Projection of a point on a straight plane. Construction of projections of points belonging to the surfaces of geometric bodies
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 performance needs to be checked following conditions:
1) horizontal and front projection 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 straight 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 ).
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Word form |
Graphic form |
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1. Set aside on the X, Y, Ζ axes the corresponding coordinates of point A. We get points A x , A y , A z | |
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2. Horizontal projection A 1 is located at the intersection of communication lines from points A x and A y drawn parallel to the X and Y axes |
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3. Frontal projection A 2 is located at the intersection of communication lines from points A x and A z, drawn parallel to the axes X and z |
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4. Profile projection A 3 is located at the intersection of communication lines from points A z and A y drawn parallel to the axes Ζ and Y |
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3.2. Point position relative to projection planes
The position of a point in space relative to the projection planes is determined by its coordinates. The X coordinate determines the distance of the point from the P 3 plane (projection to P 2 or P 1), the Y coordinate - the distance from the P 2 plane (projection to P 3 or P 1), the Z coordinate - the distance from the P 1 plane (projection to P 3 or P 2). Depending on the value of these coordinates, a point can occupy both a general and a particular position in space with respect to the projection planes (Fig. 3.1).
Rice. 3.1. Point classification
Tpointsgeneralprovisions. The coordinates of a point in general position are not equal to zero ( x≠0, y≠0, z≠0 ), and depending on the sign of the coordinate, the point can be located in one of eight octants (Table 2.1).
On fig. 3.2 drawings of points in general position are given. An analysis of their images allows us to conclude that they are located in the following octants of space: A(+X;+Y; +Z( Ioctant;B(+X;+Y;-Z( IVoctant;C(-X;+Y; +Z( Voctant;D(+X;+Y; +Z( IIoctant.
Private position points. One of the coordinates of a particular position point is equal to zero, so the projection of the point lies on the corresponding projection field, the other two lie on the projection axes. On fig. 3.3 such points are points A, B, C, D, G.A P 3, then the point X A \u003d 0; AT P 3, then the point X B \u003d 0; With P 2, then point Y C \u003d 0; D P 1, then point Z D \u003d 0.
A point can belong to two projection planes at once, if it lies on the line of intersection of these planes - the projection axis. For such points, only the coordinate on this axis is not equal to zero. On fig. 3.3, such a point is the point G(G OZ, then point X G =0, Y G =0).
3.3. Mutual position of points in space
Consider three options relative position points depending on the ratio of the coordinates that determine their position in space.
On fig. 3.4 points A and B have different coordinates.
Their relative position can be estimated by the distance to the projection planes: Y A > Y B, then point A is located farther from the plane P 2 and closer to the observer than point B; Z A >Z B, then point A is located farther from the plane P 1 and closer to the observer than point B; X A On fig. 3.5 shows points A, B, C, D, in which one of the coordinates is the same, and the other two are different. Their relative position can be estimated by their distance to the projection planes as follows: Y A \u003d Y B \u003d Y D, then points A, B and D are equidistant from the plane P 2, and their horizontal and profile projections are located respectively on the lines [A 1 B 1 ]llOX and [A 3 B 3 ]llOZ. The locus of such points is a plane parallel to П 2 ; Z A \u003d Z B \u003d Z C, then points A, B and C are equidistant from the plane P 1, and their frontal and profile projections are located respectively on the lines [A 2 B 2 ]llOX and [A 3 C 3 ]llOY. The locus of such points is a plane parallel to П 1 ; X A \u003d X C \u003d X D, then points A, C and D are equidistant from the plane P 3 and their horizontal and frontal projections are located respectively on the lines [A 1 C 1 ]llOY and [A 2 D 2 ]llOZ . The locus of such points is a plane parallel to П 3 . 3. If the points have two coordinates of the same name, then they are called competing. Competing points are located on the same projecting line. On fig. 3.3 three pairs of such points are given, in which: X A \u003d X D; Y A = Y D ; Z D > Z A; X A = X C ; Z A = Z C ; Y C > Y A ; Y A = Y B ; Z A = Z B ; X B > X A . There are horizontally competing points A and D located on the horizontally projecting line AD, frontally competing points A and C located on the frontally projecting line AC, profile competing points A and B located on the profile projecting line AB. Conclusions on the topic 1. A point is a linear geometric image, one of the basic concepts of descriptive geometry. The position of a point in space can be determined by its coordinates. Each of the three projections of a point is characterized by two coordinates, their name corresponds to the names of the axes that form the corresponding projection plane: horizontal - A 1 (XA; YA); frontal - A 2 (XA; ZA); profile - A 3 (YA; ZA). Translation of coordinates between projections is carried out using communication lines. From two projections, you can build projections of a point either using coordinates or graphically. 3. A point in relation to the projection planes can occupy both a general and a particular position in space. 4. A point in general position is a point that does not belong to any of the projection planes, i.e., lies in the space between the projection planes. The coordinates of a point in general position are not equal to zero (x≠0,y≠0,z≠0). 5. A point of private position is a point belonging to one or two projection planes. One of the coordinates of a point of particular position is equal to zero, so the projection of the point lies on the corresponding field of the projection plane, the other two - on the axes of the projections. 6. Competing points are points whose coordinates of the same name are the same. There are horizontally competing points, frontally competing points, and profile competing points. Keywords Point coordinates General point Private position point Competing points Methods of activity necessary for solving problems – construction of a point according to the given coordinates in the system of three projection planes in space; – construction of a point according to the given coordinates in the system of three projection planes on the complex drawing. Questions for self-examination 1. How is the connection of the location of coordinates on the complex drawing in the system of three projection planes P 1 P 2 P 3 with the coordinates of the projections of points established? 2. What coordinates determine the distance of points to the horizontal, frontal, profile projection planes? 3. What coordinates and projections of the point will change if the point moves in the direction perpendicular to the profile plane of the projections П 3 ? 4. What coordinates and projections of a point will change if the point moves in a direction parallel to the OZ axis? 5. What coordinates determine the horizontal (frontal, profile) projection of a point? 7. In what case does the projection of a point coincide with the point in space itself, and where are the other two projections of this point located? 8. Can a point belong to three projection planes at the same time, and in what case? 9. What are the names of the points whose projections of the same name coincide? 10. How can you determine which of the two points is closer to the observer if their frontal projections coincide? Tasks for independent solution 2. Construct projections of points A and B according to their coordinates on a visual image and a complex drawing: A (13.5; 20), B (6.5; -20). Construct a projection of point C, located symmetrically to point A relative to the frontal plane of projections П 2 . 3. Build projections of points A, B, C according to their coordinates on a visual image and a complex drawing: A (-20; 0; 0), B (-30; -20; 10), C (-10, -15, 0 ). Construct point D, located symmetrically to point C with respect to the OX axis. An example of solving a typical problem Task 1. Given the coordinates X, Y, Z of points A, B, C, D, E, F (Table 3.3) In some cases, for the convenience of solving problems, it is necessary to use additional projection planes perpendicular to the existing projection planes. If the horizontal and frontal projections of a point are given, 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 the projection connection, we postpone the segment BUT 1
BUT X
=A Z BUT 3
.
Using this rule, it is possible to build projections of points onto additional projection planes (method of plane replacements). Let a point be given A(A 2
,BUT 1
)
and a new additional projection plane P 4
P 1
.
Build BUT 4
– point projection BUT on the P 4
.
Decision
a) We build a line of intersection of planes P 1
and P 4
=
x 1,4
;
b) Through a point BUT we draw a line of projection communication
x 1,4
.
c) Building a projection BUT 4
,
I use line segments BUT 2
BUT X =A 4
BUT X .
Two point projections BUT 1
and BUT 4
lie on the same line of projection connection perpendicular to the axis X 1,4
.
Distance from the “new” point projection BUT 4
to the “new” axle x 1,4
is equal to the distance from the “old” point projection BUT 2
to the “old” axis x 1,2
. competing points
call a pair of points lying on the same projecting ray. Of the two competing points, the visible point is the point that is further from the projection plane. points BUT and AT called horizontally competitive. points With and
D are called frontal competitors. Enter an additional plane so that the points BUT and AT became competitive. Solution plan: 1 Building an axis x 1,4
A 1
,
B 1
;
2 We build a line of projection connection x 1,4
;
3 On the line of the projection connection, lay off the segments A x A 2
=
A /
x A 4
,
B x B 2
=
B /
x B 4
.
The KOMPAS-3D-V8 system is launched similarly to 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 with the mouse pointer on the desktop field and double-click the left mouse button. To open a document, click the button Open
on the panel Standard
. To start a new document, click 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. Select menu to finish. File\Output, the Alt-F4 key combination, or click the Close button. 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. A 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. A fragment differs from a drawing by the absence of a frame, title block, and other design objects of the design document. Fragments store 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); A point, as a mathematical concept, has no dimensions. Obviously, if the object of projection is a zero-dimensional object, then it is meaningless to talk about its projection. Fig.9 Fig.10 In geometry under a point, it is advisable to take a physical object that has linear dimensions. Conventionally, a ball with an infinitely small radius can be taken as a point. With this interpretation of the concept of a point, we can talk about its projections. When building orthogonal projections points should be guided by the first invariant property of orthogonal projection: the orthogonal projection of a point is a point.
The position of a point in space is determined by three coordinates: X, Y, Z, showing the distances at which the point is removed from the projection planes. To determine these distances, it is enough to determine the meeting points of these lines with the projection planes and measure the corresponding values, which will indicate the values of the abscissa, respectively. X, ordinates Y and appliques Z points (Fig. 10). The projection of a point is the base of the perpendicular dropped from the point to the corresponding projection plane. Horizontal projection points a call the rectangular projection of a point on the horizontal plane of projections, frontal projection a /- respectively on the frontal plane of projections and profile a // – on the profile projection plane. Direct Aa, Aa / and Aa // are called projecting lines. At the same time, direct Ah, projecting point BUT on the horizontal plane of projections, called horizontally projecting line, Аa / and Aa //- respectively: frontally and profile-projecting straight lines. Two projecting lines passing through a point BUT define the plane, which is called projecting. When converting the spatial layout, the frontal projection of the point A - a / remains in place as belonging to a plane that does not change its position under the considered transformation. Horizontal projection - a together with the horizontal projection plane will turn in the direction of clockwise movement and will be located on one perpendicular to the axis X with front projection. Profile projection - a // will rotate together with the profile plane and by the end of the transformation will take the position indicated in Figure 10. At the same time - a // will be perpendicular to the axis Z drawn from the point a / and will be removed from the axis Z the same distance as the horizontal projection a away from axis X. Therefore, the connection between the horizontal and profile projections of a point can be established using two orthogonal segments aa y and a y a // and a conjugating arc of a circle centered at the point of intersection of the axes ( O- origin). The marked connection is used to find the missing projection (for two given ones). The position of the profile (horizontal) projection according to the given horizontal (profile) and frontal projections can be found using a straight line drawn at an angle of 45 0 from the origin to the axis Y(this bisector is called a straight line) k is the Monge constant). The first of these methods is preferable, as it is more accurate. Therefore: 1. Point in space removed: from the horizontal plane H Z,
from the frontal plane V by the value of the given coordinate Y,
from profile plane W by the value of the coordinate. x. 2. Two projections of any point belong to the same perpendicular (one connection line): horizontal and frontal - perpendicular to the axis x, horizontal and profile - perpendicular to the Y axis, frontal and profile - perpendicular to the Z axis. 3. The position of a point in space is completely determined by the position of its two orthogonal projections. Therefore - from any two given orthogonal projections of a point, it is always possible to construct its missing third projection. Rice. 11 Fig. 12 Figure 11 shows a spatial drawing of points of particular position, Figure 12 shows a complex drawing (diagrams) of these points. Dot BUT belongs to the frontal projection plane, the point AT– horizontal plane of projections, point With– profile plane of projections and point D– abscissa axis ( X). POINT PROJECTIONS. ORTHOGONAL SYSTEM OF TWO PLANES OF PROJECTIONS. The essence of the orthogonal projection method lies in the fact that the 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 other V is placed vertically. Plane H is called the horizontal plane of projections, V - frontal. The planes H and V are infinite and opaque. The line of intersection of the projection planes is called the coordinate axis and is denoted OX.
Projection planes divide 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 constructing projections, it is necessary to remember that point orthogonal projectionon a plane is called the base of the perpendicular dropped from a given pointto this plane. The figure shows the dot BUT and its orthogonal projections a 1 and a 2 . point a 1 called plan view points BUT, point a 2- her front projection. Each of them is the base of the perpendicular dropped from the point BUT respectively on the plane H and V. It can be proved that point projectionalways located on straight lines, perpendicularcular axisOH and crossing this axisat the same point. Indeed, projecting rays BUTa 1 and BUTa 2 define a plane perpendicular to the planes of projections and the lines of their intersection - axes 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 vertex at a point ax.
The opposite is also true, i.e. if points are given on the projection planesa 1
and a 2
,
located on straight lines intersecting axis OXat this point at a right angle,then they are projections of somepoints A. This point is determined by the intersection of the 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 in this case, the above 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 aligned 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 rear 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 French epure - drawing). The figure shows a diagram of a point BUT. 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. At the same time, the distance a 1
a x —
from the horizontal projection of the point to the axis OX BUT up to the plane V, and the distance a 2
a x —
from the frontal projection of the point to the axis OX equal to the distance from the point BUT up to the plane H. Straight lines connecting opposite projections of a point on the diagram, we agree to call projection communication lines. The position of the projections of the points on the diagram depends on which quarter it is in. given point. So if the point AT is located in the second quarter, then after the alignment of the planes, both projections will lie above the axis OX. If point With is in the third quarter, then its horizontal projection after the alignment of the planes will be above the axis, and the frontal projection will be below the axis OX.
Finally, if the point D located in the fourth quarter, then both 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 be lying 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 case when both projections of the 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 letters) completely determine its shape. However, in the practice of depicting 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 and H and V, denoted by the letter W and called profile. The projections of points on this plane will also be called profile, and they are denoted by capital letters or numbers with index 3 (ah,bh,ch, ...1h, 2h, 3 3 ...). 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 the point O. The system of signs indicated in the figure corresponds to the “right 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 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 rear half-plane H- with upper half-plane V. When rotated 90° around the axis OZ front half-plane W coincides with the right half-plane V, and the rear half-plane W- with the left half-plane V. The final view of all combined projection planes is given in the figure. In this drawing, the axes OX and OZ,
lying in a fixed 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 diagram 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 indicated. THREE COORDINATES AND THREE PROJECTIONS OF A POINT AND ITS RADIUS-VECTOR. Coordinates are numbers thatput in correspondence with a point to determineniya of its position in space or onsurfaces. AT three-dimensional space point position is set using rectangular Cartesian coordinates x, y and z. Coordinate X called abscissa, at— ordinate and z — applique. Abscissa X defines the distance from a given point to a plane W, ordinate y - up to the plane V and applique z -
up to the plane H. Having adopted the system shown in the figure for counting the coordinates of a point, we will compile a table of signs of coordinates in all eight octants. Any point in space BUT, 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 BUT(5, 4, 6). This point BUT, all coordinates of which are positive, is in the first octant Point coordinates BUT are, at the same time, the coordinates of its radius-vector OA with respect to the origin of coordinates. If a i,
j,
k are unit vectors directed respectively along the coordinate axes x, y,z(picture), then OA =OA x i+OAyj +
OAzk ,
where OA X, OA U, OA g - vector coordinates OA It is recommended to build 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 put off segments, respectively equal 5, 4 and 6 units of length. On these segments (Oa x , Oa y ,
Oa z ),
as on the edges, build a rectangular parallelepiped. Its vertex, opposite the origin, will determine the given point BUT. It is easy to see that in order to determine the point BUT it is enough to construct only three edges of the parallelepiped, for example Oa x ,
a x a 1
and a 1
BUT 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 us to determine not only the point BUT, but also all three of its orthogonal projections. Rays projecting a point on a plane H,
V,
W are the three edges of the parallelepiped that intersect at the point BUT. Each of the orthogonal projections of the point BUT, being located on a plane, is determined by only two coordinates. Yes, the horizontal projection a 1
determined by coordinates X and y, front 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 the projection planes are combined, the projections a 1
and a 2
will be on the same perpendicular to the axis OX,
and projections a 2
and a 3
—
one perpendicular to the axis oz.
As for 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 OY.
But since this axis occupies two positions on the diagram, the segment a 1
a y cannot be a continuation of a segment a 3
a y .
Construction of point projections A (5, 4, 6) on the diagram at the given coordinates, they are performed in the following sequence: first of all, on the abscissa axis from the origin, a segment is laid Oa x = x(in our case x =5),
then through the dot 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 the 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
direct a 2
a z ^ oz.
Finally, the last question arises: at what distance from the axis OZ should be a 3 ? Considering the coordinate box (see figure), the edges of which a z a 3
=O a y = a x a 1
= y we conclude that the desired distance a z a 3
equals y. Line segment a z a 3
set aside 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 starts 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 defined by these coordinates, i.e. a 2
will not change his 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
). 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 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 plot will be above the axis OX,
and the profile projection, being on the back half-plane W, on the diagram will be to the left of the axis OZ.
As always, cut a za 3
3
= y. In subsequent diagrams, 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 essentialobject, not its position relative toprojection 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 indefinite, the formation of a diagram in this case does not need to be associated with the rotation of the planes around the coordinate axis. When switching to a plane plot H and V are combined so that opposite projections of points are located on vertical lines. Axisless plot of points A and B(picture) notdetermines their position in space,but allows us to judge their relative orientation. So, the segment △x characterizes the displacement of the point BUT in relation to the point AT in a direction parallel to the H and V planes. In other words, △x indicates how much the point BUT located to the left of the point AT. 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 AT, a distance equal to △y. Finally, the segment △z shows the excess of the point BUT over the dot AT. Proponents of the axisless study of the course of descriptive geometry rightly point out that when solving many problems, one 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 tasks, among which problems of spatial statics and mechanics occupy not the last place. And for this it is necessary to cultivate 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 the coordinate axes. Such drawings determine not only the shape of the object, but also its location relative to the projection planes.
Competing points
Self-study material Modeling 2d graphics objects in the compass graphics system Starting the compass system and shutting down
The main types of documents of the compass graphic system
If a point has three definite coordinates, then such a point is called point in general position. If a point has one or two coordinates equal to zero, then such a point is called private position point.
