Drawing reversibility, i.e. determination of a point in space by its projections can be determined by projection onto three projection planes. (Figure 2.1) Plane p 1 , is called horizontal, p 2 - frontal, p 3 - profile. The lines of intersection of the projection planes form the coordinate axes (x, y, z). The point of intersection of the coordinate axes is taken as the origin of coordinates and is denoted by the letter O. The positive direction of the coordinate axes is considered for the axis X- to the left of the origin, for the axis at- towards the observer from the plane p 2 , axis z- up from p-plane 1 .
Let a point be given BUT in space (Figure 2.1). Point position BUT defined by three coordinates ( X, at, z), showing the distances at which the point is removed from the projection planes.
Figure 2.1
points BUT¢, BUT¢¢, BUT¢¢¢, in which the perpendicular lines drawn from this point intersect, are called orthogonal projections of the point BUT.
BUT¢ - horizontal projection of a point BUT;
BUT¢¢ – frontal projection of a point BUT;
BUT¢¢¢ – profile projection of a point BUT.
Direct ( AA¢), ( AA¢¢), ( AA¢¢¢) are called projecting straight lines or projecting rays. At the same time, the direct ( AA¢) is called a horizontally projecting line, ( AA¢¢) – front projecting, ( AA¢¢¢) - a profile projecting straight line.
Two projecting lines passing through a point BUT, form a plane, which is called the projecting plane.
Using the spatial layout shown in Figure 2.1 to display orthogonal projections of geometric figures is inconvenient due to its bulkiness, and also due to the fact that the shape and size of the projected figure is distorted on the p 1 and p 3 planes. Therefore, instead of an image in the drawing of a spatial layout, a plot is used, i.e. a drawing composed of two or more interconnected orthogonal projections geometric figure.
The transformation of the spatial layout into diagrams is carried out by combining the p 1 and p 3 planes with the frontal projection plane p 2 . To align the plane p 1 with p 2, it is rotated 90 ° around the axis X clockwise, and to align the plane p 3 with p 2 it is rotated around the axis z counterclockwise (Figure 2.1). After the transformation, the spatial layout will take the form shown in Figure 2.2.
Since the planes do not have boundaries, then in the combined position (on the diagram) these boundaries are not shown, there is no need to leave inscriptions indicating the name of the projection planes. Then, in the final form of the diagrams, replacing the drawing of the spatial layout (Figure 2.1) will take the form shown in Figure 2.3.
On the diagram, straight lines perpendicular to the projection axes and connecting opposite projections of points are called lines. projection connection. Note that the horizontal projection of the point BUT determined by the abscissa X and ordinate at; its frontal projection - abscissa X and applique z, and the profile projection is the ordinate at and applique z, i.e. BUT¢ ( X, at), BUT¢¢
(X, z), A¢¢¢
(y, z).
Figure 2.2 Figure 2.3
Let it be necessary to build a rectangular projection of the object given in Figure 43. Let's choose a vertical projection plane (denoting it with the letter V). This plane, located in front of the viewer, is called frontal(from the French word "frontal", which means "facing the viewer"). We will now build a projection of the object on this plane, considering the object from the front. To do this, we mentally draw through some points, for example, the tops of the object and the points of the hole projecting rays perpendicular to the projection plane V (Fig. 43. a). We mark the points of their intersection with the plane and connect them with straight lines, and the points of the circle with a curved line. We will get the projection of the object on the plane.
Rice. 43. Projection onto one plane of projections
Note that the object was positioned in front of the projection plane so that two of its surfaces were parallel to this plane and were projected without distortion. According to the resulting projection, we will be able to judge only two dimensions of the object in this case- height and width and about the diameter of the hole (Fig. 43. b). What is the thickness of the object? Using the resulting projection, we cannot say this. This means that one projection does not reveal the third dimension of the object. In order to fully judge the shape of the part from such an image, it is sometimes supplemented with an indication of the thickness (s) of the part, as in Figure 44. This is done if the object is of a simple shape, does not have protrusions, depressions, etc., i.e., it is conditionally can be considered flat. You saw examples of detail drawings containing one rectangular projection in Figures 34 and 36.
Rice. 44. Detail drawing
4.2. Projecting onto multiple projection planes. One projection does not always unambiguously determine the geometric shape of an object. For example, according to one projection given in Figure 45, a, it is possible to represent objects as they are shown in Figure 45, b and c. You can mentally pick up other objects that will also have the image given in Figure 45, a, as their projection. In addition, as we found out, the third dimension of the object is not reflected in such an image.
Rice. 45. The uncertainty of the shape of the object in the image
All these shortcomings can be eliminated if we build not one, but two rectangular projections of the object onto two mutually perpendicular planes(Fig. 46): frontal and horizontal (it is denoted by the letter H).
Rice. 46. Projection onto two projection planes
To obtain a projection on the frontal plane V, the object is viewed from the front, and on the horizontal plane H - from above.
The line of intersection of these planes (it is indicated by X) is called projection axis(Fig. 46. b).
The constructed projections turned out to be located in space in different planes (horizontal and vertical). Images of the same object are usually performed on one sheet, that is, in the same plane. Therefore, to obtain a drawing of an object, both planes are combined into one. To do this, rotate the horizontal plane of projections around the X axis down by 90 ° so that it coincides with the vertical plane. Both projections will be located in the same plane (Fig. 47).
Rice. 47. Two projections of an object
The boundaries of the projection planes in the drawing can not be shown; the projections of the projecting rays and the line of intersection of the projection planes, i.e., the projection axis, are also not applied, if this is not necessary.
On the combined planes, the frontal and horizontal projection of the object are located in the projection relationship, i.e. the horizontal projection will be exactly under the frontal one.
Rice. 48. The uncertainty of the shape of the object in the image
Note that the bottom ledge of the object is invisible in the plan view, so it is shown with dashed lines.
Let's consider one more example. According to figure 48, we can easily imagine general form details. But the shape of the notch in the vertical part remains unrevealed. To see what it is, you need to build a projection onto another plane. It is placed perpendicular to the projection planes H and V.
Rice. 49. Projection on three projection planes
The third projection plane is called profile, and the projection obtained on it - profile projection subject (from the French word "profile", which means "side view"). It is denoted by the letter W (Fig. 49, a). The projected object is placed in the space of a trihedral angle formed by the planes V, H and W. and is viewed from three sides - front, top and left. Projecting rays are passed through the characteristic points of the object until they intersect with the projection planes. Intersection points are connected by straight or curved lines. The resulting figures will be projections of the object on the planes V, H and W.
The profile plane of the projections is vertical. At the intersection with the H plane, it forms the y-axis, and with the V-plane it forms the z-axis.
To obtain a drawing of an object, the W plane is rotated 90 ° to the right, and the H plane 90 ° down (Fig. 49, b). The drawing obtained in this way contains three rectangular projections of the object (Fig. 50, a): frontal, horizontal and profile. The projection axes and projecting rays in the drawing are also not shown here (Fig. 50. b).
Rice. 50. Three projections of an object
The profile projection is placed in projection connection with the frontal one, to the right of it at the same height.
A drawing consisting of several rectangular projections is called drawing in the system of rectangular projections. Depending on the complexity of the geometric shape of an object, it can be represented by one, two or more projections.
The method of rectangular projection onto mutually perpendicular planes was developed by the French geometer Gaspard Monge at the end of the 18th century. Therefore, this method is often called the Monge method (method). G. Monge laid the foundation for the development of the science of depicting objects - descriptive geometry. Descriptive geometry is theoretical basis drawing
Rice. 51. Task for exercises
- Is it always enough in the drawing of one projection of the object?
- What are projection planes called? How are they designated?
- What are the projections obtained by projecting an object onto three projection planes called? How should these planes be located relative to each other?
Figure 51 shows a visual image and a drawing of a part - a square. Arrows show projection directions in the illustration. The projections of the part are indicated by the numbers 1, 2, 3, you need, without redrawing the drawing, write in workbook: a) which projection (indicated by a number) corresponds to each direction of projection (indicated by a letter); b) names of projections 1, 2 and 3.
Goals and objectives of the lesson:
educational: show students how to use the rectangular projection method when making a drawing;
The need to use three projection planes;
Create conditions for the formation of skills to project an object onto three projection planes;
developing: develop spatial representations, spatial thinking, cognitive interest and Creative skills students;
nurturing: responsible attitude to drawing, to cultivate a culture of graphic work.
Methods, teaching techniques: explanation, conversation, problem situations, research, exercises, frontal work with the class, creative work.
Material support: computers, presentation "Rectangular projection", tasks, exercises, exercise cards, presentation for self-examination.
Type of lesson: a lesson for consolidating knowledge.
Vocabulary work: horizontal plane, projection, projection, profile, research, project.
During the classes
I. Organizational part.
Message about the topic and purpose of the lesson.
Let's spend competition lesson, for each task you will receive a certain number of points. The class will be graded based on the points scored.
II. A review of projection and its types.
Projection is the mental process of constructing images of objects on a plane.
Repetition is carried out using the presentation.
1. The students are given problem situation . (Presentation 1)
Analyze the geometric shape of the detail on the frontal projection and find this detail among the visual images.
From this situation, it is concluded that all 6 parts have the same frontal projection. This means that one projection does not always give a complete picture of the shape and design of the part.
What is the way out of this situation? (Look at the detail from the other side).
2. There was a need to use another projection plane. (Horizontal projection).
3. The need for a third projection arises when even two projections are not enough to determine the shape of an object.
Sizing:
- in frontal projection length and height;
- on a horizontal projection - lenght and width;
- on a profile projection - width and height.
Conclusion: it means that in order to learn how to make drawings, you need to be able to project objects onto a plane.
Exercise 1
Insert the missing words in the definition text.
1. There is _______________ and ______________ projection.
2. If ______________ rays come out from one point, the projection is called ______________.
3. If ______________ rays are directed in parallel, the projection is called _____________.
4. If ______________ rays are directed parallel to each other and at an angle of 90 ° to the projection plane, then the projection is called ______________.
5. A natural image of an object on the projection plane is obtained only with ______________ projection.
6. Projections are located relative to each other______________________________.
7. The founder of the rectangular projection method is _______________
Task 2. Research project
Match the main types, indicated by numbers, with the details, indicated by letters, and write down the answer in a notebook.
Fig.4
Task 3
An exercise to repeat the knowledge of geometric bodies.
By verbal description find a visual representation of the part.
Description text.
The base of the part has the shape of a rectangular parallelepiped, in the smaller faces of which grooves are made, having the shape of a regular quadrangular prism. A truncated cone is located in the center of the upper face of the parallelepiped, along the axis of which there is a through cylindrical hole.
Rice. 5
Answer: part number 3 (1 point)
Task 4
Find the correspondence between the technical drawings of the parts and their frontal projections (the projection direction is marked with an arrow). From the scattered images of the drawing, make a drawing of each part, consisting of three images. Write down the answer in the table (Fig. 129).
Rice. 6
| Technical drawings | frontal projection | Horizontal projection | Profile projection |
| BUT | 4 | 13 | 10 |
| B | 12 | 9 | 2 |
| AT | 14 | 5 | 1 |
| G | 6 | 15 | 8 |
| D | 11 | 3 | 7 |
III. Practical work.
Task number 1. research project
Find the frontal and horizontal projections to this visual image. Write down the answer in a notebook.
Evaluation of work in the classroom. Self-test. (Presentation 2)
The scores for grading the first part of the work are written on the board:
23-26 points “5”
19-22 points “4”
15 -18 points “3”
Task number 2. creative work and checking its implementation
(creative project)
Redraw the frontal projection in a workbook.
Draw a horizontal projection, changing the shape of the part in order to reduce its mass.
If necessary, make changes on the frontal projection.
To check the completion of the task, call one or two students to the board in order to explain their version of solving the problem.
(10 points)
IV. Summing up the lesson.
1. Evaluation of work in the lesson. (Checking the practical part of the work)
V. Homework.
1. Research project.
Work on the table: determine which drawing, indicated by a number, corresponds to the drawing indicated by a letter.
A point in a system of two projection planes.
To obtain projections of a point in a system of two projection planes, it is necessary to lower the perpendiculars from this point to the corresponding projection planes, the bases of these perpendiculars will be the projections of the point on the corresponding projection planes.
Fig 7. Projections of a point in a system of two projection planes.
Point A' - projection onto the plane π 1 - is called the horizontal projection of point A. Point A'' - projection of point A onto the plane π 2 - frontal projection of point A. Similarly, the projection of point A onto the profile plane of projections (π 3 ), we get the profile projection of the point A – A'''.
The segments AA’ and AA’’ are perpendicular to the projection planes π 1 and π 2, respectively, and belong to some plane α intersecting the projection axis at some point Ax . The plane α is perpendicular to the projection planes π 1 and π 2 and to the projection axis X, crossing it at the point Ax .
If the position of the planes π 1 and π 2 is fixed in space, then each point in space corresponds to an ordered pair of points on the projection planes. The converse statement is also true - an ordered pair of points on the projection planes corresponds to a single point in space.
Projection on two and three projection planes
Epure Monge.
The considered image of a point in the system of two projection planes is not very convenient for drawing.
With the development of technology, the question of applying a method that ensures the accuracy and convenience of images, that is, the ability to accurately determine the location of each point of the image relative to other points or planes, and by simple methods to determine the size of line segments and figures, has become of paramount importance. The gradually accumulated separate rules and techniques for constructing such images were brought into the system and developed in the work of the French scientist Gaspard Monge, published in 1799 under the title "Geometrie descriptive".
As noted earlier, the segments AA’ and AA’’ are perpendicular to the projection planes π 1 and π 2, respectively, belong to some plane α intersecting the projection axis at some point Ax . The plane α is perpendicular to the projection planes π 1 and π 2 and to the projection axis X, crossing it at the point Ax .
The plane α intersects the projection planes π 1 and π 2 (segments A'Ax and A''Ax ). The segments A'Ax and A''Ax are perpendicular to the projection axis X. The projections of a certain point are obtained located on straight lines perpendicular to the projection axis and intersecting this axis at the same point (in our example, at the point
Gasprard Monge proposed a method for transforming the drawing by rotating the horizontal projection plane π 1 around the projection axis X until it coincides with the front projection plane π 2 (Fig. 9.).
Projection on two and three projection planes
Rice. 10. Transformation of the drawing according to the Monge method.
After such a transformation, the π 1 plane in the drawing is aligned with the π 2 plane, and as a result, we obtain a drawing in the form of π 1 and π 2 planes superimposed on each other. This way of depicting was called "Epure Monge" (from the French Épure - drawing, project).
Rice. 11. The position of the point projections on the Monge diagram.
When considering the converted drawing, it must be taken into account that the projection planes π 1 and π 2 occupy the entire space, and we see an overlap of two planes.
On the Monge diagram, the projections of the points A - A 'and A '' on the projection planes π 1 and π 2 are located on one straight line perpendicular to the projection axis X. The segment
A'A'' is called communication line. Thus, two projections of a point are always located on the same connection line perpendicular to the projection axis.
If we carefully analyze the original drawing of the position of the point in the system of two projection planes and Monge diagrams, we can see that the value of the segment Ax A'= AA'' determines the distance of the point A from the projection plane π 2 , and the value of the segment Ax A''= AA ' - determines the distance of point A from the plane π 1 .
Projection on two and three projection planes
Two mutually perpendicular planes π 1 and π 2 divide the entire space into four quadrants (recall how two perpendicular axes X and Y on the plane divide this plane into four quarters).
Rice. 12. Division of space by two planes into 4 quadrants.
Depending on in which quadrant of space the point of its projection is located, they occupy a certain position on the Monge diagram.
E'=Ex |
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Rice. 13. The position of the points on the Monge diagram.
According to the Monge diagram, it can be determined that the points occupy the following positions in space:
Point A is located in the first quadrant; Point B is located in the second quadrant; Point C is located in the third quadrant; Point D is located in the fourth quadrant;
Point E is located directly in the plane π 2 .
Projection on two and three projection planes
A point in the system of three projection planes.
Along with projection onto two projection planes, a system of three planes is used. The position of any point in space, and hence any geometric figure, can be determined in any coordinate system.
The most convenient is the Cartesian coordinate system in space, which consists of three mutually perpendicular axes. This system can be obtained as lines of intersection of three mutually perpendicular planes - horizontal π 1, frontal π 2 and profile π 3.
The intersection lines of these three planes form a system of three mutually perpendicular axes in space: the abscissa is the X axis, the ordinate is the Y axis and the applicate is the Z axis. coordinate axes (Fig. 14), the arrows show the positive direction of the coordinate values. The X, Y and Z axes are called projection axes.
A''Az
A A'''
Rice. 14. The position of a point in the system of three projection planes.
The value of the segment AA’ = A’’Ax is the distance from point A to the plane π 1 . The value of the segment AA '' \u003d A'Ax is the distance from point A to the plane π 2. The value of the segment AA''' = A'Ay is the distance from point A to the plane π 3 .
Projection on two and three projection planes
Three intersecting planes divide the entire space into eight octants.
Rice. 15. The division of space into eight octants. |
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By the signs of the coordinates of a point, you can determine in which octant of space it is located.
Coordinate sign |
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Descriptive geometry. Engineering graphics. Levchenko S.V. |
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Projection on two and three projection planes
Transformation of a drawing in a system of three projection planes.
As in the case of a projection in a two-plane system, in a three-plane system, the drawing transformation method proposed by Gaspard Monge is used.
This is due to the fact that in this form the drawing turns out to be cumbersome and on the planes π 1 and π 2 there is a distortion of the shapes and sizes of the figures.
Rice. 16. Transformation of planes in a system of three projection planes.
In Figure 16, the arrows show the direction of rotation of the planes around the projection axes.
During the transformation, the π 2 plane remains in place, the π 1 plane rotates around the X axis until it coincides with the π 2 plane, the π 3 plane rotates around the Z axis until it coincides with the π 2 plane. After such a transformation, all three planes are superimposed on each other (Fig.
Projection on two and three projection planes
Rice. 17. View of the drawing after the conversion.
The X and Y axes lie in the π 1 plane. The X and Z axes lie in the π 2 plane. The Y and Z axes lie in the π 3 plane.
At π 1, the X-axis stays in place, while the Y-axis in the drawing points down.
AT the result of the transformation of the planeπ 3 the Z axis remains in place, and the Y axis in the drawing is directed to the right.
Thus, after the transformation of the drawing, the Y-axis occupies two positions in the drawing: the Y-axis directed downward belongs to the plane π 1 ; the Y axis directed to the left belongs to the plane π 3 .
The position of the projections of a point in the drawing depends on the octant of space in which it is located.
Projections of any point can be built directly on the drawing: the position of the horizontal projection is determined by a pair of X,Y coordinates (the Y axis is directed downward); the position of the frontal projection is determined by the pair coordinates X,Z; the position of the profile projection is determined by the pair Y,Z coordinates(Y-axis directed to the right).
If the point is located in the first octant, then the values of all three coordinates (X,Y,Z) are positive.
Projection on two and three projection planes
Building the missing projection in the system of three projection planes:
Rice. 18. The procedure for constructing the missing point projection.
Let horizontal (A’) and frontal (A’’) projections of point A be given.
It is necessary to build the missing profile projection (A'''). When constructing, performing constructions, it is necessary to remember the following rules of descriptive geometry:
1. The horizontal and frontal projections of a point are always on the same line of communication perpendicular to the X axis.
2. The frontal and profile projections of a point are always on the same connection line, perpendicular to the Z axis.
3. The horizontal and profile projections of a point are always on the same a horizontal-vertical link line perpendicular to the Y-axis.
Building order:
Projection on two and three projection planes
Let's draw a line perpendicular to the Z axis through point A''. The desired profile projection must be on this line.
To build a horizontal-vertical connection line perpendicular to the Y-axis, we will use the constant straight line of the drawing.
A. The constant straight line of the drawing is the bisector of the angle formed by the axes Y. Usually denoted by the letter k.
Through the horizontal projection of the point, we draw a perpendicular to the vertical Y axis until it intersects with the constant line of the drawing (point 1), then from point 1 we draw a perpendicular to the vertical Y axis until it intersects with the connection line perpendicular to the Z axis.
The point of intersection of the communication line perpendicular to the Z axis and horizontally
vertical connection line perpendicular to the Y-axis and is the profile projection of point A.
Once again, we note that the horizontal projection of a point is determined by its abscissa and ordinate, the frontal projection - by the abscissa and applicate, the profile - by the ordinate and applicate.
A point in space is removed from the plane:
π 1 by a value equal to the value of the segment A''Ax or A'''Ay.
π 2 by a value equal to the value of the segment A'Ax or A'''Az.
π 3 by a value equal to the value of the segment A'Ay or A''Az.
There are many parts whose shape information cannot be conveyed by two projections of the drawing. In order for information about the complex shape of the part to be presented quite fully, projection is used on three mutually perpendicular projection planes: frontal - V, horizontal - H and profile - W (read "double ve").
Complex drawing A drawing represented by three views or projections, in most cases gives a complete picture of the shape and design of the part (object and object) and is also called a complex drawing. basic drawing. If a drawing is built with coordinate axes, it is called an axis drawing. axisless If the drawing is built without coordinate axes, it is called axisless profile If the plane W is perpendicular to the frontal and horizontal projection planes, then it is called profile
An object is placed in a trihedral angle so that its shaping face and base are parallel to the frontal and horizontal projection planes, respectively. Then, projecting beams are passed through all points of the object, perpendicular to all three projection planes, on which frontal, horizontal and profile projections of the object are obtained. After projection, the object is removed from the trihedral angle, and then the horizontal and profile projection planes are rotated by 90° around the Ox and Oz axes, respectively, until they coincide with the frontal projection plane and a detail drawing is obtained containing three projections.
The three projections of the drawing are interconnected with each other. Frontal and horizontal projections preserve the projection connection of images, i.e., projection connections are established between frontal and horizontal, frontal and profile, as well as horizontal and profile projections. Projection relationship lines define the location of each projection on the drawing field. The shape of most objects is a combination of various geometric bodies or their parts. Therefore, in order to read and complete drawings, you need to know how geometric bodies in the system of three projections in production
1. Facets parallel to the projection plane are projected onto it without distortion, in full size. 2. Faces perpendicular to the projection plane are projected in a segment of straight lines. 3. Edges located obliquely to the projection planes, images on it with distortion (reduced)
& 3. page questions writing task 4.1. pp pp, & 5, pp. 37-45, writing assignment questions
