = a? A = Oa x = a y a = a z a ?;

y = a? A = Oa 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 = a z a? = Oa x = a y a,

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

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

y = Oa y = a x a

and two times - located horizontally:

y = Oa y = a z a ?.

This difference appeared due to the fact that the y-axis is present on the plot 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 NS 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 plot.

If point A is specified by coordinates, their record is determined as follows: A ( NS; y; z).

When constructing projections of the point A you need to check the fulfillment of the following conditions:

1) horizontal and frontal projection a and a? NS NS;

2) frontal and profile projection a? and a? must 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 NS like a profile projection a removed from the axis z since the projection ah? and huh? have a common coordinate at.

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

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

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

In some cases, for the convenience of solving problems, it is necessary to use additional projection planes perpendicular to the existing projection planes.

If horizontal and frontal projections of a point are specified, then the profile projection is determined by the following algorithm.

We draw a line of projection connection perpendicular to the axis Oz.

On this line of projection communication, we postpone the segment A 1 A X = A Z A 3 .

Using this rule, it is possible to construct projections of points onto additional projection planes (the method of replacing planes).

Let a point be given A (A 2 ,A 1 ) and a new additional projection plane NS 4 NS 1 . Build A 4 - point projection A on NS 4 .

Solution

a) We build a line of intersection of planes NS 1 and NS 4 = x 1,4 ;

b) Through point A we draw a line of projection communication x 1,4 .

c) Build a projection A 4 , I use the equality of the segments A 2 A X = A 4 A X .

Two point projections A 1 and A 4 lie on one line of the projection connection perpendicular to the axis X 1,4 .

Distance from the “new” point projection A 4 to the “new” axis x 1,4 is equal to the distance from the “old” projection of the point A 2 to the "old" axis x 1,2 .

Competing points

Competing points call a pair of points lying on one projection ray.

Of the two competing points, the visible point is the point that is farther from the projection plane.

Points A and V called horizontally competing.

Points WITH and D are called frontally competing.

Introduce an additional plane so that the points A and V became competitive.

Solution plan:

1 Building an axis x 1,4 A 1 , B 1 ;

2 We build a line of projection communication x 1,4 ;

3 On the line of projection communication, we postpone the segments A x A 2 = A / x A 4 , B x B 2 = B / x B 4 .

Self-study material Modeling 2D graphics objects in the compass graphics system Starting the compass system and shutting down

The KOMPAS-3D-V8 system starts up in the same way as other programs. To start the system, select the menu \ Start\ All Pprograms\ ASCON \KOMPAS-3D- V8 and run COMPASS... You can select the program shortcut on the desktop field with the mouse pointer and double-click the left mouse button. To open the document, you must click the button Open on the panel Standard ... To start a new document press the button Create on the panel Standard or run the command File > Create and in the dialog box that opens, select the type of document to be created and click OK.

To complete the work, select the menu File\Output, the Alt-F4 key combination, or click the Close button.

Basic types of compass graphics documents

The type of document created in the KOMPAS system depends on the type of information stored in this document. Each document type has a file name extension and its own icon.

1 Drawing- the main type of graphic document in KOMPAS. The drawing contains a graphic image of the product in one or more views, a title block, a frame. The KOMPAS drawing always contains one sheet of a user-defined format. The drawing file has the extension .cdw.

2 Fragment- auxiliary type of graphic document in KOMPAS. The fragment differs from the drawing by the absence of a frame, title block and other design objects of the design document. The fragments store the created standard solutions for later use in other documents. The snippet file has the extension .frw.

3 Text Document(file extension . kdw);

4 Specification(file extension . spw);

5 Assembly(file extension . a3 d);

6 Detail- 3D modeling (file extension . m3 d);

POINT PROJECTION.

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 projection plane, V - frontal. The H and V planes are infinite and opaque. The line of intersection of the projection planes is called the coordinate axis and is denoted OX. The projection planes divide the space into four dihedral angles - quarters.

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 building projections, it must be remembered that point orthogonal projectionon the plane is called the base of the perpendicular dropped from a given pointonto this plane.

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

Point a 1 are called horizontal projection points A, point a 2- her frontal projection... Each of them is the base of the perpendicular dropped from the point A respectively on the plane H and V.

It can be proved that point projectionalways located on straight lines, perpendicurly axesOH and crossing this axisat the same point. Indeed, the projecting rays Aa 1 and Aa 2 define a plane perpendicular to the projection planes and the line of their intersection - axes OH. This plane crosses H and V by direct a 1 ax and a 1 ax, which form with the axis OX and right angles to each other with apex at the point ax.

The converse is also true, i.e. if points are given on the 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 the perpendiculars retrieved from the points a 1 and a 2 to the planes H and V.

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

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

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

With this method of aligning the 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 the point to the axis OX A to plane V and the distance a 2 a x — from the frontal projection of the point to the axis OX is equal to the distance from the point itself A to plane H.

Straight lines connecting opposite projections of a point on the plot, we agree to call projection communication lines.

The position of the projections of points on the plot depends on which quarter the given point is in. So if the point V is located in the second quarter, then after aligning the planes, both projections will be lying above the axis OX.

If point WITH is in the third quarter, then its horizontal projection, after aligning the planes, will be above the axis, and the frontal one will be below the axis OX. Finally, if the point D is located in the fourth quarter, then both projections of it 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 reflected in the designation: near the projection with which the point itself coincides, a capital letter is written without an index.

It should be noted that both projections of a point coincide. This will be the case if the point is in the second or fourth quarter at the same distance from the projection planes. Both projections are aligned 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 practice, images building structures, machines and various engineering structures, it becomes necessary to create additional projections. They do this for the sole purpose of making the projection drawing clearer, more readable.

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

The projections of points onto this plane will also be called profile, and denote them by capital letters or numbers with an index of 3 (ah,bh,cs, ...1h, 2h, 3 3 ...).

The projection planes, intersecting in pairs, define three axes: OX, OY and OZ, which can be considered as a system of rectangular Cartesian coordinates in space with the origin at point O. The system of signs indicated in the figure corresponds to the "right system" of coordinates.

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

To get a plot of a 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 aligned with the lower half-plane V, and the back half-plane H- with the upper half-plane V... When rotated 90 ° around the axis OZ front half-plane W will be aligned with the right half-plane V, and the back half-plane W- with the left half-plane V.

The final view of all aligned projection planes is shown in the figure. In this drawing, the axes OX and OZ, lying in a non-movable plane V, are shown only once, and the axis OY shown twice. This is explained by the fact that, rotating with the plane H, axis OY on the plot is aligned with the axis OZ, while rotating with the plane W, the same axis is aligned with the axis OX.

In the future, when designating the axes on the diagram, the negative semiaxes (- OX, — OY, — OZ) will not be specified.

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

Coordinates are numbers thatmatch the point for the definitionits position in space or onsurface.

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

Coordinate NS are called abscissa, at— ordinate and z — applicate. Abscissa NS determines the distance from a given point to the plane W, ordinate y - to plane V and applicate z - to plane H... Having adopted the system shown in the figure for the reference point coordinates, 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 record will take the following form A(5, 4, 6). This point A, all coordinates of which are positive are in the first octant

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

OA relative to the origin. If i, j, k- unit vectors, respectively directed along the coordinate axes x, y,z(figure), then

OA =OA 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 O set aside segments, respectively equal 5, 4 and 6 units of length. On these segments (Oa x , Oa y , Oa z ), like on the edges, build rectangular parallelepiped... Its vertex, opposite to the origin, will determine the given point A. It is easy to see that to define a point A it is enough to construct only three edges of the parallelepiped, for example Oa x , a x a 1 and a 1 A or Oa y , a y a 1 and a 1 A and so on. These edges form a coordinate polyline, the length of each link of which is determined by the corresponding coordinate of the point.

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

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

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

So, horizontal projection a 1 defined by coordinates NS and y, frontal projection a 2 - coordinates x andz, profile projection a 3 — coordinates at and z... But any two projections are defined by three coordinates. This 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 aligned, projections a 1 and a 2 will be on the same perpendicular to the axis OX, and the projection a 2 and a 3 — on one perpendicular to the axis OZ.

As for projections a 1 and a 3 , then they are also connected by straight lines a 1 a y and a 3 a y , perpendicular to the axis OY. But since this axis occupies two positions on the diagram, the segment a 1 a y cannot be a continuation of the segment a 3 a y .

Point projection A (5, 4, 6) on the plot along the given coordinates, perform in the following sequence: first of all, a segment is laid on the abscissa axis from the origin of coordinates Oa x = x(in our case x =5), then through the point a x draw perpendicular to the axis OX, on which, taking into account the signs, we postpone 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 projection of the points must be located on the same perpendicular to the axis OZ , then through a 3 conduct a direct a 2 a z ^ OZ.

Finally, the last question arises: at what distance from the axis OZ should there 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 at. Section a z a 3 laid to the right of the axis OZ, 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, point A (5, 4, 6) will move in a straight line perpendicular to the plane V... With this movement, only one coordinate will change y, showing the distance from a point to a plane V... Coordinates will remain constant x andz , and the projection of the point defined by these coordinates, i.e. a 2 will not change its position.

As for projections a 1 and a 3 , then the first will begin to approach the axis OX, the second - to the axis OZ. In the figures, the new position of the point corresponds to the designations a 1 (a 1 1 a 2 1 a 3 1 ). The moment 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.

Moving from I octant in II, the point will start moving away from the plane V, coordinate at becomes negative, its absolute value will increase. The horizontal projection of this point, being located on the back half-plane H, on the diagram will be above the axis OX, and the profile projection, being on the back half-plane W, on the plot will be to the left of the axis OZ. As always, the segment a za 3 3 = y.

In the subsequent plots, we will not denote by letters the points of intersection of the coordinate axes with the lines of the projection connection. This will simplify the drawing to some extent.

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

In this case, the projection planes 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 undefined, 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 switching to a plot of a plane H and V are combined so that opposite projections of points are located on vertical lines.

Axleless plot of points A and B(drawing) notdetermines their position in space,but allows one to judge their relative orientation. So, the segment △ x characterizes the displacement of the point A with respect to the point V in the direction parallel to the planes H and V. In other words, △ x indicates how much the point A located to the left of the point V. The relative offset of the point in the direction perpendicular to the plane V, is defined by the segment △ y, i.e., the point And in our example is closer to the observer than the point V, by a distance equal to △ y.

Finally, the segment △ z shows the elevation of the point A over point V.

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

Projection apparatus

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

π 1 - horizontal plane of projections;

π 2 - frontal projection plane;

π 3- profile plane of projections .

The projection planes are located mutually perpendicular ( π 1^ π 2^ π 3), and their lines of intersection 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 point of intersection of the axes (ОХ∩OY∩OZ = 0) is considered the origin point (point 0).

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

The projection planes are divided into eight octants (in Fig. 1 they are designated by Roman numerals). The 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 the horizontal, frontal and profile projection planes.

A.

From the centers of projection S 1, S 2 and S 3 the projection beams come out l 1, l 2 and l 3 A

- A 1 A;

- A 2- frontal projection of a point A;

- A 3- point profile projection 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 designations and shows the connections between the point A space, its projections and coordinates.

Point plots

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, before aligning it with the plane π 2... Direction of rotation of planes π 2 and π 3 shown in fig. 1 arrows. At the same time, direct A 1 A x and A 2 A x 0X perpendicular A 1 A 2 and straight A 2 A x and A 3 A x will be located on a common to the axis 0Z perpendicular A 2 A 3... In what follows, these lines will be called, respectively. vertical and horizontal link lines.

It should be noted that in the transition from the projection apparatus to the diagram, the projected object disappears, but all information about its shape, geometric dimensions and its position in space are retained.

A(x A, y A, z Ax A, y A and z A in the following sequence (Fig. 2). This sequence is called the point plotting technique.

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

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

3. Through point A x perpendicular to axis OX

A x in the direction of the axis OY the numerical value of the coordinate is postponed y A points A A 1 on the diagram.

A x in the direction of the axis OZ the numerical value of the coordinate is postponed 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. Intersection of this line and axis OZ will give the position of the point A z.

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

Point characteristics

All points in space are subdivided into points of particular and general provisions.

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

Metamathematical - one, two or all numerical values ​​of coordinates are equal to zero and (or) infinity;

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

Points of general position. Points of general position are points that do not belong to the projection apparatus. For example, point 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: the coordinate NS 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 at the numerical values ​​of the coordinates indicate the direction of the point moving away from the projection planes. Depending on the combination of signs at the numerical values ​​of the coordinates of a point, it depends in which of the octane it is located.

Two-image method

In practice, in addition to the full projection method, the method of two images is used. It differs in that this method excludes the third projection of the object. To obtain a projection apparatus of the method of two images, the profile plane of projections with its projection center is excluded from the apparatus of full projection (Fig. 3). In addition, on the axis 0X the origin 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 apparatus, the entire space is divided into four quadrants. In fig. 3 they are marked with Roman numerals.

The 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 projection of a point A.

From the centers of projection S 1 and S 2 the projection beams 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 device (Fig. 3) the plane π 1 with the obtained projection of the point A 1 rotate clockwise around the axis 0X, before aligning it with the plane π 2... Direction of rotation of the plane π 1 shown in Fig. 3 arrows. In this case, on the plot 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 by the numerical values ​​of its coordinates x A, y A and z A in the following sequence (fig. 4).

1. An axis is drawn OX and the origin is assigned (point 0 ).

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

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

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

5. On the vertical communication line from the point A x in the direction of the axis OZ the numerical value of the coordinate is postponed 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 on the diagram the axis OZ is not plotted, but it is assumed that its positive values ​​are located above the axis OX and the negative ones are lower.

Competing points

Points on one projection ray are called competing points. They have a common projection in the direction of the projection ray, i.e. their projections are identical. A characteristic feature competing points on the diagram is the identical coincidence of their projections of the same name. Competition lies in the visibility of these projections relative to the observer. In other words, in space for the 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) of two competing points A and V point visible A according to two mutually complementary characteristics. Judging by the chain S 1 → A → B point A closer to the observer than a point V... And, accordingly, - farther from the projection plane π 1(those. z A > z A).

Rice. Fig. 5 Fig. 6

If the point itself is visible A, then its projection is also visible A 1... With respect 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 V... 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 the 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.

Consider the competing points in a similar way WITH and D apparently relative to the plane π 2. Since the common projection ray of these points l 2 parallel to axis 0Y, then the sign of visibility of competing points WITH and D is defined by the inequality y C> y D... Therefore, the 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.

Consider how the visibility of competing points in a composite drawing is determined (Figure 6).

Based on the coincident projections A 1≡IN 1 the points themselves A and V are on one projection ray parallel to the axis 0Z... This means that the coordinates are subject to comparison z A and z B these points. For this we use the frontal projection plane with separate point images. In this case z A > z B... From this it follows that the visible projection A 1.

Points C and D in the considered complex drawing (Fig. 6) are also located on one projecting ray, but only parallel to the axis 0Y... Therefore, from the comparison y C> y D we conclude that the projection C 2 is visible.

General rule. The visibility for coincident projections of competing points is determined by comparing the coordinates of these points in the direction of the common projection ray. The projection of the point at which this coordinate is greater is visible. In this case, the comparison of coordinates is carried out on the projection plane with separate images of points.