Job analysis: the part has two planes of symmetry - frontal and profile, respectively, the image on the horizontal plane of projections has two axes of symmetry - vertical and horizontal, and on the front and profile - one, vertically located. We begin constructing images by drawing the axes of symmetry. We place them on the drawing field, taking into account the overall dimensions of the part. L x B x H– 100x80x75 mm.
The part is hollow, i.e. Inside there is a through hole of a complex geometric shape, the frontal projection is depicted by lines of an invisible contour.
External form . The part is a base in the form of a rectangular prism with dimensions 100x80x15mm. At all four corners of the prism, cuts are made, the so-called chamfers, measuring 10x10mm. In the middle of the prismatic base there is a hexagonal straight prism, 75-15mm high. On a horizontal projection, the base of the prism is a hexagon inscribed in a circle with a diameter of 70 mm. On top of the prism there is a recess - a through prismatic groove - to a depth of 20mm and a width of 26mm. In the middle of the prismatic base, on the left and right, two stiffening ribs 10 mm thick are adjacent to the side faces of the prism to their entire height.
Internal shape of the part. A through cylindrical hole (diameter 48 mm) is made in the part from top to bottom. At a distance of 35 mm from the upper base of the part, on the right and left, symmetrically located prismatic-shaped protrusions are adjacent to the side surface of the cylindrical hole, the distance between which is 32 mm.
The right and left faces (planes) of the prismatic groove intersect with the side faces of the hexagonal prism along horizontally projecting straight lines. We mark their horizontal projection, measure the depth and construct a profile projection of the groove. The lower plane of the groove intersects the faces of the hexagon along straight, parallel sides of the base of the prism. The right and left faces (planes) of the prismatic groove intersect with the side surface of the internal cylindrical hole along straight generatrices. We construct their profile projection by measuring the depth of the straight line segment.
The upper face of the internal prismatic protrusion intersects the side surface of the cylindrical hole in a circle, and the left and right faces parallel to each other, located parallel to the axis of rotation of the cylindrical surface, intersect it along straight generatrices (horizontally projecting straight lines). We construct frontal and profile projections of these lines taking into account the projection connection.
According to GOST 2.102-68 “Types of products and completeness of design documents”, theoretical drawing – this is a document that defines the geometric shape (contours, contours) of the product and the coordinates of the location of the main components. Document code – PM.
The geometric shape of a given part is a combination, a certain arrangement of simple geometric bodies (surfaces) or their elements and is divided into external and internal. We depict the external shape of the part using views, and the internal shape using two vertical (frontal and profile) sections. We perform all images of the part using the method of rectangular projection onto three mutually perpendicular projection planes - we construct three orthogonal rectangular projections (three main types).
GOST 2.305-2008 “Images - views, sections, sections” gives the following definitions:
Item type (type): An orthogonal projection of the visible part of the surface of an object facing the observer, located between it and the projection plane.
Vertical section: A section made by planes perpendicular to the horizontal plane of projections. In the section we show what is obtained in the secant plane (shaded) and what is located behind it (not shaded).
Frontal (profile) section: A vertical section made by secant planes parallel to the frontal (profile) projection plane.
We place the sections in place of the corresponding main types. Since the cutting planes coincide with the planes of symmetry of the part as a whole, we do not mark the positions of the cutting planes and do not accompany the cuts with inscriptions.
If the view or section is a symmetrical figure, it is allowed to draw half of the image. In this case, the dividing line is the axis of symmetry. It is allowed to draw a little more than half of the image of a view or section, in this case drawing a break line - a wavy line.
Features of performing images of sections of a part
To construct frontal and profile sections, we use cutting planes parallel to the corresponding projection planes. Since both images of sections have vertically located axes of symmetry, we use the provisions of GOST 2.305-2008, which in this case allows us to save ½ of the view (to the left of the axis of symmetry) and perform only ½ of the section (to the right of the axis of symmetry). In this case, the border between half of the view and half of the section is a dash-dotted line. We note that in the frontal projection in the image of the half section, the projection of the edge of the hexagonal prism - the contour line - coincides with the boundary line. In this case, in accordance with the provisions of the standard, we draw the contour line in place of the dash-dotted line, and the boundary between the larger part of the section and the smaller part of the view is drawn with a wavy line. A wavy line can be drawn either in a limited place, running it a little further than the contour line, or along the entire axis of symmetry.
When making a frontal section, we take into account the instructions of GOST 2.305-2008 that stiffeners, like some other elements of parts, are shown conditionally not dissected, i.e. no shading is applied.
Features when applying part dimensions
To apply the dimensions of the part, we use all the images. First of all, we mark the dimensions of the parts on the side of the images of the types. Dimensions related to images of the external shape of the part are placed on the side of the views image (to the left of the axis of symmetry). We mark the dimensions of the internal shape of the part from the side of the image of the sections (to the right of the axis of symmetry). If possible, dimensions related to the same part element are placed side by side (close to each other). The dimensions of the chamfers made at the corners of a rectangular prism are set as shown in GOST 2.307-68.
Features when constructing a visual image of a part (isometric projection)
An axonometric drawing is formed by parallel projection of an object along with the natural coordinate system attached to it onto one projection plane. Such drawings are remarkable for their clarity.
Unlike projection drawings, in the cutout of an axonometric projection, stiffeners, inclined walls, spokes of flywheels and pulleys, axles, balls and other similar elements are hatched.
When applying dimensions, extension lines are drawn parallel to the axonometric axes, and dimension lines are drawn parallel to the measured segment.
Lesson topic: Analysis of the geometric shape of an object.
Lesson objectives:
Lesson objectives:
Equipment:
Lesson structure:
DURING THE CLASSES
Announce the topic and goals of the lesson.
– I want to start the lesson with information that, at first glance, has nothing to do with drawing. In order for the churches not to get lost in space and to be clearly visible from afar, it was necessary to find an expressive silhouette for them. The search for it led the architects to a compositional solution for churches with a tiered top made of a series of decreasing octagons.
The prototype of the ancient bell towers was a military defense watchtower, which was built according to the traditional scheme - an eight on a four.
The leading role in the architectural appearance of the Ural city of Nevyansk, located in our region, is played by the famous “leaning” tower<Annex 1
>. It was built in 1725 and can be seen from any street in the city. It is believed that it was originally a watchtower. The height of the tower is 57.5 meters. The tower consists of four parts: a “quadrangle”, which occupies half the height. On the quadrangle, one on top of the other, there are three “octagons”. The tower is crowned with a “tent”. In drawing language, a tower is a combination of geometric bodies. But we must find out which ones by the end of the lesson.
(Write the topic of the lesson in your notebook)
Geometric body- this is a closed part of space, limited by flat and curved surfaces.
The shape of each body has its own characteristic features.
On your desks you have cards describing these geometric bodies. Let's get to know them better.<Appendix 2
>
(The teacher shows a model of a geometric body, one of the students reads out the definition and essential features of the body from the card)
In engineering, the shape of a part is often compared with simpler shapes - geometric bodies, and also the shapes of geometric bodies are used to describe the shape of more complex parts.
Any simple form technical details can be represented as geometric body shape(for example, the shape of a technical part “axle” can be represented as a cylinder shape - see Figure 73 in the textbook), and shape of a complex product- How combination of shapes of geometric bodies(for example, the “fork” part - see Fig. 73 in the textbook, ..., the tower, which we talked about at the beginning of the lesson). The considered approach to the study of parts is based on an analysis of its geometric shape.
Geometric shape analysis of an object is a mental dissection of an object into its constituent geometric bodies. (Write in notebook)
Let's consider how the geometric shape of an object is analyzed using a visual image of the part. We mentally divide the part into simple geometric bodies, name them and tell how they are located relative to each other in space.
For example, the “Support” part (poster on the board) consists of a rectangular parallelepiped (1) with five through cylindrical holes. In the center of the upper face of the rectangular parallelepiped there is a quadrangular prism (2) with a through cylindrical hole, the axis and diameter of which coincide with the axis and diameter of the hole of the part (1). The parallelepipeds are connected to each other by two stiffening ribs (3) in the shape of triangular prisms, which ensures their stable fastening.
By using the method of dividing a part into simple geometric bodies, you can learn to quickly, correctly read drawings and execute them competently.
Exercise: use a visual image of the part to analyze its shape (visual image of the part - poster on the board).
Answer: At the base of the part lies a rectangular parallelepiped with a through cylindrical hole in the center. Two more rectangular parallelepipeds are adjacent to it at the ends. One has a through cylindrical hole, the other has a rectangular cutout.
Names of elements of geometric bodies. Base, faces, edges, vertex, generatrix (the teacher shows models of geometric bodies, see the picture in the textbook).
- Now let's go back to the beginning of the lesson. As noted by the study, the Nevyansk Tower “is related to the tiered towers and bell towers of Ancient Rus', but is distinguished by its emphasized severity.” I'll remind you of her (read information from the board). <Appendix 6 >
– Let’s get acquainted with the definitions of “quadruple”, “octagon”, “tent” - I gave several children the task of finding the meanings of these words in dictionaries. (read out, post on the board)
So how can you now, having become familiar with geometric bodies, analyze the geometric shape of the Nevyansk Tower?
Answer: the tower consists of four parts - a regular quadrangular prism and three octagonal prisms standing on top of each other. The tower is crowned by an octagonal pyramid.
– What other geometric bodies did you meet today? (Ball, cube, cone, cylinder)
– Why do we need to analyze the geometric shape of an object? ( To read and execute quickly and correctly).
Homework: in the textbook §10, pp. 58 – 61. Come up with and create a visual representation of a toy, the shape of which consists of simple geometric bodies (show example). If it is difficult to complete the drawing, you can sculpt a toy from plasticine.
Literature:
Analysis of the geometric shape of objects. Bodies of rotation. Group of geometric bodies
|
Equipment for the student: |
|
|
Accessories, textbook “Drawing”, ed. A. D. Botvinnikova §10, 11, 16, colored pencils. |
|
|
Rules for making drawings of geometric bodies. Sequence of reading a group of geometric bodies. |
|
Fixing the material
Working with cards
Fixing the material
Using colored pencils, complete the task on the card.
Geometric shape analysis -
Drawing of a part according to these two types
|
Equipment for the student: |
tools, |
|
f A4, tools |
|
|
Analyze drawings, give an accurate verbal description of the object depicted in the drawing. |
Obtaining axonometric projections of plane figures
Homework:
Repeat paragraph 7-7.2; complete the construction of table 1.
Equipment for students:
textbook "Drawing" ed. Botvinnikova A.D., workbook, drawing accessories.
Square in dimetric projection
Exercise:
Construct a square in isometric projection
Triangle in dimetry Triangle in isometry
Hexagon in dimetry and isometry
Exercise:
Construct a hexagon in isometric projection
Exercise:
Axonometric projections volumetric bodies
|
Equipment for the student: |
Textbook "Drawing" ed. A.D. Botvinnikova, notebook, instruments. |
|
Accessories, textbook “Drawing”, ed. A. D. Botvinnikova page 49 table No. 2, §7-8. |
|
|
Rules for constructing axonometric projections. Methods for constructing a volumetric part in isometry. |
|
|
Construct images in axonometry starting from flat figures lying at the base of the part. Learn to analyze the resulting images. |
Review task:
Construct a geometric figure on a horizontal projection plane.
Amount (increasing)
Clipping
Reinforcement task
Axonometric projection of a part with cylindrical elements
|
Equipment for the student: |
Textbook "Drawing" ed. A. D. Botvinnikova, accessories, notebook. |
|
Accessories, textbook “Drawing”, ed. A. D. Botvinnikova § 7-8. |
|
|
Rules for constructing a part with a curved surface. The general concept of “axonometry of a part.” |
|
|
Analyze the shape of the part and the resulting image. |
Ellipse –
Oval -
Algorithm for constructing an oval
1. Let's construct an isometric projection of a square - a rhombusABCD
2. Let us denote the points of intersection of the circle and the square 1 2 3 4
3. From the top of the rhombus (D) draw a straight line to the point4 (3). We get the segmentD4, which will be equal to the arc radiusR.
4. Let's draw an arc that connects the points3 And4 .
5. When crossing a segmentAT 2AndACwe get a pointO1.
When crossing a line D4 AndACwe get a pointO2.
6. From the received centersO1AndO2let's draw arcsR1 , which will connect points 2 and 3, 4 and 1.
Consolidating new material
! work in the workbook
Make isometric projections of the circle parallel to the frontal and profile projection planes.
Drawing and visual representation of the part
|
Equipment for the student: |
F A4, tools, textbook |
|
§12, tracing paper |
|
|
Analyze the shape of the part, build 3 types of parts and apply dimensions. |
Technical drawing
|
Equipment for the student: |
Textbook "Drawing" ed. A. D. Botvinnikova§9, accessories, notebook. |
|
Accessories, textbook “Drawing”, ed. A. D. Botvinnikova § 9 |
|
|
Rules for making technical drawings and techniques for making parts. |
|
|
Perform axonometric projections depicting flat figures. Perform technical drawing. |
Technical drawing –
Hatching methods:
Fixing the material
Complete a technical drawing of the part, two views of which are shown in Fig. 62
Projections of vertices, edges and faces of an object
|
Equipment for the student: |
Textbook "Drawing" ed. A.D. Botvinnikova, accessories, notebook, colored pencils. |
|
Accessories, textbook “Drawing”, ed. A. D. Botvinnikova §12, fA4, colored pencils. |
|
|
Methods for selecting a point on a plane. Principles of constructing edges and faces. |
|
|
Construct projections of points and faces. |
? Problem
What is a rib?
What is the top of an object?
What is the edge of an object?
Point projection
Practical work:
Label the projections
points on the part drawing, marked in the visual image.
Graphic work No. 9
Part sketch and technical drawing
|
Equipment for the student: |
|
|
Tools, graph paper, fA4, § 18 |
|
|
What is a sketch? Sketch rules |
|
|
Complete the sketch in the required number of types. Draw according to the sketch. |
What's called sketch?
Fixing the material
Exercise tasks
Applying dimensions taking into account the shape of the object
|
Equipment for the student: |
tools, textbook, notebook, tracing paper. |
|
Rice. 113 (1, 2, 3, 5, 8, 9) |
|
|
General rule for drawing dimensions in a drawing. |
|
Repetition and consolidation of the material covered.
Oral exercise
Practical work:
Cutouts and slices on geometric bodies
Parts elements
SLOT- a groove in the form of a slot or groove on machine parts. For example, a slot in the head of a screw or screw into which the end of a screwdriver is inserted when screwing it in.
GROOVE- an oblong depression or hole on the surface of a part, limited on the sides by parallel planes.
LYSKA– a flat cut on one or both sides of cylindrical, conical or spherical sections of a part. The flats are designed to be grabbed with a wrench, etc.
GROWTH- this is an annular groove on the rod, technologically necessary for the exit of a threaded tool during the manufacture of a part or for other purposes.
KEYWAY GROOVE- a slot in the form of a groove, which serves to install a key, which transmits rotation from the shaft to the bushing and vice versa.
CENTER HOLE- an element of a part that serves to reduce its mass, supply lubricant to rubbing surfaces, connect parts, etc. The holes can be through or blind.
CHAMFER– turning a cylindrical edge of a part onto a truncated cone.
Exercise: Instead of numbers, write the names of the part elements
Exercise: Perform an axonometric projection of the part
Practical work No. 7
"Reading Blueprints"
|
Equipment for the student: |
Textbook, notebook, sheet. |
|
Graph paper, §17 |
|
|
Master the methods of constructing 3 types, analyze the geometric shape of an object, know the names of the elements of a part. |
|
|
Analyze the drawing, determine dimensions, give an accurate verbal description |
Graphic dictation
“Drawing and technical drawing of a part based on a verbal description”
|
Equipment for the student: |
Format (notebook), tools |
|
Tools, graph paper. |
|
|
Rules for sketching |
|
|
Determine the necessary and sufficient number of types for a given part. Select the main view. Dimension. |
Option #1
Frame is a combination of two parallelepipeds, of which the smaller one is placed with a larger base in the center of the upper base of the other parallelepiped. A through stepped hole runs vertically through the centers of the parallelepipeds.
The total height of the part is 30 mm.
The height of the lower parallelepiped is 10 mm, length 70 mm, width 50 mm.
The second parallelepiped has a length of 50 mm and a width of 40 mm.
The diameter of the bottom step of the hole is 35 mm, height 10 mm; diameter of the second stage is 20 mm.
Note:
Option No. 2
Support is a rectangular parallelepiped, to the left (smallest) face of which is attached a half-cylinder, which has a common lower base with the parallelepiped. In the center of the upper (largest) face of the parallelepiped, along its long side, there is a prismatic groove. At the base of the part there is a through hole of a prismatic shape. Its axis coincides in the top view with the axis of the groove.
The height of the parallelepiped is 30 mm, length 65 mm, width 40 mm.
Half-cylinder height 15 mm, base R 20 mm.
The width of the prismatic groove is 20 mm, the depth is 15 mm.
Hole width 10 mm, length 60 mm. The hole is located at a distance of 15 mm from the right edge of the support.
Note: When drawing dimensions, consider the part as a whole.
Option No. 3
Frame is a combination of a square prism and a truncated cone, which stands with its large base in the center of the upper base of the prism. A through stepped hole runs along the axis of the cone.
The total height of the part is 65 mm.
The height of the prism is 15 mm, the size of the sides of the base is 70x70 mm.
The height of the cone is 50 mm, the lower base is Ǿ 50 mm, the upper base is Ǿ 30 mm.
The diameter of the lower part of the hole is 25 mm, height 40 mm.
The diameter of the upper part of the hole is 15 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 4
Sleeve is a combination of two cylinders with a stepped through hole that runs along the axis of the part.
The total height of the part is 60 mm.
The height of the lower cylinder is 15 mm, the base is Ǿ 70 mm.
The base of the second cylinder is 45 mm.
Bottom hole Ǿ 50 mm, height 8 mm.
Upper part of the hole Ǿ 30 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 5
Base is a parallelepiped. In the center of the upper (largest) face of the parallelepiped, along its long side, there is a prismatic groove. There are two through cylindrical holes in the groove. The centers of the holes are spaced from the ends of the part at a distance of 25 mm.
The height of the parallelepiped is 30 mm, length 100 mm, width 50 mm.
Groove depth 15 mm, width 30 mm.
Hole diameters are 20 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 6
Frame It is a cube, along the vertical axis of which there is a through hole: semi-conical at the top, and then turning into a stepped cylindrical one.
Cube edge 60 mm.
The depth of the semi-conical hole is 35 mm, the upper base is 40 mm, the bottom is 20 mm.
The height of the bottom step of the hole is 20 mm, the base is 50 mm. The diameter of the middle part of the hole is 20 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 7
Support is a combination of a parallelepiped and a truncated cone. The cone with its large base is placed in the center of the upper base of the parallelepiped. In the center of the smaller side faces of the parallelepiped there are two prismatic cutouts. A through hole of cylindrical shape Ǿ 15 mm is drilled along the axis of the cone.
The total height of the part is 60 mm.
The height of the parallelepiped is 15 mm, length 90 mm, width 55 mm.
The diameters of the cone bases are 40 mm (lower) and 30 mm (upper).
The length of the prismatic cutout is 20 mm, width 10 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 8
Frame is a hollow rectangular parallelepiped. In the center of the upper and lower base of the body there are two conical tides. A through hole of cylindrical shape Ǿ 10 mm passes through the centers of the tides.
The total height of the part is 59 mm.
The height of the parallelepiped is 45 mm, length 90 mm, width 40 mm. The thickness of the walls of the parallelepiped is 10 mm.
The height of the cones is 7 mm, the base is Ǿ 30 mm and Ǿ 20 mm.
Note: When drawing dimensions, consider the part as a whole.
Option No. 9
Support is a combination of two cylinders with one common axis. A through hole runs along the axis: at the top it is prismatic in shape with a square base, and then cylindrical in shape.
The total height of the part is 50 mm.
The height of the lower cylinder is 10 mm, the base is Ǿ 70 mm. The diameter of the base of the second cylinder is 30 mm.
The height of the cylindrical hole is 25 mm, the base is Ǿ 24 mm.
The base side of the prismatic hole is 10 mm.
Note: When drawing dimensions, consider the part as a whole.
Test
Graphic work No. 11
“Drawing and visual representation of the part”
Using the axonometric projection, construct a drawing of the part in the required number of views on a scale of 1:1. Add dimensions.
Graphic work No. 10
“Sketch of a part with design elements”
|
Equipment for the student: |
tools, textbook, graph paper |
|
Tools, graph paper. |
|
|
Sketch rules |
|
|
Make a sketch, put down the dimensions correctly |
Draw a drawing of a part from which parts have been removed according to the markings applied. The projection direction for constructing the main view is indicated by an arrow.
Graphic work No. 8
"Part drawingctransforming its form"
|
Equipment for the student: |
tools, fA4, textbook |
|
Tools, graph paper. |
|
|
Execute drawing |
General concept of shape transformation. Relationship between drawing and markings
|
Equipment for the student: |
Textbook, notebook, graph paper, accessories |
|
Textbook fig. 151 (get to know each other), fA4 |
|
|
Analyze the form. Draw the drawing in orthogonal rectangular projection. |
Graphic work
Making a drawing of an object in three views with transforming its shape (by removing part of the object)
Complete the technical drawing of the part, making, instead of the protrusions marked with arrows, notches of the same shape and size in the same place.
Logical thinking task
Subject"Design of drawings"
Subject"Drawing tools and accessories"
Crossword"Projection"
1.The point from which the projecting rays emanate during central projection.
2. What is obtained as a result of modeling.
3. Cube face.
4. The image obtained during projection.
5. In this axonometric projection, the axes are located at an angle of 120° to each other.
6. In Greek, this word means “double dimension.”
7. Side view of a person or object.
8. Curve, isometric projection of a circle.
9. The image on the profile projection plane is a view...
Rebus on the topic"View"
Rebus
Subject"Developments of geometric bodies"
Crossword"Axonometry"
Vertically:
Translated from French as “front view”.
The concept in drawing on which the projection of a point or object is obtained.
The boundary between the halves of a symmetrical part in the drawing.
Geometric body.
Drawing tool.
Translated from Latin, “throw, throw forward.”
Geometric body.
The science of graphic images.
Unit of measurement.
Translated from Greek “double dimension”.
Translated from French as “side view”.
In the drawing, “she” can be thick, thin, wavy, etc.
Working programmFrom "____" _________ 2014 Working program By drawing Grades 8 and 9 Modified based on the program... separate A4 sheets, exercises in notebooks.) 1. Sketch of the part with the required cut...
In Figure 72 you see images of some geometric bodies. The shape of each of them has its own characteristic features. By these characteristics we distinguish a cylinder from a cone, and a cone from a pyramid. You are familiar with most of these bodies. We say “cube” and everyone imagines its shape. We say “ball”, and again the image of a certain geometric body appears in our minds.
Take a closer look at the objects around us. They have the shape of geometric bodies or are combinations thereof.
Rice. 72. Geometric bodies
The shape of machine parts and mechanisms is also based on geometric bodies. Take a look at Figure 73. Various parts are shown here. Some of them are of the simplest form. Tell me what shape the axle and roller have. What is the shape of the gasket?
Rice. 73. Various details are based on geometric bodies
About such parts as the axle and roller, we will say that they are cylindrical, and about the gasket - that it is prismatic.
Other parts have a more complex shape. They are a collection of geometric bodies. For example, a roller (Fig. 73) is formed by adding another smaller cylinder to a cylinder. A bushing is a cylinder from which another cylinder of smaller diameter has been removed.
It is more difficult to understand the shape of a more complex part, such as a fork, from a drawing.
What is the easiest way to determine the shape of an object from a drawing? To do this, a complex-shaped part is mentally dissected into its individual constituent parts, which have the shape of various geometric bodies. Let's look at an example.
Figure 74a shows an image of a support. What is its shape? It is composed of a rectangular parallelepiped, two half-cylinders and a truncated cone. The part has a cylindrical hole (Fig. 74. b). After such “dismemberment” the shape of the part is easier to determine.
Rice. 74. Analysis of the geometric shape of the support
The mental division of an object into its constituent geometric bodies is called analysis of geometric shape.
Determine which surfaces of geometric bodies form the shape of the objects shown in Figure 75.
Rice. 75. Exercise task
§ 11. Drawings and axonometric projections of geometric bodies
So, you already know that the shape of most objects is a combination of various geometric bodies or their parts. Therefore, to read and complete drawings you need to know how geometric bodies are depicted.
11.1. Projecting a cube and a cuboid. The cube is positioned so that its edges are parallel to the projection planes. Then they will be depicted on projection planes parallel to them in full size - as squares, and on perpendicular planes as straight segments (Fig. 76).
Rice. 76. Cube and parallelepiped: a - projection: b, d - drawings in a system of rectangular projections: c, d - isometric projections
The projections of a cube are three equal squares.
In the drawing of a cube and a parallelepiped, three dimensions are indicated: length, height and width.
In Figure 77, the part is formed by two rectangular parallelepipeds, each having two square faces. Pay attention to how the dimensions are shown on the drawing. Flat surfaces are marked with thin intersecting lines.
Rice. 77. Image of a part in one view
Thanks to the symbol, the shape of the part is clear even from one view.
11.2. Projection of regular triangular and hexagonal prisms. The bases of the prisms, parallel to the horizontal projection plane, are depicted on it in full size, and on the frontal and profile planes - as straight segments. The side faces are depicted without distortion on those projection planes to which they are parallel, and in the form of straight segments on those to which they are perpendicular (Fig. 78). Edges. inclined to the projection planes are depicted distorted on them.
Fig 78. Prisms: a. g - projection; b, d - drawings in a rectangular projection system: c, c - isometric projections
The dimensions of the prisms are determined by their height and the size of the base figure. The dash-dot lines in the drawing indicate the axes of symmetry.
The construction of isometric projections of the prism begins from the base. Then perpendiculars are drawn from each vertex of the base, on which segments equal to the height are laid, and straight lines parallel to the edges of the base are drawn through the resulting points.
A drawing in a system of rectangular projections also begins with a horizontal projection.
11.3. Projecting a regular quadrangular pyramid. The square base of the pyramid is projected onto the horizontal plane H in full size. On it, diagonals depict the lateral ribs running from the tops of the base to the top of the pyramid (Fig. 79).
Rice. 79. Pyramid: projection: b drawing in a system of rectangular projections; in isometric projection
The frontal and profile projections of the pyramid are isosceles triangles.
The dimensions of the pyramid are determined by the length b of the two sides of its base and the height h.
The isometric projection of the pyramid begins to be built from the base. A perpendicular is drawn from the center of the resulting figure, the height of the pyramid is plotted on it and the resulting point is connected to the vertices of the base.
11.4. Projecting a Cylinder and a Cone. If the circles lying at the bases of the cylinder and cone are located parallel to the horizontal plane H, their projections onto this plane will also be circles (Fig. 80, b and d).
Rice. 80. Cylinder and cone: a, d - projection; b, d drawings in a system of rectangular projections; V. e - isometric projections
The frontal and profile projections of the cylinder in this case are rectangles, and the cones are isosceles triangles.
Please note that on all projections the axes of symmetry should be drawn, with which the drawings of the cylinder and cone begin.
The frontal and profile projections of the cylinder are the same. The same can be said about cone projections. Therefore, in this case, profile projections in the drawing are unnecessary. In addition, thanks to the “diameter” icon, you can imagine the shape of a cylinder from one projection (Fig. 81). It follows that in such cases there is no need for three projections.
Rice. 81. Image of a cylinder in one view
The dimensions of the cylinder and cone are determined by their height h and base diameter d. The methods for constructing an isometric projection of a cylinder and a cone are the same. To do this, draw the x and y axes, on which a rhombus is built. Its sides are equal to the diameter of the base of the cylinder or cone. An oval is inscribed in the rhombus (see Fig. 66).
11.5. Projections of a ball. All projections of the ball are circles, the diameter of which is equal to the diameter of the ball (Fig. 82). Center lines are drawn on each projection.
Rice. 82. Projections of a ball
Thanks to the "diameter" sign, the ball can be depicted in one projection. But if it is difficult to distinguish a sphere from other surfaces from the drawing, add the word “sphere”, for example: “Sphere with a diameter of 45”.
11.6. Projections of a group of geometric bodies. Figure 83 shows the projections of a group of geometric bodies. Can you tell how many geometric bodies are included in this group? What bodies are these?
Rice. 83. Drawing of a group of geometric bodies
Having examined the images, we can establish that it contains a cone, a cylinder and a rectangular parallelepiped. They are located differently relative to the projection planes and each other. How exactly?
The axis of the cone is perpendicular to the horizontal plane of projections, and the axis of the cylinder is perpendicular to the profile plane of projections. Two faces of the parallelepiped are parallel to the horizontal projection plane. On a profile projection, the image of a cylinder is to the right of the image of a parallelepiped, and on a horizontal projection it is below. This means that the cylinder is located in front of the parallelepiped, therefore part of the parallelepiped in the front projection is shown by a dashed line. From horizontal and profile projections it can be established that the cylinder touches the parallelepiped.
The frontal projection of the cone touches the projection of the parallelepiped. However, judging by the horizontal projection, the parallelepiped does not touch the cone. The cone is located to the left of the cylinder and parallelepiped. In profile projection, it partially covers them. Therefore, invisible sections of the cylinder and parallelepiped are shown with dashed lines.
How will the profile projection in Figure 83 change if a cone is removed from the group of geometric bodies?
Entertaining tasks
Rice. 84. Exercise tasks
§ 12. Projections of vertices, edges and faces of an object
12.1. How elements of objects are depicted. Any point or segment on the image of an object is a projection of one or another element: a vertex, an edge, a face, a curved surface, etc. (Fig. 85). Therefore, the image of any object is reduced to the image of its top, edges, edges and curved surfaces.
Rice. 85. Elements of the surface of an object
Let's consider this process using the example of constructing rectangular projections of an object (Fig. 86).
Let's place the object in space so that each of the two sides parallel to each other is parallel to one of the projection planes. Then these faces will be depicted on the corresponding projection planes without distortion.
Let us draw projecting rays through the vertices of the object, perpendicular to the projection planes, and mark the points of their intersection with the planes V, H and W.
The object is so located relative to the projection planes that there are two vertices on one projecting ray, so their projections merge into one point. Thus, vertices A and B lie on the same ray perpendicular to the horizontal plane of projections H. Their horizontal projections a and b coincide. Vertices A and C lie on the same ray, which projects these points onto the frontal projection plane. Their frontal projections a" and c" also coincided. On the profile plane of projections W, vertices B and D were projected into one point (b" and d").
Of the two points coinciding in the image, one is the image of a visible vertex, the other is a closed (invisible) one. On the horizontal projection, the vertex that is located in space above will be visible. So, vertex A is visible, vertex B is invisible. On the frontal projection, the vertex that is closest to us will be visible. Hence a" is the image of the visible vertex A, c" is the image of the invisible vertex C, it is covered when projected by vertex A. In the image, the designation of projections of invisible points is sometimes taken in brackets.
By connecting pairs of points on the frontal, horizontal and profile projections, we obtain images of the edges of the object. For example, ac is the horizontal projection of the edge AC, and "b" is the frontal projection of the edge AB
Rice. 86. Images of the subject
Figure 86 shows that if an edge is parallel to the projection plane, then it is depicted on this plane without distortion, or, as they say, in its true (natural) size. In this case, the projection of the edge and the edge itself are equal to each other. For example, projection a"b" is the true size of edge AB on the frontal plane, and projection a"b" is on the profile plane of projections.
If an edge is perpendicular to the projection plane, it is projected onto it to a point. Thus, edge AC was projected onto the frontal plane of projections into a point, edge AB onto the horizontal plane, edge BD onto the profile plane, etc.
Having constructed the projections of the edges, we see that in the image they limit the projections of the faces. Like an edge, a face parallel to the projection plane is projected onto it without distortion. For example, the face in which points A, B and C lie was projected onto the profile projection plane without distortion. The lower and upper faces, etc., were projected onto the horizontal projection plane without distortion. Find these faces on the drawing of the object in the system of rectangular projections.
If a face is perpendicular to the projection plane, it is projected onto it into a line segment.
Thus, each line segment in the image is a projection of an edge or a projection of a plane perpendicular to the plane of projections. The edges and faces of an object, inclined to the projection plane, are projected onto it with distortion. Find such edges and faces in Figure 86.
When constructing a drawing, you need to clearly imagine how each vertex, edge and face of the object will be depicted on it. When reading a drawing, you need to imagine which part of the object is hidden behind each point, segment or figure.
It should be remembered that each view is an image of the entire object, and not just one side of it. The only difference is that some faces are projected into a true figure, others into straight segments.
1. In what case do the projections of the points on the image coincide? Which of the two points whose projections on the horizontal plane coincide will be visible?
2. In what case is a straight line segment (edge) projected to its true value? exactly?
3. In what case is a face (part of a plane) projected onto a line segment? In what case will it be projected into its true value?
Rice. 87. Exercise tasks
1. Figure 87a shows a visual image and three projections of the part. The drawing shows the projections of point A, which is one of the vertices of the part.
2. Figure 87, b shows an image of the part.
Rice. 88. Image of surface elements of a part
3. In Figure 88, the edges of the object are highlighted in color. Label the vertices with letters or numbers. Analyze how the edges of the object are located relative to the projection planes. Write the answer in your workbook.
4. Redraw or transfer Figure 89 onto tracing paper and highlight the corresponding edges on all projections in the same color as in the visual images.
Rice. 89. Exercise tasks
5. Figure 90 shows images of three objects. The projections of their faces are indicated by letters. Write how these faces are located in each case relative to the frontal plane of projections. Example of a recording: A - parallel, B - perpendicular, C - oblique.
Rice. 90. Exercise tasks
12.2. Constructing projections of points on the surface of an object. Now let's look at ways to construct projections of points lying on the surfaces of objects.
Figure 91 shows a hexagonal pyramid. On a line that is a projection of an edge, a frontal projection a of point A is given. How to find its other projections?
Rice. 91. Construction of projections of a point lying on the edge of a pyramid
They reason like this. The point is on the edge of the object. The projections of the point must lie on the projections of this edge. Therefore, you must first find the projections of the edge, and then, using communication lines, find the projections of the point.
To construct a profile projection of an object and, in particular, a profile projection of the edge on which point A is located, it is convenient to use a constant straight line. This is the name of the line that is drawn to the right of the top view at an angle of 45° to the drawing frame (Fig. 91). The communication lines coming from the top view are brought to a constant straight line. From the points of their intersection, perpendiculars are drawn to the horizontal line and a profile projection is constructed.
Rice. 92. Construction of a constant line
The location of the constant straight line determines the location of the view under construction (Fig. 91). But if three views have already been constructed, as in Figure 92, a, you need to find a point through which a constant straight line will pass. To do this, it is enough to continue the horizontal and profile projections of the symmetry axis until they intersect each other. Through the resulting point k (Fig. 92, b) a straight line is drawn at an angle of 45° to the axes. This will be a constant straight line.
If there are no axes of symmetry in the drawing, then the horizontal and profile projections of the faces, projected in the form of straight segments, are continued until they intersect at point k 1. A constant straight line is drawn through point k 1.
Now let's return to Figure 91. The projections of the edge on which point A lies are highlighted in blue. The horizontal projection of point A must lie on the horizontal projection of the rib. Therefore, we draw a vertical connection line from point a. At the point where it intersects with the projection of the edge, there is point a - the horizontal projection of point A.
The profile projection a" of point A lies on the profile projection of the edge. It can also be defined as the point of intersection of communication lines.
We looked at how to find projections of points lying on the edges of objects in a drawing. However, it is often necessary to construct projections of points that lie not on the edges, but on the faces. For example, to drill a hole in a part, you need to determine where its center is.
In order to find the others using one projection of a point lying on the edge of an object, you must first find the projections of this face. You have already performed such exercises (see Fig. 89). Then, using connection lines, you need to find the projections of the point that should lie on the projections of the face.
The connection line is first drawn to the projection on which the face is depicted as a straight segment.
Rice. 93. Construction of projections of a point lying on the surface of an object
In Figure 93, the projections of the faces containing the projections of point A are highlighted in color. Point A is specified by the frontal projection a". The horizontal projection a of this point must lie on the horizontal projection of the face. To find it, draw a vertical connection line from point a".
To find a profile projection, you need to draw a horizontal connection line from point a". At the point of its intersection with a straight segment - the projection of the face - point a" lies.
The construction of projections of point B, issued by horizontal projection b, is also shown by connection lines with arrows.
1. Figure 94, a, b shows drawings in a system of rectangular projections and visual images of objects. In the views, letters indicate the projections of the vertices. Redraw or transfer the given images onto tracing paper. Label the remaining projections of the vertices with letters. Find these vertices in the visual images and label them with letters.
Rice. 94. Exercise tasks
2. Redraw or transfer the given images to tracing paper (Fig. 95) and construct the missing projections of the points specified on the edges of the object. Color the projections of the edges (each edge has its own color) containing points. Draw the points on the axonometric projection and highlight the edges on which the points lie with the same colors.
Rice. 95. Exercise task
3. Redraw or transfer Figure 96 onto tracing paper. Construct the missing projections of the points specified on the visible surfaces of the object. Color the projections of the surfaces on which the points lie (each surface has its own color). Highlight the surfaces of the object in the visual image with the same colors as in the drawing, and apply dots.
Rice. 96. Exercise task
4. Redraw or transfer Figure 97 onto tracing paper. Construct the missing projections of the points and label them with letters. Highlight in color, as in the previous task, the projections of the surfaces on which these points lie.
Rice. 97. Exercise task
The development of the lesson is recommended for teaching a lesson in 8th grade “Analysis of the geometric shape of an object” with a presentation attached to the lesson. Study and initial awareness of new educational material, understanding of connections and relationships in the objects of study. Formation and development of skills: remember geometric bodies, learn to find simple geometric bodies, read and draw drawings.
Drawing lesson in 8th grade.
Subject : "Analysis of the geometric shape of an object"
Bagomolova Lidiya Serafimovna teacher of fine arts and drawing,
GBOU Secondary School No. 416, Peterhof
year 2014
Lesson topic : Analysis of the geometric shape of an object.
1. Didactic rationale for the lesson
Lesson Objectives : study and initial awareness of new educational material. Understanding connections and relationships in objects of study.
To promote the formation and development of skills and abilities: remember geometric bodies, give the concept of analyzing the shape of an object, teach students to find simple geometric bodies in any technical detail.
Teach students to confidently distinguish models of geometric bodies and name them correctly.
Promote the development of students' speech.
Help develop spatial thinking.
To promote the formation and development of students’ cognitive interest in the subject.
Continue developing logical thinking techniques (comparison, analysis, synthesis).
Equipment:
For the teacher: three-dimensional models of geometric bodies: cube, prism, pyramid, ball, cylinder, cone; technical means: computer with MS Windows operating system, multimedia projector, screen. Presentation for the lesson.
For students: handouts in the form of cards - tasks containing visual images of geometric bodies; parts consisting of geometric bodies.
Lesson structure:
During the classes
Teacher:
Creating a problem situation: Please look at the drawing of the part, (slide) can you determine the shape of the part?
Students: Hard enough.
The topic of our lesson will help us with this. Write down the topic of today's lesson in your notebook (slide) “Analysis of the geometric shape of an object.” Read the topic again and try to determine the objectives of the lesson: What do you want to learn about? What questions have arisen?
Students: 1. What is analysis of the geometric shape of an object?
2. Why is it needed?
3. What geometric shapes exist?
Today in the lesson we must learn to analyze the geometric shape of objects, and for this we need the ability to listen, analyze, and be able to highlight the most important and essential.
It will help to reveal the topic of our lesson - the plan of our work. (slide-3)
We will consider the following questions:
I suggest you remember what geometric bodies are familiar to you from the subject “geometry”, and from our previous topics, when we built axonometric projections of flat figures and flat-sided objects?
Students: cylinder, cube, parallelepiped, etc.
Teacher: What is a geometric body? A geometric body is a closed part of space, limited by flat and curved surfaces.
All geometric bodies can be divided into two groups: Polyhedra - which have flat faces, and bodies of rotation, which have curved surfaces (slide) (write in a notebook).
Each geometric body has its own characteristics (slide)
By these characteristics we distinguish a ball from a cube, etc. You are already familiar with most of these bodies. We say "cube" and everyone imagines its shape. We say “ball” and again the image of a certain geometric body appears in our minds. Let's get to know them better. (slides)
Now let's check how well you can imagine images of geometric bodies. There are cards on your tables. Assignment: Write down in a notebook in one column the numbers of images of faceted geometric bodies and their names, and in the other column - bodies of revolution. (slide)
Let's check how the guys coped with the task.
(If necessary, everyone together corrects errors in the answers)
Faceted geometric bodies include: 1. hexagonal prism, 2. hexagonal pyramid, 3. parallelepiped, 4. cube, 5. hexagonal truncated pyramid, 6. hexagonal prism, 7. hexagonal truncated prism.
To geometric bodies of revolution. 1. cylinder, 2. cone, 3. frustum. 4. ball, 5. Thor.
Take a close look at the objects around us.
They also take the form of geometric solids or a combination thereof. I name bodies, and you give examples of objects:
Ball-pyramid - prism-cone-cylinder-torus.
In engineering, the shape of a part is often compared with simpler shapes - geometric bodies, and also the shapes of geometric bodies are used to describe the shape of more complex parts (slide).
Any simple shape of a technical part can be represented as the shape of a geometric body (for example, the shape of a technical part “axle” can be represented as a cylinder shape - (slide), and the shape of a complex product can be represented as a combination of shapes of geometric bodies (for example, a part “fork”)
The considered approach to the study of parts is based on an analysis of its geometric shape.
Analysis of the geometric shape of an object is the mental division of an object into its constituent geometric bodies. (write in notebook) (slide).
Let's consider how the geometric shape of an object is analyzed using a visual image of the part. We mentally divide the part into simple geometric bodies, name them and tell how they are located relative to each other in space (slide).
An image of the part is given. What is its shape? It is composed of a rectangular parallelepiped, two half-cylinders and a truncated cone located on top. The part has a cylindrical hole.
By using the method of dividing a part into simple geometric bodies, you can learn to quickly, correctly read drawings and execute them competently.
Task: analyze the shape of the part you looked at at the beginning of the lesson (slide).
The “Support” part consists of a rectangular parallelepiped with five through cylindrical holes. In the center of the upper face of the rectangular parallelepiped there is a quadrangular prism with a through cylindrical hole, the axis and diameter of which coincide with the axis and diameter of the hole of the part. The parallelepipeds are connected to each other by two stiffening ribs in the shape of triangular prisms, which ensures their stable fastening.
