There are 2 ways to build a cone sweep:

• Divide the base of the cone into 12 parts (we enter a regular polyhedron - a pyramid). You can divide the base of the cone into more or less parts, because. the smaller the chord, the more accurate the construction of the sweep of the cone. Then transfer the chords to the arc of the circular sector.
• Construction of a sweep of the cone, according to the formula that determines the angle of the circular sector.

Since we need to plot the lines of intersection of the cone and the cylinder on the development of the cone, we still have to divide the base of the cone into 12 parts and inscribe the pyramid, so we will immediately follow the 1st path for constructing the development of the cone.

Algorithm for constructing a sweep of a cone

• We divide the base of the cone into 12 equal parts (we enter the correct pyramid).
• We build the lateral surface of the cone, which is a circular sector. The radius of the circular sector of the cone is equal to the length of the generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone. We transfer 12 chords to the arc of the sector, which will determine its length, as well as the angle of the circular sector.
• We attach the base of the cone to any point of the arc of the sector.
• Through the characteristic points of intersection of the cone and the cylinder we draw generators.
• Find the natural size of the generators.
• We build data generators on the development of the cone.
• We connect the characteristic points of intersection of the cone and the cylinder on the sweep.

More details in the video tutorial on descriptive geometry in AutoCAD.

During the construction of the sweep of the cone, we will use the Array in AutoCAD - a Circular array and an array along the path. I recommend watching these AutoCAD video tutorials. Video course AutoCAD 2D at the time of writing contains classic way building a circular array and interactive when building an array along a path.

You will need

• Pencil Ruler square compasses protractor Formulas for calculating the angle from the length of the arc and radius Formulas for calculating the sides of geometric shapes

Instruction

On a sheet of paper, build the base of the desired geometric body. If you are given a box or , measure the length and width of the base and draw a rectangle on a piece of paper with the appropriate parameters. To build a sweep of a or a cylinder, you need the radius of the base circle. If it is not specified in the condition, measure and calculate the radius.

Consider a parallelepiped. You will see that all its faces are at an angle to the base, but the parameters of these faces are different. Measure the height of the geometric body and use a square to draw two perpendiculars to the length of the base. Set aside the height of the parallelepiped on them. Connect the ends of the resulting segments with a straight line. Do the same on the opposite side of the original.

From the points of intersection of the sides of the original rectangle draw perpendiculars and to its width. Set aside the height of the parallelepiped on these straight lines and connect the obtained points with a straight line. Do the same on the other side.

From the outer edge of any of the new rectangles, the length of which is the same as the length of the base, build the upper face of the box. To do this, draw perpendiculars from the intersection points of the length and width lines located on the outside. Set aside the width of the base on them and connect the points with a straight line.

To build a sweep of a cone through the center of the base circle, draw a radius through any point on the circle and continue it. Measure the distance from the base to the top of the cone. Set aside this distance from the point of intersection of the radius and the circle. Mark the top point of the side surface. Based on the radius of the side surface and the length of the arc, which is equal to the circumference of the base, calculate the angle of development and set aside it from the straight line already drawn through the top of the base. Using a compass, connect the intersection point of the radius and circle found earlier with this new point. The reaming of the cone is ready.

To build a pyramid sweep, measure the heights of its sides. To do this, find the middle of each side of the base and measure the length of the perpendicular dropped from the top of the pyramid to this point. Having drawn the base of the pyramid on the sheet, find the midpoints of the sides and draw perpendiculars to these points. Connect the obtained points with the points of intersection of the sides of the pyramid.

The development of a cylinder consists of two circles and a rectangle located between them, the length of which is equal to the length of the circle, and the height is equal to the height of the cylinder.

We eat perpendiculars to each segment, on them we set aside the actual values ​​\u200b\u200bof the generatrix of the cylinder, taken from the frontal projection. By connecting the obtained points together, we get a curve.

To obtain a complete development, add a circle (base) and a full-scale section (ellipse) to the development of the lateral surface, built along its major and minor axes or points.

5.3.4. Building a development of a truncated cone

V special case, the development of the cone is flat figure, consisting of a circular sector and a circle (the base of the cone).

V In the general case, the surface is unfolded according to the principle of unfolding a polyhedral pyramid (i.e., by the method of triangles) inscribed in a conical surface. How more faces of a pyramid inscribed in a conical surface, the smaller the difference between the actual and approximate scans of the conical surface.

The construction of the development of the cone begins with drawing from the point S 0 an arc of a circle with a radius equal to the length of the generatrix of the cone. On this arc, 12 parts of the circumference of the base of the cone are laid, and the resulting points are connected to the top. An example of an image of a full sweep of a truncated cone is shown in fig. 5.7.

Lecture 6 (beginning)

MUTUAL INTERCECTION OF SURFACES. METHODS FOR CONSTRUCTING MUTUAL INTERCECTION OF SURFACES.

METHOD OF AUXILIARY CUT PLANES AND SPECIAL CASES

6.1. Mutual intersection of surfaces

Intersecting each other, the surfaces of the bodies form various broken or curved lines, which are called lines of mutual intersection.

To construct lines of intersection of two surfaces, you need to find points that simultaneously belong to two given surfaces.

When one of the surfaces completely penetrates the other, 2 separate lines of intersection are obtained, called branches. In the case of a tie-in, when one surface partially enters another, the line of intersection of the surfaces will be one.

6.2. Intersection of faceted surfaces

The line of intersection of two polyhedra is a closed spatial broken line. Its links are the lines of intersection of the faces of one polyhedron with the faces of another, and the vertices are the points of intersection of the edges of one polyhedron with the faces of another. Thus, to build a line of intersection of two polyhedra, you need to solve the problem either for the intersection of two planes (face method), or for the intersection of a straight line with a plane (edge ​​method). In practice, both methods are usually used in combination.

The intersection of the pyramid with a prism. Consider the case of crossing

of a pyramid with a prism, the lateral surface of which is projected onto π3 on the outline bases (quadrilateral). We start the construction with a profile projection. When drawing points, we will use the edge method, that is, when the edges of a vertical pyramid intersect the faces of a horizontal prism (Fig. 6.1).

Analysis of the condition of the problem shows that the line of intersection of the pyramid and the prism splits into 2 branches, one of the branches is a flat polygon, points 1, 2, 3, 4 (points of intersection of the edges of the pyramid with the face of the prism). Their horizontal, frontal and profile projections are on the projections of the corresponding edges and are determined by communication lines. Similarly, points 5 , 6 , 7 and 8 belonging to another branch can be found. Points 9, 10, 11, 12 are determined from the condition that the upper and lower faces of the prism are parallel to each other, i.e. 1 "2" is parallel to 5 "10", etc.

You can use the method of auxiliary cutting planes. The auxiliary plane intersects both surfaces along broken lines. The mutual intersection of these lines gives us the points that belong to the desired line of intersection. We choose α""" and β""" as auxiliary planes. Using the plane α"""

we find the projections of points 1 ", 2" , 3 ", 4" , and the planes β """ - points 5" , 6" , 9 " , 10" , 11 " , 12 ". Points 7 and 8 are determined as in the previous method .

6.3. Intersection of faceted surfaces

With surfaces of revolution

Most technical details and objects consist of a combination of various geometric bodies. Intersecting with each other

the surfaces of these bodies form various straight or curved lines, which are called lines of mutual intersection.

To construct a line of intersection of two surfaces, you need to find such points that would simultaneously belong to two surfaces.

When a polyhedron intersects with a surface of revolution, a spatial curve line of intersection is formed.

If a complete intersection (penetration) occurs, then two closed curved lines are formed, and if an incomplete intersection occurs, then one closed spatial intersection line.

To construct a line of mutual intersection of a polyhedron with a surface of revolution, the method of auxiliary cutting planes is used. The auxiliary plane intersects both surfaces along a curve and along a broken line. The mutual intersection of these lines gives us the points that belong to the desired line of intersection.

Let it be required to construct projections of the line of intersection of the surfaces of the cylinder and the triangular prism. As can be seen from fig. 6.2, all three faces of the prism participate in the intersection. Two of them are directed at some angle to the axis of rotation of the cylinder, therefore, they intersect the surface of the cylinder in ellipses, one face is perpendicular to the axis of the cylinder, that is, it crosses it in a circle.

Solution plan:

1) find the points of intersection of the edges with the surface of the cylinder;

2) find the lines of intersection of the faces with the surface of the cylinder. As can be seen from fig. 6.2, the side surface of the cylinder is horizontal

tally-projecting, i.e. perpendicular to the horizontal plane of projections. The lateral surface of the prism is profile-projecting, i.e., each of its faces is perpendicular to the profile projection plane. Consequently, the horizontal projection of the line of intersection of the bodies coincides with the horizontal projection of the cylinder, and the profile projection coincides with the profile projection of the prism. Thus, in the drawing, only a frontal projection of the intersection line needs to be built.

We begin the construction by drawing characteristic points, i.e., points that can be found without additional constructions. These are points 1, 2 and 3. They are located at the intersection of the outline generatrix of the frontal projections of the cylinder with the frontal projection of the corresponding edge of the prism with the help of communication lines.

Thus, the points of intersection of the edges of the prism with the surface of the cylinder are constructed.

In order to find intermediate points (there are four such points, but we will designate one of them A) of the lines of intersection of the cylinder with the faces of the prism, we intersect both surfaces with some projecting plane or level plane. Take, for example, the horizontal plane α. The plane α intersects the faces of the prism along two straight lines, and the cylinder - along a circle. These lines intersect at point A "(one point is signed, but the rest is not), which simultaneously belongs to the surface of the cylinder (lies on a circle that belongs to the cylinder) and the surface of the prism (lies on straight lines that belong to the faces of the prism).

The straight lines along which the faces of the prism intersect with the plane α were first found on the profile projection of the polyhedron (there they were projected to the point A """ and the symmetrical point), and then, using the connection lines, were constructed on the horizontal projection of the prism. Point A and the symmetrical points were obtained at the intersection of the horizontal projection of the intersection lines (plane α with a prism) with a circle and with the help of communication lines found on the frontal projection.

In the manufacture of reamers on metal, a meter ruler, a scriber, a compass for metal, a set of patterns, a hammer and a core are used to mark the nodal points.

The circumference is calculated by the formula:

Or

Where:
- radius of the circle,
- circle diameter,
- circumference,
- Pi (),
As a rule, the value () up to the second sign (3,14) is used for calculation, but in some cases, this may not be enough.

• Truncated cone with accessible vertex: A cone that can be used to determine the position of the vertex.
• A truncated cone with an inaccessible vertex: A cone, during the construction of which the position of the vertex is difficult to determine, in view of its remoteness.
• Triangulation: a method for constructing unfolded surfaces of non-developing, conical, general form and with a cusp.
• It should be remembered: Regardless of whether the surface under consideration is developable or non-developable, only an approximate development can be plotted graphically. This is due to the fact that in the process of removing and postponing dimensions and performing other graphic operations, errors are inevitable due to design features drawing tools, the physical capabilities of the eye and errors from replacing arcs with chords and angles on the surface with flat angles. Approximate developments of curves of non-developing surfaces, in addition to graphical errors, contain errors obtained due to the mismatch of the elements of such surfaces with flat approximating elements. Therefore, in order to obtain a surface from such a development, in addition to bending, it is necessary to partially stretch and compress its individual sections. Approximate scans, when carefully performed, are accurate enough for practical purposes.

The material presented in the article implies that you have an idea about the basics of drawing, know how to divide a circle, find the center of a segment with a compass, take / transfer dimensions with a compass, use patterns, and relevant reference material. Therefore, the explanation of many points in the article is omitted.

Construction of a cylinder sweep

Cylinder

A body of revolution with the simplest unfolding, having the shape of a rectangle, where two parallel sides correspond to the height of the cylinder, and the other two parallel sides correspond to the circumference of the bases of the cylinder.

Truncated cylinder (fish)

truncated cylinder

Training:

• To create a sweep, draw a quadrilateral ACDE full size (see drawing).
• Let's draw a perpendicular BD, out of plane AC exactly D, cutting off from the construction the straight part of the cylinder ABDE which can be adjusted as needed.
• From the center of the plane CD(dot O) draw an arc with a radius of half the plane CD, and divide it into 6 parts. From the resulting points O, draw perpendicular lines to the plane CD. From points on a plane CD, draw straight lines perpendicular to the plane BD.

Building:

• Section BC transfer, and turn it into a vertical. From a point B, vertical BC, draw a ray perpendicular to the vertical BC.
• Take the size with a compass C-O 1 B, point 1 . We remove the size B1-C1 1 .
• Take the size with a compass O 1 -O 2, and set aside on the beam, from the point 1 , point 2 . We remove the size B2-C2, and set aside the perpendicular from the point 2 .
• Repeat until point is delayed D.
• The resulting verticals, from the point C, vertical BC, to the point D- connect with a curved curve.
• The second half of the sweep is mirrored.

Any cylindrical slices are constructed in a similar way.
Note: Why "Rybina"- if you continue building a sweep, while building half from the point D, and the second in reverse side from the vertical BC, then the resulting pattern will look like a fish, or a fish tail.

Construction of a development of a cone

Cone

The reaming of the cone can be done in two ways. (See drawing)

1. If the size of the side of the cone is known, from the point O, an arc is drawn with a compass, with a radius equal to the side of the cone. Two points are plotted on the arc ( A 1 and B1 O.
2. A life-size cone is built, from a point O, exactly A, a compass is placed, and an arc is drawn passing through the points A and B. Two points are plotted on the arc ( A 1 and B1), at a distance equal to the circumference and connected to a point O.

For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.
A cone with a displaced apex is constructed in the same way as a truncated cone with displaced bases.

1. Construct the circumference of the base of the cone in top view, full size. Divide the circle into 12 or more equal parts, and put them on a straight line one by one.

A cone with a rectangular (polyhedral) base.

Cones with polyhedral base

1. If the cone has an even, radial base: ( When constructing a circle in a top view, by setting the compass to the center, and outlining the circle along an arbitrary vertex, all the vertices of the base fit on the arc of the circle.) Construct a cone, by analogy with the development of an ordinary cone (build the base in a circle, from a top view). Draw an arc from a point O. Put a point in an arbitrary part of the arc A 1, and alternately put all the faces of the base on the arc. The end point of the last face will be B1.
2. In all other cases, the cone is built according to the principle of triangulation ( see below).

Truncated cone with accessible apex

Frustum

Construct a truncated cone ABCD full size (See drawing).
Parties AD and BC continue until the intersection point appears O. From the point of intersection O, draw arcs, with radius OB and OC.
On the arc OC, set aside the circumference DC. On the arc OB, set aside the circumference AB. Connect the resulting points with segments L1 and L2.
For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.

How to plot the circumference of an arc:

1. With the help of a thread, the length of which is equal to the circumference.
2. With the help of a metal ruler, which should be bent “in an arc”, and put the appropriate risks.

Note: It is not at all necessary that the segments L1 and L2, if they continue, will converge at a point O. To be completely honest, they should converge, but taking into account the corrections for the errors of the tool, material and eye, the intersection point may be slightly lower or higher than the top, which is not a mistake.

Truncated cone with a transition from a circle to a square

Cone with a transition from a circle to a square

Training:
Construct a truncated cone ABCD full size (see drawing), build a top view ABB 1 A 1. Divide the circle into equal parts (in the above example, the division of one quarter is shown). points AA 1-AA 4 connect segments with a dot A. Hold Axis O, from the center of which draw a perpendicular O-O 1, with a height equal to the height of the cone.
Below, the primary dimensions are taken from the top view.
Building:

• Remove size AD and build an arbitrary vertical AA0-AA1. Remove size AA0-A, and put an "approximate point" by making a go-ahead with a compass. Remove size A-AA 1, and on the axis O, from the point O O 1 AA 1, to the expected point A. Connect dots with line segments AA 0 -A-AA 1.
• Remove size AA 1-AA 2, from the point AA 1 put a "approximate point", making a go-ahead with a compass. Remove size A-AA 2, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point A, to the expected point AA 2. Draw a segment A-AA 2. Repeat until the segment is delayed A-AA 4.
• Remove size A-AA 5, from the point A set a point AA5. Remove size AA 4-AA 5, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point AA 4, to the expected point AA5. Draw a segment AA 4-AA 5.

Build the rest of the segments in the same way.
Note: If the cone has an accessible vertex, and SQUARE foundation - then the construction can be carried out according to the principle truncated cone with an accessible vertex, and the base is cones with a rectangular (polyhedral) base. The accuracy will be lower, but the construction is much simpler.

It is necessary to build a development of surfaces and transfer the line of intersection of the surfaces to the development. This problem is based on surfaces ( cone and cylinder) with their line of intersection given in previous task 8.

To solve such problems in descriptive geometry, you need to know:

- the order and methods of constructing surface developments;

- mutual correspondence between the surface and its development;

- special cases of constructing sweeps.

Solution orderhadachi

1. Note that a sweep is a figure obtained in
as a result of cutting the surface along some generatrix and gradually unbending it until it is completely aligned with the plane. Hence the development of a right circular cone - a sector with a radius equal to the length of the generatrix, and a base equal to the circumference of the base of the cone. All sweeps are built only from natural values.

Fig.9.1

- the circumference of the base of the cone, expressed in natural value, is divided into a number of shares: in our case - 10, the accuracy of the sweep construction depends on the number of shares ( fig.9.1.a);

- we postpone the received shares, replacing them with chords, on the length
an arc drawn with a radius equal to the length of the generatrix of the cone l=|Sb|. We connect the beginning and end of the count of the shares with the top of the sector - this will be the development of the side surface of the cone.

Second way:

- we build a sector with a radius equal to the length of the generatrix of the cone.
Note that in both the first and second cases, the extreme right or left generators of the cone l=|Sb| are taken as the radius, because they are expressed in natural size;

- at the top of the sector, we set aside the angle a, determined by the formula:

Fig.9.2

where r- the value of the radius of the base of the cone;

l is the length of the generatrix of the cone;

360 is a constant value converted to degrees.

To the sweep sector we build the base of the cone of radius r.

2. According to the conditions of the problem, it is required to move the line of intersection
surfaces of the cone and cylinder on the development. To do this, we use the properties of one-to-one between the surface and its development, in particular, we note that each point on the surface corresponds to a point on the development and to each line on the surface there corresponds a line on the development.

From this follows the sequence of transferring points and lines
from the surface to the development.

Fig.9.3

For reaming a cone. Let us agree that the cut of the surface of the cone is made along the generatrix S’ a’ . Then the points 1, 2, 3,…6
will lie on circles (arcs on the sweep) with radii correspondingly equal to the distances taken along the generatrix S’ A’ from the top S’ to the corresponding cutting plane with points 1’ , 2’, 3’…6’ -| S1|, | S2|, | S3|….| S6| (Fig.9.1.b).

The position of the points on these arcs is determined by the distance taken from the horizontal projection from the generatrix Sa along the chord to the corresponding point, for example, to the point c, ac=35 mm ( fig.9.1.a). If the distance along the chord and the arc are very different, then to reduce the error, you can divide a larger number of shares and put them on the corresponding sweep arcs. In this way, any points are transferred from the surface to its development. The resulting points will be connected by a smooth curve along the pattern ( fig.9.3).

For cylinder reaming.

The development of a cylinder is a rectangle with a height equal to the height of the generatrix and a length equal to the circumference of the base of the cylinder. Thus, to construct a sweep of a right circular cylinder, it is necessary to construct a rectangle with a height equal to the height of the cylinder, in our case 100mm, and a length equal to the circumference of the base of the cylinder, determined by the known formulas: C=2 R=220mm, or by dividing the circumference of the base into a series of shares, as indicated above. We attach the base of the cylinder to the upper and lower parts of the obtained sweep.

Let us agree that the cut is made along the generatrix AA 1 (A’ A’ 1 ; AA1) . Note that the cut should be made along the characteristic (reference) points for a more convenient construction. Given that the length of the sweep is the circumference of the base of the cylinder C, from point A’= A’ 1 section of the frontal projection, we take the distance along the chord (if the distance is large, then it is necessary to divide it into shares) to the point B’ (in our example, 17mm) and put it aside on the scan (along the length of the base of the cylinder) from point A. From the resulting point B we draw a perpendicular (generatrix of the cylinder). Dot 1 should be on this perpendicular) at a distance from the base, taken from the horizontal projection to the point. In our case, the point 1 lies on the axis of symmetry of the sweep at a distance 100/2=50mm (fig.9.4).

Fig.9.4

And so we do to find all other points on the sweep.

We emphasize that the distance along the length of the sweep to determine the position of the points is taken from the frontal projection, and the distance along the height is taken from the horizontal, which corresponds to their natural values. We connect the obtained points with a smooth curve along the pattern ( fig.9.4).

In problem variants, when the intersection line splits into several branches, which corresponds to the complete intersection of surfaces, the methods for constructing (transferring) the intersection line to the development are similar to those described above.

Section: Descriptive geometry /