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

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

- the procedure and methods for constructing unfolded surfaces;

- mutual correspondence between the surface and its development;

- special cases of building sweeps.

Decision proceduresproblems

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 sweep of a straight 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.

Figure 9.1

- the circumference of the base of the cone, expressed in natural size, we divide by a number of shares: in our case - 10, the accuracy of building the sweep depends on the number of shares ( figure 9.1.a);

- we postpone the received shares, replacing them with chords, on the length
arc drawn with a radius equal to the length of the generatrix of the cone l \u003d | Sb |. We connect the beginning and end of the counting of the shares to the top of the sector - this will be the sweep of the lateral 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 both in the first and in the second case the extreme right or left generators of the cone l \u003d | Sb | are taken as the radius, since they are expressed in natural size;

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

Figure 9.2

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

l - the length of the generatrix of the cone;

360 - constant value converted to degrees.

To the unfolded 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 intersection line
surfaces of the cone and cylinder for a scan. To do this, we use the properties of one-to-one between the surface and its flat pattern, in particular, note that each point on the surface corresponds to a point on the flat pattern and each line on the surface corresponds to a line on the flat pattern.

Hence follows the sequence of transferring points and lines
from the surface to the sweep.

Figure 9.3

For sweeping the 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 respectively equal to the distances taken along the generatrix S’ A’ from the top S’ to the corresponding secant plane with points 1’ , 2’, 3’…6’ -| S1|, | S2|, | S3|….| S6 | (Figure 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 \u003d 35 mm ( figure 9.1.a). If the distance along the chord and the arc is very different, then to reduce the error, you can divide a larger number of fractions and put them on the corresponding sweep arcs. In this way, any points are transferred from the surface to its flat pattern. The resulting points will be connected by a smooth curve along the pattern ( figure 9.3).

For cylinder sweep.

A cylinder sweep is a rectangle with a height equal to the generatrix height and a length equal to the circumference of the cylinder base. Thus, to construct a sweep of a straight 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\u003d 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 resulting scan.

Let us agree that the cut is made along the generatrix AA 1 (A’ A’ 1 ; AA1) ... Note that the cut should be made using characteristic (control) points for more convenient construction. Considering 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 must be divided into shares) to the point B’ (in our example - 17mm) and put it on the sweep (along the length of the base of the cylinder) from point A. From the resulting point B, draw a perpendicular (generatrix of the cylinder). Point 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 \u003d 50mm (fig. 9.4).

Figure 9.4

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

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

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

There are 2 ways to construct a flat pattern of a cone:

• Divide the base of the cone into 12 parts (we enter the correct 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.
• Constructing a sweep of a cone, using the formula that determines the angle of the circular sector.

Since we need to plot the intersection lines of the cone and the cylinder on the sweep of the cone, we still have to divide the base of the cone into 12 parts and inscribe a pyramid, so we will go straight along 1 path of constructing the sweep 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.
• We draw generators through the characteristic points of intersection of the cone and the cylinder.
• Find the actual size of the generators.
• We build the data generators on the unfolded cone.
• We connect the characteristic points of intersection of the cone and the cylinder on the flat pattern.

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

During the construction of the unfolding of the cone, we will use the Array in AutoCAD - Circular array and array along the path. I recommend viewing these AutoCAD video tutorials. At the time of this writing, the AutoCAD 2D video course contains a classic way of constructing a circular array and an interactive one when constructing an array along a path.

we take the perpendiculars to each segment, on them we lay down the real values \u200b\u200bof the generatrices of the cylinder, taken from the frontal projection. Connecting the obtained points together, we get a curve.

To obtain a full sweep, add a circle (base) and the actual size of the section (ellipse) to the sweep of the side surface, built along its major and minor axes or along points.

5.3.4. Creation of a flattened cone flat pattern

IN in a particular case, the sweep of a cone is a flat figure consisting of a circular sector and a circle (the base of the cone).

IN in the general case, the unfolding of the surface is carried out according to the principle of unfolding a polyhedral pyramid (that is, by the method of triangles) inscribed in a conical surface. The greater the number of faces of the pyramid inscribed in the conical surface, the smaller the difference between the actual and approximate sweeps of the conical surface will be.

The construction of the sweep of the cone begins with drawing from the point S 0 a circular arc 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 scan of a truncated cone is shown in Fig. 5.7.

Lecture 6 (beginning)

MUTUAL CROSSING OF SURFACES. METHODS FOR CONSTRUCTION OF MUTUAL CROSSING OF SURFACES.

METHOD OF AUXILIARY SECTIONAL PLANES AND SPECIAL CASES

6.1. Mutual intersection of surfaces

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

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

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

6.2. Intersection of faceted surfaces

The intersection line of two polyhedrons is a closed spatial polyline. Its links are the lines of intersection of the faces of one polyhedron with the faces of another, and the vertices are the intersection points of the edges of one polyhedron with the faces of another. Thus, to construct a line of intersection of two polyhedra, you need to solve the problem either on the intersection of two planes (facet method), or on the intersection of a straight line with a plane (edge \u200b\u200bmethod). In practice, both methods are usually used in combination.

Intersection of a pyramid with a prism. Consider the case of intersection

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

Analysis of the problem statement 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 located on the projections of the corresponding edges and are determined by the communication lines. Similarly, points 5, 6, 7 and 8 can be found belonging to another branch. Points 9, 10, 11, 12 are determined from the condition that the upper and lower edges of the prism are parallel to each other, that is, 1 "2" is parallel to 5 "10", etc.

You can use the construction clipping planes method. The construction plane intersects both surfaces along the broken lines. The mutual intersection of these lines gives us the points belonging to the desired intersection line. Select α "" "and β" "" as auxiliary planes. Using the plane α "" "

we find projections of points 1 ", 2", 3 ", 4", and 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

from surfaces of revolution

Most of the technical parts and objects are composed 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 build a line of intersection of two surfaces, you need to find points that would simultaneously belong to two surfaces.

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

If there is a complete intersection (penetration), then two closed curved lines are formed, and if an incomplete intersection, 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 construction plane intersects both surfaces along curved lines and along broken lines. The mutual intersection of these lines gives us the points belonging to the desired intersection line.

Let it be required to construct the projection of the line of intersection of the surfaces of the cylinder and the triangular prism. As seen from Fig. 6.2, all three faces of the prism participate in the intersection. Two of them are directed at a certain 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, i.e., intersects 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 seen from Fig. 6.2, the lateral surface of the cylinder is horizontal

tally-projecting, that is, perpendicular to the horizontal plane of the projections. The lateral surface of the prism is profile-projection, that is, each face of it is perpendicular to the profile plane of the projections. Consequently, the horizontal projection of the line of intersection of the bodies coincides with the horizontal projection of the cylinder, and the profile projection - with the profile projection of the prism. Thus, in the drawing, you only need to build a frontal projection of the intersection line.

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

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

In order to find intermediate points (there are four such points in total, but let us designate one of them as A) of the intersection lines of the cylinder with the prism faces, we intersect both surfaces with a projection plane or a level plane. Take, for example, the horizontal plane α. The α plane intersects the prism faces along two straight lines, and the cylinder intersects in a circle. These lines intersect at point A "(one point is signed, and the rest are not), which belongs to both the surface of the cylinder (lies on the 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 found first on the profile projection of the polyhedron (where they were projected to point A "" "and a symmetric point), and then, using communication lines, were constructed on the horizontal projection of the prism. Point A and symmetric points were obtained at the intersection of the horizontal projection of the intersection lines (plane α with the prism) with the circle and using the communication lines are found on the frontal projection.

You will need

• Pencil Ruler Square Compass Protractor Formulas for calculating the angle by arc length and radius Formulas for calculating the sides of geometrical figures

Instructions

Draw the base of the desired geometric body on a piece of paper. If you are given a parallelepiped or, measure the length and width of the base and draw a rectangle with the appropriate parameters on a sheet of paper. To build a flat pattern 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 of 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 to its width. Set aside the height of the parallelepiped on these lines and connect the resulting 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 coincides with the length of the base, draw the top face of the parallelelepiped. 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 create a flat pattern 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. Build this distance from the intersection of the radius and the circle. Mark the vertex point of the side surface. By 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 the sweep and set it aside from the straight line already drawn through the top of the base. Using a compass, connect the previously found intersection point of the radius and circle with this new point. The sweep of the cone is ready.

To build a flat pattern of a pyramid, 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. Center the obtained points with the intersection points of the sides of the pyramid.

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

Or

Where:
- radius of the circle,
- the diameter of the circle,
- circumference,
- Pi (),
As a rule, the value () up to the second digit (3.14) is used for the calculation, but in some cases this may not be enough.

• Truncated cone with accessible vertex: A cone that can be used to define the position of the vertex.
• Truncated cone with 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 of constructing unfolding surfaces of non-developing, conical, general view and with a cusp edge.
• Remember: Regardless of whether the surface in question is developable or non-deployable, only an approximate flat pattern 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 the design features of the drawing tools, the physical capabilities of the eye and errors from replacing arcs with chords and angles on the surface with flat angles. Approximate sweeps of curves of non-developable surfaces, in addition to graphic errors, contain errors obtained due to the mismatch of the elements of such surfaces with flat approximating elements. Therefore, to obtain a surface from such a scan, in addition to bending, it is necessary to carry out partial stretching and compression of its individual sections. Close-up sweeps, when carefully executed, are accurate enough for practical purposes.

The material presented in the article implies that you have an idea of \u200b\u200bthe basics of drawing, know how to divide a circle, find the center of a segment using a compass, remove / transfer dimensions with a compass, use templates, and the corresponding reference material. Therefore, the explanation of many points in the article is omitted.

Building a cylinder unfolded

Cylinder

A body of revolution with the simplest sweep, in 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 flat pattern, draw a quadrilateral ACDE in full size (see drawing).
• Let's draw a perpendicular BD, from the plane AC exactly D, cutting off the straight part of the cylinder from the construction ABDE, which can be completed as needed.
• From the center of the plane CD (point 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.

Build:

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

Any cylindrical sections are constructed in the same way.
Note: Why "Rybina" - if you continue to build the sweep, while building half from the point D, and the second in the opposite direction from the vertical BC, then the resulting drawing will look like a fish, or a fish tail.

Building a sweep of a cone

Cone

Unfolding 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, a compass draws an arc with a radius equal to the side of the cone. Two points are laid on the arc ( A 1 and B 1 ABOUT.
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 laid on the arc ( A 1 and B 1), at a distance equal to the circumference and are connected to the point ABOUT.

For convenience, you can set aside half the length of the circle, on either side of the centerline of the cone.
A cone with an offset top is constructed in the same way as a truncated cone with offset bases.

1. Construct the circumference of the base of the cone in top view, in 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.

Cone with a polyhedral base

1. If the cone has an even, radial base: ( When constructing a circle in the top view, by placing a compass in the center, and outlining a circle along an arbitrary vertex, all the vertices of the base are placed on an arc of a circle.) Construct a cone, by analogy with the development of an ordinary cone (build the base in a circle, from the top view). Postpone arc from point O... Put a point in an arbitrary part of the arc A 1, and one by one put all the edges of the base on the arc. The end point of the last face will be B 1.
2. In all other cases, the cone is constructed according to the triangulation principle ( see further).

Truncated cone with accessible top

Frustum

Construct a truncated cone ABCD life-size (See drawing).
Parties AD and BC continue until the intersection point appears O... From the point of intersection O, draw arcs, with a radius OB and OC.
On an arc OC, set aside the circumference DC... On an arc OB, set aside the circumference AB... Connect the obtained points with segments L 1 and L 2.
For convenience, you can set aside half the length of the circle, on either side of the centerline of the cone.

How to plot the length of a circle on an arc:

1. Using 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 L 1 and L 2if they continue, they converge at the point O... To be completely honest, they should converge, but taking into account the corrections for the errors of the instrument, material and eye gauge - the intersection point may be slightly below or above the top, which is not an error.

Truncated cone with a transition from circle to square

Cone with a transition from circle to square

Training:
Construct a truncated cone ABCD life-size (see drawing), build a top view ABB 1 A 1... Divide the circle into equal parts (in the given example, division of one quarter is shown). Points AA 1 -AA 4 connect with a point A... Draw 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.
Build:

• Take off size AD and build an arbitrary vertical AA 0 -AA 1... Take off size AA 0 -A, and put an "approximate point", making the go-ahead with a compass. Take off size A-AA 1, and on the axis O, from point O O 1 AA 1, to the expected point A... Connect points AA 0 -A-AA 1.
• Take off size AA 1 -AA 2, from point AA 1 put an "approximate point" by making a go-ahead with a compass. Take off size A-AA 2, and on the axis O, from point O, postpone the segment, take the dimension from the obtained point to the point O 1... Make a waveform with a compass from a point A, to the expected point AA 2... Draw segment A-AA 2... Repeat until the segment is delayed A-AA 4.
• Take off size A-AA 5, from point A put a "rough point" AA 5... Take off size AA 4 -AA 5, and on the axis O, from point O, postpone the segment, take the dimension from the obtained point to the point O 1... Make a waveform with a compass from a point AA 4, to the expected point AA 5... Draw 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 accessible vertexand the base is cone with a rectangular (polyhedral) base... The accuracy will be lower, but the construction is much easier.