When performing engineering drawings, the most common case is the intersection of two cylindrical surfaces, the axes of which are located at an angle of 90 °. Let's analyze an example of constructing a line of intersection of the surfaces of two straight circular cylinders, the axes of which are perpendicular to the projection planes (Figure 201). At the beginning of the construction, as is known, the projections of obvious points /, 3 and 5 are found. The construction of the projection of intermediate points is shown in Figure 201. If in this example we apply the general method of constructing intersection lines using auxiliary mutually parallel planes that intersect both cylindrical surfaces along the generators, then at the intersection of these generators the desired intermediate points of the intersection line will be found (for example, points 2, 4 in Figure 201). However, in this case, such a construction is not necessary for the following reasons. The horizontal projection of the desired line of intersection of the surfaces coincides with the circle - the horizontal projection of the large cylinder. The profile projection of the intersection line also coincides with the circle - the profile projection of the small cylinder. Thus, the frontal projection of the desired intersection line is easy to find by the general rule for constructing a curved line from points, when two projections of points are known. For example, on the horizontal projection of the point 2 "find the profile projection 2"". On two projections 2" and 2"" determine the frontal projection 2" of point 2 belonging to the line of intersection of the cylinders. The construction of an isometric projection of intersecting cylinders (Figure 202) begins with the construction of an isometric projections of a vertical cylinder.Further through the point O, parallel to the axis l-, the axis of the horizontal cylinder is drawn. The position of the point 0) is determined by the value //, taken from the complex drawing (Figure 201). A segment equal to L is laid from the point O up along the z axis. Putting aside from the point O, along the axis of the horizontal cylinder, the segment /, we get the point 02 - the center of the base of the horizontal cylinder. The isometric projection of the line of intersection of the surfaces is constructed by points using three coordinates. However, in this example, the desired points can be constructed in a slightly different way. For example, the point 2 is built as follows: From the center 02 upwards, parallel to the z axis, lay the segment t taken from the complex drawing tezha. Through the end of this segment, a straight line is drawn parallel to the axis\\ to the intersection with the base of the horizontal cylinder at point 2V. frontal or horizontal projection of the complex drawing. The endpoints of these segments will belong to the line of intersection. Through the points obtained, a curve is drawn along the pattern, highlighting its visible and invisible parts. If the diameters of the intersecting cylindrical surfaces are the same, then the frontal projection of the intersection line is two intersecting straight lines. If the intersecting cylindrical surfaces have axes located at an angle other than a right angle, then the line of their intersection is built using auxiliary secant planes or in other ways (for example, the method of spheres).

4. Intersection of polyhedra

1 MUTUAL INTERCECTION OF CURVED SURFACES

1.1 General

Curved surfaces intersect in the general case along a spatial curved line, the projections of which are usually constructed from points. To find these points, the given surfaces are crossed by the third auxiliary secant surface, the lines of intersection of the auxiliary surface with each of the given ones are determined, then the common points of the constructed intersection lines are found. By repeating such constructions many times, the required number of points is obtained to determine the line of intersection.

The general algorithm for constructing a line of intersection of surfaces:

1) Select the type of auxiliary surfaces. When choosing an auxiliary secant surface, one should choose surfaces that would intersect the given surfaces along the simplest lines for constructing - straight lines or circles. As auxiliary surfaces - intermediaries, planes and spheres are most often used.

2) Build lines of intersection of auxiliary surfaces with given surfaces.

3) Find the intersection points of the obtained lines and connect them together.

4) Determine the visibility of the intersection line with respect to the considered surfaces and projection planes.

Constructions begin with the definition characteristic (reference) points(points located on sketch generatric surfaces, which usually divide the line of intersection into visible and invisible parts (visibility boundaries), the highest and lowest points of the line of intersection, extreme points (right and left).

When constructing, methods of converting a drawing are used if this simplifies and thins the construction.

1.2 Construction of a line of intersection of surfaces using auxiliary cutting planes

A task. Construct a line of intersection of the cone and the cylinder of revolution (Fig. 186).

First of all, we define characteristic points intersection lines:

Projections of the highest and lowest points A2 and E2 defined using the auxiliary frontal plane Q, which intersects the surface of the cylinder and the cone along the extreme generators. The horizontal projections of the points are on a horizontal trace Qπ2 auxiliary plane.

Points C and C are found using a horizontal plane S drawn through the axis of the cylinder. The plane S intersects the surface of the cylinder along the extreme generators (front and back), and the surface of the cone - along the circle. The intersections of the horizontal projections of the extreme generators and the circle give points C 1 and C 1 - horizontal projections of points C and C. The frontal projections of these points lie on the frontal trace of the plane S.

intermediate points intersection lines are found using horizontal planes P and R.

Figure 186

Figure 187

In the considered example, the points of the intersection line are found using auxiliary planes of particular position. Sometimes the introduction of planes of particular position does not give the desired effect and it is more expedient to use planes of general position.

1.3 Construction of a line of intersection of surfaces using auxiliary secant spheres with a constant center

It is known that if the axis of the surface of revolution passes through

the center of the sphere and the sphere intersects this surface, then the line of intersection of the sphere and the surface of revolution is a circle, the plane of which is perpendicular to the axis of the surface of revolution. In this case, if the axis of the surface of revolution is parallel to the plane of projections, then the line of intersection onto this plane is projected into a segment of a straight line.

On fig. 187 shows a frontal projection of the intersection of a sphere of radius R and surfaces of revolution - a cone, a torus, a cylinder, a sphere, the axes of which pass through the center of the sphere

radius R and parallel to the plane π 2 . The circles along which the indicated surfaces of revolution intersect with the surface of the sphere are projected onto the plane in the form of straight line segments. This property is used to construct a line of mutual intersection of two surfaces of revolution using auxiliary spheres.

The method of cutting spheres with a constant center is used under the following conditions:

1) both surfaces are surfaces of revolution;

2) both surfaces of revolution intersect; the intersection point is taken as the center of the auxiliary (concentric) spheres;

3) the plane formed by the axes of the surfaces (plane of symmetry) must be parallel to the plane of projections. In the event that this condition is not met, resort to methods of converting the drawing.

Radius sphere (R min )

Figure 188

Example. Construct a line of intersection of the cone of revolution and the cylinder of revolution (Fig. 188).

The axes of the given surfaces of revolution intersect (point O) and are parallel to the plane of projections π 2, therefore, the conditions necessary for applying the method of spheres are available.

We define the frontal projections of reference points 1 2 and 2 2 as the points of intersection of the frontal projections of the outlines of the cylinder and the cone. The horizontal projections of these points are determined using the lines of the projection connection.

Sphere radius of maximum radius (Rmax )

is equal to the distance from the frontal projection of the center of the spheres O 2 to the most distant point of the projection of the intersection point of the outlines (point 1 2 ).

minimum

This is a sphere that can be inscribed in one geometric body and intersecting another.

The sphere of minimum radius only touches the surface of the cone and, therefore, intersects it but the circle, the frontal projection of which is the straight line A 2 B 2 . Cylinder surface

the sphere R min also intersects in a circle, the frontal projection of which is a straight line C 2 D 2 . The intersection of these lines - point 4 2 is the frontal projection of one of the points of the desired intersection line.

In a similar way, using a sphere of intermediate radius R i, a frontal projection 3 2 of another point belonging to the line of intersection is constructed. Horizontal projections of the found points can be constructed as projections of points lying on the surface of the cone.

2 SPECIAL CASES OF SURFACE CROSSING

1 Coaxial surfaces of revolution

Coaxial surfaces of revolution intersect in a circle, so the lines of intersection of the cone and cylinder (Fig. 189) are two circles that are projected onto the horizontal plane in full size, and onto the plane π 2 - into line segments.

Figure 189

2 Intersection of surfaces circumscribed around one sphere

As noted earlier, the line of intersection of two curved surfaces in the general case is a spatial curve. However, in some special cases, this line can split into flat curves.

Monge's theorem: two second-order surfaces circumscribed about (or inscribed in) a third second-order surface intersect each other along two second-order curves

Figure 188

3 INTERCEPTION OF A CURVED SURFACE WITH THE SURFACE OF A POLYHEDRON

Each face of a polyhedron generally intersects a curved surface along a plane curve. These curves intersect each other at the meeting points of the edges of the polyhedron with the surface. In this way, the task of constructing a line of intersection of a curved surface with a polyhedron is reduced to finding the line of intersection of the surface with a plane and the points of intersection of the line with the surface.

Example. Construction of the line of intersection of the surfaces of the hemisphere

performed by the method of auxiliary cutting planes.

Each face of the prism intersects the surface of the hemisphere along semicircles that intersect each other at the meeting points of the edges of the prism with the surface of the hemisphere.

In the given example, one of the faces of the prism is located parallel to the frontal projection plane, so the circle along which this face intersects the surface of the hemisphere is projected onto the frontal projection plane without distortion. The frontal projections of the remaining two arcs of semicircles will obviously be arcs of semi-ellipses. Building them on the diagram should begin with finding reference points. For this, frontal planes (P and Q) are drawn through each edge of the prism, which intersect the surface of the hemisphere along circles.

The points of intersection of the frontal projections of the ribs with the corresponding

semicircles are frontal projections of the meeting points of the edges of the prism with the hemisphere (points 1, 2, 3).

Points 4 and 5, dividing the curves into visible and invisible parts, were obtained using the frontal plane S drawn through the center of the hemisphere.

Intermediate points are found by a similar construction (using the frontal planes R and T).

4 Mutual intersection of polyhedra

The line of intersection of the surfaces of two polyhedra is a closed spatial polygonal line (or two closed polygonal lines) that passes through the intersection points of the edges of one of the polyhedra with the faces of the other and the edges of the other with the faces of the first.

The construction of the line of intersection of polyhedra can be done in two ways, combining or choosing from them the one that, depending on the conditions, gives simpler constructions:

1 way. Determine the points at which the edges of one of the polyhedra intersect the faces of the other and the edges of the second intersect the faces of the first. Through the obtained points in a certain sequence, a broken line is drawn, which is a line of intersection of the given surfaces. In this case, it is possible to connect by direct projections only those points that lie on the same face.

2 way. Determine the line segments along which the faces of one of the polyhedra intersect the faces of the other; these segments are the links of the broken line obtained by crossing the polyhedra.

	Example. Construction of the line of intersection of the surfaces of the prism and
pyramids (Fig. 189)
	As can be seen from Fig. 189,
	surface		pyramids
intersects only
front edge of the prism. So
perpendicular to the plane
π1 ,		horizontal
projections
exit (points 1 and 2)
are celebrated		directly
but on the plot.
		finding
frontal			projections
through the top of the pyramid and
anterior
carried out		auxiliary
horizontally
quoting		plane
She crossed the surface						Figure 189
pyramids in a straight line				SD and

SE, in the intersection of the frontal projections of which with the frontal projection of the front edge of the prism, the frontal projections 1 2 , 2 2 of the entry and exit points 1 and 2 are marked. Since the faces of the prism are horizontal

Projecting planes, then the construction of the meeting points of the edges of the pyramid with the faces of the prism (points 3, 4, 5, 6) does not present any difficulties and is clear from the drawing. By connecting in series the frontal projections of the found points, we obtain the frontal projection of the intersection line. Its horizontal projection coincides with the horizontal projection of the prism.

When determining the visibility of points, belonging to the line of intersection are guided by the following rule: the projection of a point obtained at the intersection of two visible lines is visible. The intersection point of two invisible or one visible and another invisible line is invisible.

In the drawings of machine parts, there are often lines of intersection of surfaces, or, in other words, transition lines. Therefore, it is necessary to study the methods of constructing these lines.

Mutual intersection of polyhedra. On fig. 177, and there are three images of two intersecting prisms - quadrangular and triangular. The construction of the frontal projection in the figure is not completed; the projection of the intersection line is not shown on it. It is required to build projections of the intersection line on all images of the drawing.

Considering the horizontal and profile projections, it can be established that the side faces of a vertically located prism are perpendicular to the horizontal projection plane; the projection of the intersection line onto this plane coincides with the projections of the side faces, i.e., with segments of straight lines. The profile projection of the intersection line also coincides with the profile projection of the triangular prism. There will be no additional lines on these projections (Fig. 177, b). Consequently, the solution of the problem is reduced to the construction of a frontal projection of the intersection line. To do this, you need to find the point of intersection of the edges of one prism with the faces of another.

When solving the problem, first determine the edges of each of the prisms that do not intersect the faces of the other (these edges in Fig. 177, b are not marked with numbers). Then, considering the profile and horizontal projections, we see that edges 1 - 2 and 3-4 intersect the inclined faces of the triangular prism. The points of intersection, the meeting points of the ribs 1-2 and 3-4 with the contour of the profile projection of a triangular prism, i.e. a", b", c", d" are visible in the drawing. Projections of invisible points are enclosed in brackets.

Horizontal projections a, b, c, d points A, B, C, D are located on the horizontal projections of the ribs 1-2 and 3-4. Edge projections are shown as points. Frontal projections - points a "b", c", a" are determined using communication lines. Next, it is established that the edges 5-6 and 7-8 of the triangular prism intersect the faces of the quadrangular one. Horizontal projections of the intersection points e, f, g, h are visible in the drawing. Frontal projections of points E, F, G, H are found by drawing communication lines to the projections of the corresponding edges. To get a line of intersection, you need to connect the obtained points with straight lines. Connect those points that are on the same faces of each prism. Then you need to sequentially connect the points a", b", g", h", d", c", f", e". The segments e"f" and g"h" - the lines of intersection on the frontal projection - are invisible, since they are covered by the inclined faces of the triangular prism, so they are outlined with a dashed line.

A visual representation of intersecting prisms is given in Fig. 177, c.

On fig. 178 shows the construction of the line of intersection of a quadrangular truncated pyramid and a quadrangular prism. The construction is carried out similarly to that shown in Fig. 177. On the frontal projection, the line of intersection coincides with the projection of the side faces of the prism, since they are perpendicular to the frontal projection plane (see Fig. 178). The upper and lower edges of the prism intersect with the front and rear edges of the pyramid at points 1, 2, 3, 4, the projections of which 1", 2", 3", 4" are at the intersection points of the corresponding edges. Having frontal and profile projections of points 1, 2, 3, 4, their horizontal projections are found using communication lines, as shown by arrows in the drawing.

The intersection points of the other two edges of the prism with the faces of the pyramid cannot be obtained without additional construction. To determine these points, the prism and the pyramid are crossed by a horizontal cutting plane P. When the plane P intersects with the pyramid, a rhombus is formed, the sides of which will be parallel to the sides of the bases of the pyramid. A rhombus is easy to build by projecting the point a "on the horizontal plane of projections and drawing straight lines parallel to the sides of the base. When the plane P intersects with the prism, a rectangle is formed equal to the horizontal projection of the prism. Points 5, 6, 7, 8 of the intersection of the contours of the rhombus and the rectangle will be the desired points lines of intersection of both bodies.

Profile projections 5", 6", 7", 8" were obtained using communication lines. The projections of invisible points are given in brackets. Connecting the projections of points located on the same faces of the pyramid and prism, i.e. points 1, with straight lines, 6, 2, 5, points 3, 8, 4, 7, points 1", 5", 2" and points 3", 7", 4", receive the missing projections of the intersection line.

Mutual intersection of bodies of revolution.

On fig. 179 shows the construction of the line of intersection of two cylinders of different diameters; the axes of the cylinders are mutually perpendicular and intersect.

On fig. 179, a shows a part intended for connecting pipes - a tee, and its simplified model - two intersecting cylinders. Intersecting, the cylindrical surfaces form a spatial curved line. The horizontal projection of the line of intersection coincides with the horizontal projection of a vertically located cylinder, that is, with a circle (Fig. 179, b). The profile projection of the intersection line coincides with the circle, which is the profile projection of a horizontally located cylinder. Having marked characteristic points 1, 2, 3 on the horizontal projection, their profile projections 1", 2", 3", which are located on the arc of the circle, are found. According to the horizontal and profile projections of points 1, 2, 3, their frontal projections 1", 2 are found ", 3". Thus, the projections of the points that define the transition line are found.

In some cases, this number of points is not enough, and in order to obtain additional points, apply auxiliary cutting planes method. This method consists in the fact that the surface of each body is crossed by an auxiliary plane that forms sectional figures, the contours of which intersect. The points obtained by crossing the section contours are the points of the intersection line. In this case, both cylinders are crossed by an auxiliary horizontal cutting plane (Fig. 179, c). When crossing a vertically located cylinder, a circle is formed, and a horizontally located cylinder - a rectangle. The intersection points 4 and 5 of the circle and the rectangle belong to both cylinders and, therefore, determine the line of intersection of both bodies (see Fig. 179, a). Having marked the profile, and then the horizontal projections of points 4 and 5, with the help of communication lines, frontal projections are found (see Fig. 179, c). The resulting points are connected by a smooth curve.

If it is necessary to increase the number of points that define the line of intersection, several more parallel auxiliary cutting planes are drawn.

If both cylinders have same diameters, then one of the projections of the intersection lines is an intersecting straight line (Fig. 179, d and e), and the intersection lines are ellipses.

The line of intersection of the ball and a right circular cylinder, the axis of which passes through the center of the ball, is shown in Fig. 180. As can be seen from the drawing, on one projection the line of intersection is depicted as a circle, and on the other it is projected into a straight line.

Projection of bodies with holes. In technology, there are parts with holes of a cylindrical, rectangular or some other shape (Fig. 181). When the holes intersect with the surfaces of the parts, intersection lines are formed, the shape of which in some cases must be reproduced in the drawing. This problem is solved in general terms in the same ways as the construction of lines of intersection of geometric bodies.

On fig. 182, a shows a cylinder with a cylindrical side hole. The axes of the cylinder and the hole intersect at right angles. The line of intersection is a spatial curve. The construction of the line of intersection was shown in fig. 179, and obtaining the characteristic points of this curve is given in fig. 182, a.

The line of intersection of a cylinder with a rectangular hole when the axes intersect at a right angle is shown in Fig. 182b. To construct the line of intersection on the horizontal projection, characteristic points 1, 2, 3, 4, 5, 6 are selected. . Frontal projections 1, 2", 3", 4", 5", 6" are found from the obtained horizontal and profile projections. By connecting the points 1", 2", 3", 4", 5", 6" with straight lines, a broken line is obtained intersections in the form of a rectangular cavity.

On fig. 182c shows the line of intersection of a cylinder with a hole formed by a quadrangular prism and two half-cylinders. The keyway has this shape. The line of intersection is a rectilinear depression (see Fig. 182, b) with curved edges (see Fig. 182, a).

Answer the questions

1. What is the method of auxiliary cutting planes? What is it used for?

2. What shape does the line of intersection of two cylinders of different diameters and two cylinders of the same diameter have if the axes of the cylinders intersect?

Assignments to § 25 and chapter IV

Exercise 83

According to these two projections of the part, draw a third (Fig. 183). Build the missing projections of points A and B, given by the projections a and b "located on the visible faces. Perform an axonometric projection, mark the dimensions on it and plot points A and B.

Answer the questions

1. What projections are given in the drawing?

2. What are the overall dimensions of the part?

3. What are the dimensions of the rectangular groove on the part?

4. What is the roughness of the surface shown by the dashed line in the main view?

5. Do I need to machine the base of the part and the sides?

6. Do I need to machine the upper inclined plane of the part?

Exercise 84

On two projections of the part, draw a third (Fig. 184). Build the missing projections of a point located on the visible surface of the part and given by the frontal projection d.

Answer the questions for Fig. 184

1. What is the original shape of the part?

2. What projections are given in the drawing?

3. What do the dashed lines on the frontal projection mean?

4. What do the two horizontal dashed lines on the profile projection mean?

5. What caused the appearance of two concave lines on the frontal projection?

6. Is it possible, without additional constructions, to designate point B on the profile projection, given by the frontal projection b "? Where is this point on the profile projection?

7. What are the overall dimensions of the part?

8. What dimensions determine the position of a hole with a diameter of 40 mm?

9. Is it possible to turn a part to a size of 119.98 mm?

10. Is it possible to turn a part to a size of 119.8 mm? If not, is it possible to fix such a marriage?

11. Is it possible to machine a groove of 60 mm for size 60 -0.1? If not, is it possible to fix such a marriage?

12. Do I need to apply the size between the lines indicated by the number 1 in the green quadrangle? What caused these lines?

13. What should be the roughness of most of the surface of the part?

14. What is the roughness of the two parallel planes in each of the grooves?

Exercise 85

According to the visual images of the details (Fig. 185, a-c), draw the drawings in the system of rectangular projections. Take the scale of the drawings 2: 1. Determine the dimensions by measuring visual images.

Answers to the exercises of chapter IV

To exercise 50

Designation	Name
1	Communication line
2	Featured item
3	Profile projection (left view)
4	Profile projection plane (W)
5	Frontal projection plane (V)
6	Frontal projection (front view)
7	Horizontal projection plane (H)
8	Horizontal projection plane (top view)
9	projector rays
BUT	Front view (main view)
B	Left side view
IN	Communication line
G	Auxiliary line
D	View from above

Go to exercise 54

Go to exercise 56

The answers to examples 1 and 2 are as follows (no answers are given to examples 3, 4, 5):

In examples 1 and 2, the views should be arranged like this:

AB AB C B

Go to exercise 57

An example of solving the problem is given in fig. 277.

Go to exercise 58

An example of solving the problem is given in fig. 278.

Go to exercise 59

To select the correct position for the main view, you need to look at the details in the direction indicated by the arrows with the following letters.

To construct a curved line obtained when a plane intersects a cylindrical surface, in the general case, one should find the points of intersection of the generators with the cutting plane, as stated on p. 170 for ruled surfaces in general. But this does not exclude the possibility of using auxiliary planes, which each time intersect the surface and the plane.

First of all, we note that any cylindrical surface is intersected by a plane located parallel to the generatrix of this surface, along straight lines (generators). On fig. 360 shows the intersection of a cylindrical surface with a plane. In this case, this surface is an auxiliary element in constructing the point of intersection of a curved line with a plane: a cylindrical surface is drawn through a given curve (see Fig. 360, left) DMNE, projecting the curve onto the square. pi 1 . Further, the plane (in Fig. 360 - a triangle) intersects the cylindrical surface along a flat curve M 1 ... N 1. The desired point of intersection of the curve with the plane - point K - is obtained at the intersection of the curves - given and constructed.

Such a scheme for solving the problem of the intersection of a curved line with a plane coincides with the scheme for solving problems for the intersection of a straight line with a plane(see §§ 23

and 25); in both cases, an auxiliary surface is drawn through the line, which for a straight line is a plane.

The horizontal projection of the curve M 1 ... N 1, along which the cylindrical surface intersects with the plane, coincides with the horizontal projection of the curve D ... E, since this curve is a guide for the cylindrical surface at perpendicular to the square. π 1 , its generators. Therefore, from the point M "1 on the projection A"C" we can find the projection of M" 1 on A"C" and from the point N "1 - the projection N" 1. Next, in fig. 360 on the right shows the auxiliary square. α intersecting ABC along the line CF, and the cylindrical surface - along its generatrix with a horizontal projection at the point 1 ". At the intersection of this generatrix with the line CF, a point with projections 1" and 1" is obtained, belonging to the curve M 1 ... N 1 Obviously , you can not specify a trace of the plane, but simply draw a line in a triangle, as shown in relation to the line CG, on which a point with projections 2 "and 2" is obtained.

The following examples will show sweep. The unfolding of a cylindrical surface in the general case can be carried out according to the scheme of unfolding the surface of a prism. The cylindrical surface is, as it were, replaced by an inscribed or described prismatic one, the edges of which correspond to the generators of the cylindrical surface. The deployment itself, as shown in Fig. 283 is made using a normal section. But instead of a broken line, a smooth curve is drawn.

On fig. 361 shows the intersection of a right circular cylinder with a front-projecting plane. The sectional figure is an ellipse, the minor axis of which is equal to the diameter of the base of the cylinder; the magnitude of the major axis depends on the angle between the cutting plane and the axis of the cylinder.

Since the axis of the cylinder is perpendicular to the square. π 1 then the horizontal projection of the section figure coincides with the horizontal projection of the cylinder.

Usually, to construct the points of the contour of the section, evenly spaced generators are drawn, i.e., those whose projections on the square. π 1 are points equidistant from each other. It is convenient to use this "markup" not only for constructing section projections, but also for developing the side surface of the cylinder, as will be shown below.

The projection of the figure of the section on the square. π 3 is an ellipse, the major axis of which in this case is equal to the diameter of the cylinder, and the minor one is the projection of the segment 1 "7". On fig. 361 on the square. π 3 the image is constructed as if the upper part of the cylinder was taken after crossing it with a plane.

If in Fig. 361 plane α made an angle of 45 ° with the axis of the cylinder, then the projection of the ellipse on π 3 would be a circle. In this case, the segments 1""7"" and 4""10"" would be equal.

If the same cylinder is crossed by a plane in general position, which also makes an angle of 45 ° with the axis of the cylinder, then the projection of the section figure (ellipse) in the form of a circle can be obtained on an additional projection plane parallel to the axis of the cylinder and the horizontals of the cutting plane.

Obviously, with an increase in the angle of inclination of the secant plane to the axis, the segment 1 "" 7 "" decreases; if this angle is less than 45 °, the segment 1 "" 7 "" increases and becomes the major axis of the ellipse on the square. π 3 , but the segment 4 "" 10 "" becomes the minor axis of this ellipse.

The natural view of the section is, as mentioned above, an ellipse. Its axes are obtained in the drawing: large - segment 1 0 7 0 \u003d 1 "7", small - segment 4 0 10 0, equal to the diameter of the cylinder. An ellipse can be built along these axes.

On fig. 362 shows a full development of the lower part of the cylinder.

The unfolded circle of the base of the cylinder is divided into equal parts according to the divisions in Fig. 361; the segments of the generators are plotted on perpendiculars drawn at the division points of the expanded circle of the base of the cylinder. The ends of these segments correspond to the points of the ellipse. Therefore, drawing a curved line through them, we obtain a developed ellipse (this line is a sinusoid) - the upper edge of the development of the side surface of the cylinder.

To the development of the side surface in Fig. 362, a base circle and an ellipse are attached - a natural view of the section, which makes it possible to make a model of a truncated cylinder.

On fig. 363 shows an elliptical cylinder with a circular base; its axis is parallel to the square. π 2 . To determine the normal section of this cylinder, it must be cut by a plane perpendicular to the generatrix, in this case, the front-projecting plane. The figure of normal section is an ellipse with a major axis equal to the segment 3 0 7 0 and a small one equal to 1 0 5 0 = 1 "5".

If it is necessary to unfold the side surface of this cylinder, then, having a normal section, unfold the curve limiting it into a straight line and, at the corresponding points of this straight line, perpendicular to it, lay off the segments of the generators, taking them from the frontal projection. To mark the generators, divide the circumference of the base into equal parts. In this case, the ellipse (normal section) will also be divided into the same number of parts, but not all of these parts are equal

length. The development of an ellipse into a straight line can be done by successively laying off sufficiently small parts of the ellipse on the straight line.

On fig. 364 shows a right circular cylinder intersected by a plane in general position. An ellipse is obtained in the section: the cutting plane makes an acute angle with the axis of the cone.

Just as it was in Fig. 361, the horizontal projection of the section coincides with the horizontal projection of the cylinder. Therefore, the position of the horizontal projection of the point of intersection of any of the generators of the cylinder with pl. α is known (for example, point A "in Fig. 365). To find the corresponding frontal projection, you can draw a horizontal or frontal in square α, on which the desired point should be located. In Fig. 365, a frontal is drawn; in the place where the frontal projection frontal intersects the frontal projection of the corresponding generatrix, the projection A lies. The same front defines two points of the curve, A and B (Fig. 365). If we construct a frontal corresponding to point C, then

this line defines only one point of the intersection curve. The front, built on the points D and E, determines the extreme points D "and E".

Continuing similar constructions, one can find enough points to draw a frontal projection of the intersection line.

On fig. 366 the upper part of the cylinder is cut off, as it were. If the frontal projection is shown in full, then the line of intersection is drawn as shown in Fig. 364.

On fig. 365 shows the auxiliary frontal planes β, γ, δ intersecting the cylinder along the generators, and pl. α along the frontals. This is in line with what was said at the beginning of the paragraph. Auxiliary square. δ only touches the cylinder, which makes it possible to define only one point for the curve.

When constructing a frontal projection of the intersection line, in addition to points D "and E" (Fig. 365), two more extreme points should be found, namely M "and N" - the highest and lowest points of the projection of the section on the square. π 2 . To construct them, it is necessary to choose an auxiliary plane perpendicular to the trace h "0α and passing through the axis of the cylinder (Fig. 366). This plane is a common plane of symmetry of the cylinder data and the secant square a. Having found the line of intersection of the planes α and β, we mark the points M "and N" by constructing them on the frontal projection using the points M" and N".

Another way to find the points M "and N" is to draw two planes tangent to the cylinder, the horizontal traces of which are parallel to the trace h "0α. These planes will intersect with pl. α along the horizontals of the latter (Fig. 364, auxiliary planes β and γ); marking the points M" and N" we construct the points M" and N" on the frontal projections of the contour lines.

The segment MN is the major axis of the ellipse - the figure of the section of the given cylinder pl. a. This is also seen in Fig. 366, where it was built in combination with the square. π 1 ellipse - a natural view of the section. But the segment M"N" in the same figure is by no means the major axis of the ellipse - the frontal projection of the sectional figure. This major axis can be found from the conjugate diameters M "N" and F "G" (Fig. 364) by the construction indicated in § 21, or by the special construction given in § 76.

The natural view of the section can be found by combining the cutting plane with one of the projection planes, π 1 or π 2 .

On fig. 366 the ellipse in the combined position is built along the major and minor axes (in the same place, point D "is obtained by combining the front).

The development of the side surface is shown in fig. 364. Pay attention to the fact that the marking of the points - the horizontal projections of the generators - on the circumference of the base was made from the point N ". This simplified the construction, since using the same horizontal line two points are obtained on the frontal projection

ellipse. In addition, the development figure has an axis of symmetry. But at the same time, points D "and E" did not fall into the number of points marked on the circle.

Another example of constructing a figure of a section of a cylinder of revolution by a plane is given in Fig. 367. This construction was made using the method of changing projection planes. The secant plane is given by intersecting straight lines - the frontal (AF) and the profile straight line (AP). Since the profile projection of the frontal and the frontal projection of the profile line lie on the same straight line A "≡A"", A ""F"" \u003d A"P", these lines lie respectively in the planes π 2 and π 3, (see Fig. 367, top left) The π 2 /π 3 axis passes through A""F""(A"P").

We introduce a new square. π 4 so that π 4 ⊥π 3 and π 4 ⊥AP. The cutting plane turns out to be perpendicular to π 4, and the projection onto π 4 of the section figure is obtained in the form of a straight line segment 2 IV 6 IV, equal to the major axis of the ellipse - the section figure. The position of the straight line A IV 6 IV is determined by the construction of the projections of points A and 1 on the square. π 4 .

Let's trace the construction of some points. To avoid unnecessary constructions, the projection 1"" was taken on the continuation of the perpendicular drawn from O"" to π 3 / π 4 . A projection 1" was obtained from the point 1""; the segment 1"1"", laid off from the axis π 3 /π 4, determined the point IV and the point O 1 coinciding with it - the projection of the center of the ellipse. Knowing the projections 0 IV and O "", you can get O" - the center of the ellipse - the desired frontal projection of the section figure.

Based on points 2 IV and 2"" we found point 2", the least distant from π 3 , and using points 6 IV and 6"" - point 6", the most distant from π 3 .

Point 5 "" was taken from point 5 IV, and now point 5 "is found from points 5 IV and 5" "- one of the points that determine the division of the ellipse on the frontal projection of the cylinder into "visible" and "invisible" parts. The second point is located symmetrically point 5" in relation to O".

The rest is clear from the drawing. The natural view of the sectional figure (ellipse in Fig. 367, on the right) is built along the axes - large, equal to 2 IV 6 IV, and small, equal to the diameter of the cylinder.

Questions to §§ 55-56

1. How is a curved line constructed when a plane intersects a curved surface?
2. What lines intersect a cylindrical surface with a plane drawn parallel to the generatrix of this surface?
3. What technique is used in the general case to find the point of intersection of a curved line with a plane?
4. What lines are obtained when planes intersect a cylinder of revolution?
5. In which case is the ellipse obtained by crossing a cylinder of revolution, the axis of which is perpendicular to the square. π 1 , a front-projecting plane, is projected onto the square. π 3 in the form of a circle?
6. How should the additional projection plane be positioned so that the ellipse obtained by crossing the cylinder of revolution, the axis of which is perpendicular to the square. π 1 , a plane of general position, making an angle of 45 ° with the axis of the cylinder, projected onto this plane of projections in the form of a circle?