A ruled surface is a surface formed by moving a straight line in space according to some law. The nature of the movement of the rectilinear generatrix determines the type of ruled surface. Typically, the law of motion of the generatrix is ​​specified using guide lines. IN general case To define a ruled surface, three guide lines are needed, which can unambiguously define the law of movement of the guide. Let us select three lines a, b and c on the ruled surface and take them as guides (Fig. 7.17).

Rice. 7.17. Ruled surface in general

The study of a group of ruled non-developable surfaces can begin with cylinderoids - surfaces with a plane of parallelism (Catalan surfaces), surfaces formed by the movement of a straight line sliding along two curved guides that do not lie in the same plane, and remaining all the time parallel to the so-called plane of parallelism (Fig. .7.18).

Rice. 7.18. Example of a cylindroid: a - in space; b - on a complex drawing

The next surface in this group is the conoid, which is a ruled, non-developable surface, which is formed by the movement of a straight line sliding along two guides that do not lie in the same plane, and remaining all the time parallel to the so-called plane of parallelism.

At the same time, you need to know that one of these guides is a straight line (Fig. 7.19).

Rice. 7.19. Example of a conoid: a - in a complex drawing; b - in space

If both guides of the cylinderoid are replaced by straight lines (crossing), then a ruled non-developable surface with a plane of parallelism is formed - an oblique plane, or a ruled paraboloid, or hyperbolic paraboloid(Fig. 7.20).

The ruled surface got its name (hyperbolic paraboloid) due to the fact that when it is intersected by the corresponding planes in the section, parabolas and hyperbolas can be obtained

Varieties of oblique surfaces are ruled surfaces with a guide plane and their particular types are ruled surfaces with a plane of parallelism (Catalan surfaces).

In the first case (Fig. 7.20, a) the surface is uniquely defined by two rectilinear intersecting lines d, n and a guide plane γ, which replaces the third guide line. The generating straight line slides along two guides and remains parallel to the plane parallelism γ.

If the planes of parallelism are perpendicular to each other γ ⊥ π1, then the hyperbolic paraboloid is called straight.

In Fig. 7.20, b shows a complex drawing of an oblique plane. In appearance, this surface resembles a saddle.

Rice. 7.20. Hyperbolic paraboloid:
a - in space; b - on a complex drawing

Surfaces with a guide plane are called oblique cylinderoids if both guides are curved lines; oblique conoids - if one of the guides is a straight line; double oblique plane if the guides are intersecting straight lines.

A doubly oblique cylindrical, like a ruled surface with three guides, of which two are spatial curves and one is a straight line, is shown in Fig. 7.21.

In Fig. 7.22. a double oblique conoid is shown, formed by moving the generatrix of the straight line (red) along three guides, two of which are straight. The construction of one generatrix is ​​shown as a result of the intersection of an auxiliary plane passing through one of the rectilinear guides with two other guides.

Rice. 7.21. Double oblique cylinderoid

Rice. 7.22. Double oblique
conoid

We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"

INTRODUCTION

The world of surfaces is diverse and limitless. Surfaces of amazing shape and strength are found in nature. Let's pay attention to the wing and body of the bird; they have surface shapes developed by nature, the totality of which has excellent aerodynamic characteristics.

Aircraft bodies, sea ​​vessels, automobiles, overhead shells and underground structures- these are all complexes of surfaces of various very complex laws of formation. By examining ruled surfaces, it can be revealed that they are widely used in technology, engineering, in most cases used in the design of buildings, industrial and government architectural structures, highways.

The relevance is due to the demand for ruled screw surfaces V modern architecture and technology, as well as the search for new forms of helical ruled surfaces applicable for construction, combining qualities such as beauty, reliability and manufacturability.

The object of research is the formation and design of complex curved surfaces.

The subject of the study is the formation of composite ruled shells in the architecture of buildings and structures.

The purpose of this work is to study ruled surfaces and study the possibilities of their use in the architecture of buildings and structures.

During the research, the following tasks are set:

1. Analyze theoretical foundations ruled surfaces.

2. Construct a composite ruled surface applicable in the architecture of buildings and structures.

3. Make a mock-up of the developed structure.

Methods used in conducting the research:

Theoretical:

Monographic - analytical synthesis and systematization of information from literary and other sources;

Analysis - analysis of information at each stage of the work;

Synthesis - collection and synthesis of information.

Praxeological:

Graphic - geometric modeling and execution of graphic documentation;

Layout method.

LINED SURFACES

A ruled surface is a surface formed by moving a straight line in space according to some law. The nature of the movement of the rectilinear generatrix determines the type of ruled surface. Typically, the law of motion of the generatrix is ​​specified using guide lines. In general, three guide lines are needed to define a ruled surface. The nature of the movement of the rectilinear generatrix determines the type of ruled surface.

Ruled surfaces are divided into two types:

1. developing surfaces;

2. non-developing or oblique surfaces.

NON-EXPANDING LINED SURFACES

Non-developable ruled surfaces are generally formed by the movement of a rectilinear generatrix along three guide lines, which uniquely define the law of its movement. Varieties of oblique surfaces are ruled surfaces with a guide plane and their particular types are ruled surfaces with a plane of parallelism (Catalan surfaces). Surfaces with a guide plane are called oblique cylindroids if both guides are curved lines; oblique conoids - if one of the guides is a straight line; double oblique plane, if the guides are crossing straight lines (see Appendix A, Fig. 1). Surfaces with a plane of parallelism are respectively called straight cylinderoids, straight conoids and oblique planes.

The concept of a ruled surface

Ruled surface is a surface formed by moving a straight line in space according to some law. The nature of the movement of the rectilinear generatrix determines the type of ruled surface. Typically, the law of motion of the generatrix is ​​specified using guide lines. In general, to define a ruled surface, you need three guide lines . Select three lines on the ruled surface a , b And c and take them as guides. Let us show that the motion of the rectilinear generatrix l will be determined in a unique way (Fig. 11.1).

Let's take it on the guide a some point K and draw through it a bunch of straight lines intersecting the guide With . These straight lines form a conical surface with its vertex at the point K . Guide b will intersect the conical surface at some point N . Constructed point N and period K determine the straight line l , intersecting the guide c at the point M . Thus, each point TO guide a the only generator will correspond. Moving a point TO along the guide a , it is possible to obtain other positions of the generatrix of the straight line, i.e. build a frame of a ruled surface.

Depending on the shape of the guide lines, ruled surfaces with three guides are divided into:

oblique cylinder with three guides– all three leading curved lines;

cone– two guiding curved lines, and the third is straight;

single-sheet hyperboloid– all guide lines are straight.

To construct a point on a ruled surface, you must use an auxiliary line, which can be a straight generatrix or an arbitrary curved line.

In addition to the above general method of forming a ruled surface with the help of three guides, there are other methods that, by imposing additional restrictions, determine the law of motion of the rectilinear generatrix.

Ruled surfaces

A ruled surface is a surface that can be formed by the movement of a straight line in space. Depending on the nature of the motion of the generatrix, we obtain various types ruled surfaces.

Figure 1.3.37 – Ruled faceted surfaces

Polyhedra(pyramids, prisms) are closed surfaces formed by a certain number of faces. IN in this case both the surface and the body bounded by this surface bear the same name. The elements of a polyhedron are vertices, edges, and faces; the set of all edges of a polyhedron is called it mesh. Constructing projections of a polyhedron comes down to constructing projections of its mesh.

Among the many polyhedra there are correct polyhedra. In such polyhedra, all edges, faces and angles are equal to each other. Figure 1.3.38, for example, shows regular polyhedron, called octahedron.

Figure 1.3.39 – Conical and cylindrical surfaces

Conical surface formed by a straight line l m(guide) and having a fixed point S(top) in accordance with Figure 1.3.39, A. Surface determinant Q(l,m,S).

Cylindrical surface formed by a straight line l(generator) moving along a curved line m(guide) and having a constant direction s in accordance with Figure 1.3.39, b. Surface determinant S(l,m,s).

Since all straight lines having the same direction, i.e. parallel to each other intersect at an infinitely distant (improper) point, then the cylindrical surface can be considered as special case conical surface.

When specifying a conical and cylindrical surfaces in a complex drawing, a line is often chosen as a guide m intersection of the surface with one of the projection planes.

Ruled is a surface that is formed by moving a straight line (generator) in space according to some law. Depending on the type of guide lines and the nature of the movement of the generatrix, the following types of ruled surfaces are obtained: deployable and non-deployable.

A. Developable ruled surfaces(torso, cylindrical, conical).

1. Torso- a surface with a return edge m, is formed by the movement of a rectilinear generatrix ℓ, touching in all positions a certain spatial curve m - return edge, (Fig. 45). Given by the determinant Ø ( l~m).

2. Cylindrical surface. The return edge is removed to infinity. The surface is formed by the movement of a straight line ℓ, which has a constructed direction S along some curve n (Fig. 46). Surface determinant: ∑(S~n).

3. Conical surface. The cusp edge has degenerated to point S. The surface is formed by moving the straight line ℓ passing through the point S along some curve n, and may have two cavities (Fig. 47). Determinant of the surface Δ(S~n).

B. Non-developable ruled surfaces(cylindroid, conoid, oblique plane).

This type of surface is formed by moving a straight line ℓ moving along two guides and remaining parallel to a certain plane of parallelism, which is usually taken to be one of the projection planes P 1 or P 2.

1. Cylinder is formed by moving a straight line ℓ along two guides and remaining parallel to a certain plane of parallelism (Fig. 48a, b). The determinant of the surface Σ (~a, ~b) and Δ is perpendicular to P 1.

2. Conoid is formed by moving the rectilinear generatrix ℓ along two guides: a curve and a straight line, while ℓ remains parallel to a certain plane of parallelism. Its determinant is Ø(b~a,∑) (Fig. 49).

If the rectilinear generatrix n is perpendicular to the plane of parallelism, then the conoid called direct, and if the curvilinear guide m is a cylindrical helix, the conoid called a helical helicoid.

3. Oblique plane(hyperbolic paraboloid) is obtained by moving the line ℓ along two skew lines and remaining parallel to some plane of parallelism. Surface determinant

Р(a b, ∑), (Fig. 50).

In the cross section of a hyperbolic paraboloid, hyperbolas, parabolas, and straight lines can be obtained (Fig. 51).

B. Ruled helical surfaces - helicoids. A ruled helical surface is a surface in which one guide is a helical line, and the other is a straight line (the axis of the helical line). The determinants of the surface are the helix and its axis: Ø (ί, m, ℓ).

The helicoid is called straight, if the generatrix is ​​a straight line ℓ perpendicular to the axis ί of the helix and this axis ί acts as a straight guide (Fig. 52).

If the straight line is not perpendicular to the t axis, then the helicoid is called oblique or inclined - Archimedes screw(Fig. 53). Helicoids can be closed or open. Straight line ℓ when crossing the axis ί of the helix, forms closed helicoid, if ℓ does not intersect the ί axis, then a open helicoid. During the formation of the surface of an inclined helicoid, the generatrices are located parallel to the generatrices of the surface of a certain cone of rotation, the ί axis of which coincides with the ί axis of the helix, and the generatrices have the same inclination to the ί axis of the helix as the generatrices of the helicoid. This cone called a guide. Hence, the determinant of an inclined helicoid consists of guides: the helix m(m 1, m 2, the axis of the helix ί (ί 1, ί 2) and generatrix ℓ(ℓ 1, ℓ 2), which is located at an angle α to the axis of the helix. By placing a frame of generators ℓ and drawing the envelope of the family of frontal projections of generators ℓ, on P 2 we obtain the outline of an inclined helicoid. The section of the helicoid by the plane Σ (Σ 2) perpendicular to the axis of the helicoid (normal section) is an Archimedes spiral and requires a special construction (see Fig. 53).

Intersection of surfaces with a plane and a straight line.

When a plane intersects any surface, a flat figure is obtained, which is called a section. If the cutting plane is a projection plane, then constructing a section is not difficult. Since one of the projections of the cutting plane degenerates into a straight line, then, based on the collective property of projecting planes, this projection includes all points of the plane, including the section. Thus, the task comes down to constructing another projection of the section. Common points that belong to both the plane and the intersecting surface are identified. Then, based on the belonging of these points to the figure, their missing projections are constructed.

When a plane intersects a polyhedron in section, a polygon is obtained (bounded by a closed polyline). The number of its sides and vertices is equal to the number of faces and edges of the polyhedron intersected by the cutting plane.

Constructing a section of a polyhedron can be done in two ways:

Finding the vertices of a section polygon - the edge method. In this case, the construction comes down to solving the problem of finding the point of intersection of a straight line (edge) with a plane (cutting plane) several times. first positional problem

Finding the sides of a section polygon - the method of edges. In this case, the problem is solved several times - finding the line of intersection of two planes (face and cutting plane) - the second positional problem.

When a plane intersects curved surfaces, the section results in flat curved lines. As already mentioned, if the cutting plane is projected into a straight line, then the second one can be constructed from individual points (Fig. 54).

Among the points of the intersection curve there are those that are particularly located in relation to the projection planes or occupy special places on the curve. Such points are called reference points, and when constructing a section, these points are determined first. Control points include extreme points, outline points and visibility change points.

Extreme points- this is the highest and lowest point of the section, the closest and farthest relative to the projection plane P 2, the leftmost and rightmost relative to P 3.

Ocherkovykh are called points whose projections lie on the contours of the surface.

Visibility change points delimit the projection of the intersection line into the visible and invisible parts. Visibility change points are always selected from sketch points. It often happens that one and the same point is both an extreme point, a highlight point, and a point of change in visibility.

After determining the reference points when constructing a curved line, in order to more accurately determine its nature, a number of random points are determined.

Random points– these are points that are taken arbitrarily. Often the type of section is known in advance. Let's consider what sections are obtained in the most common surfaces.

Cone– a surface in which five types of different sections are obtained:

If the cutting plane passes through the vertex of the cone, then the cross-section results in a triangle (all lines are straight). If the cutting plane does not pass through a vertex, the section will produce curved lines.

If the cutting plane is located at an indirect angle to the base and is not parallel to any of the generatrices, then an ellipse (m) is obtained in the section.

If the cutting plane is parallel to any one generatrix of the cone, the section produces a parabola (n).

If the cutting plane is parallel to two generatrices, the section produces a hyperbola (k).

If the cutting plane is parallel to the base and straight cone perpendicular to the axis, a circle (e) is obtained in cross-section, the radius of the circle is measured from the axis to the outline (Fig. 55).

Cylinder- a surface in the cross section of which three types of flat figures are obtained:

If the cutting plane is parallel to the base and perpendicular to the axis, a circle is obtained in the section; the radius of the circle coincides with the radius of the base.

If the cutting plane is parallel to the axis, the section results in a rectangle.

If the cutting plane is located at an angle to the base and intersects all the forming lines, an ellipse is obtained in the section (Fig. 56).

Sphere- a surface in the cross section of which a circle is always obtained, no matter how the cutting plane is positioned. The radius of the circle is determined as follows: a perpendicular is lowered from the center of the sphere onto the cutting plane, and the radius of the circle is measured from the point of intersection of the perpendicular with the plane to the outline of the sphere (Fig. 57) for (2), for (2) the radius is taken from the axis spheres to the essay.

If the cutting plane is general, then to solve such a problem it is convenient to transform complex drawing so that the cutting plane becomes projecting, and then continue the solution according to the scheme described above (Fig. 58).

When a surface intersects a straight line, it is necessary to determine two intersection points, which are called the entry and exit points of the line.

The problem is solved according to the following scheme:

One of the projections of the straight line is placed into the projecting plane, then the problem of constructing a section of the surface by the projecting plane is solved. After the section has been constructed, the common points of the section with the projection of the straight line are found.

The missing projections of intersection points are constructed based on their belonging to a straight line using communication lines.

Visibility is determined (without determining visibility, the problem is considered unsolved) (Fig. 59).

When solving the problem of the intersection of a straight line with a surface, methods for transforming a complex drawing can be widely used, in particular, the method of replacing planes (Fig. 60).

Mutual intersection of two surfaces.

When two surfaces intersect each other, one or two closed spatial lines (transition lines) are formed, which simultaneously belong to each of the intersecting surfaces. These lines are constructed using individual points. One line is obtained in the case of insertion, i.e. when both surfaces are partially involved in the intersection. Two lines are obtained in cases of penetration, i.e. when at least one of the surfaces is completely involved in the intersection.

If two polyhedra are involved in the intersection, then the intersection line turns out to be a broken line, consisting of a number of straight segments. If a polyhedron and a curved surface intersect, then the intersection line is a broken curve. If two curved surfaces intersect, the result is a smooth curved line. There is a sequence for determining the points of the intersection line. First of all, the reference points are determined. These include extreme, outline (determined on each outline of each surface), points of change of visibility (selected among the outline). If a polyhedron is involved in the intersection, then the points of intersection of its edges with another surface also belong to the reference points.

After the reference points are found, the random points. Such points are needed if a curved surface is involved in the intersection, since if at least one of the surfaces is curved, then the result is a curved line. The more random points are taken, the more accurately the curved line is constructed.

Problems involving the mutual intersection of two surfaces are divided into three difficulty groups:

First difficulty group– both surfaces are projecting. In this case, two projections of the common element (i.e., intersection lines) are specified in the original complex drawing - they coincide with the main (degenerate) projections of the projecting surfaces. You just need to designate them. Sometimes it becomes necessary to construct a third missing projection. In this case, one of the given projections of the intersection lines is divided into points, on the second projection of the given line there are projections of the designated points, and then a third projection is constructed from two projections of points using communication lines (Fig. 61).

Second difficulty group– one surface is projecting, the other is of general position. One projection of the common element is specified in the original drawing - it coincides with the main (degenerate) projection of the projecting surface. It needs to be designated. The second projection of a general element is determined from the condition of its belonging to a generic surface. To do this, it is necessary to divide the existing projection of the intersection line into points (reference and random), and then construct the missing projections of these points from the condition that they belong to a general surface. If the cone is a surface in general position (Fig. 62a), and the prism is a projecting surface, then the frontal projection of the intersection line, coinciding with frontal projection The prism is divided into points and parallels are drawn through them. Then the radius of the parallel is measured (from the axis to the outline) and a circle of this radius is drawn on another projection, after which, using communication lines, the missing projections of the points of the intersection line are found. When all the points are found, they are connected by a smooth curve.

The problem is solved in the same way if the general figure is a sphere (Fig. 62b).

Third group of complexity– both intersecting surfaces in general position. In this case, none of the projections of the surface intersection line in the original complex drawing is specified. Such problems are solved by introducing intermediaries, which reduces the solution of each problem to the intersection of two lines obtained from the intersection of the intermediary with given surfaces.

There are two ways to solve this type of problem: the method of auxiliary cutting planes and the method of spheres.

Method of auxiliary cutting planes is used if the cross-section of both surfaces results in simple graphic construction lines (circles or straight lines). Cutting planes are required private situation, in most cases are chosen as level plane mediators. Let's look at this method of solving the problem using an example.

Example:

Construct the line of intersection of the hemisphere P and the pyramid Q (Fig. 63).

a) Analysis of the drawing shows that this is a problem of the third group of complexity (pyramid and hemisphere are figures of general position). The problem is solved with the help of intermediaries. We select horizontal planes of the level as intermediaries. They intersect P along parallels, and Q through triangles - graphically simple lines.

b) Determine the reference points on the intersection line m. We find the points of intersection of the edges of the pyramid with the hemisphere: M 1, F 1 and E 1. Point М=SBP is found using the plane ( 1) – the plane of the main meridian of the hemisphere P. Points E and F are obtained as a result of the intersection of the edges AS and SC and the hemisphere P, points are found using the plane ( 2) – plane equator of the hemisphere. Points M, E, F are extreme points, as well as sketch points on P 2, points E and F are sketch points on P 1, and they are also visibility change points on P 1.

c) Random points are determined using level planes ( 2) and Г(Г 2); P=n(n 2 ,n 1) - parallel of the hemisphere Q= l(l 2 ,l 1) – DTS triangle; nL=points 1 and 2. Similarly, using the plane Г(Г 2), points 3 and 4 are found.

d) Connect the found points of line m, taking into account visibility.

e) Determine the mutual visibility of P and Q.

Method of auxiliary spheres is based on one property of the surface of revolution: if the center spherical surface is located on the axis of the surface of revolution (the sphere and the surface of revolution in this case are called coaxial), then when they intersect, a circle is formed. Moreover, the planes of these circles are located perpendicular to the axis of the surface of rotation (Fig. 64a, b).

Thanks to this property, spherical surfaces are used as auxiliary surfaces in determining the points of intersection between the surfaces of two bodies of revolution with intersecting axes. The method where the sphere is taken as an intermediary is called method of auxiliary concentric spheres. It applies only if three conditions are met:

Both surfaces must be surfaces of revolution.

Both surfaces must have a common axis of symmetry (i.e. they must be coaxial).

The axes of symmetry of intersecting surfaces must be straight lines, and these axes must intersect.

Consider the application of this method using a practical example.

Example:

Construct a line of intersection between the surfaces of the cylinder and cone, the axes of which intersect at an angle (Fig. 65). The common plane of symmetry of both bodies P(P 1) is parallel to the plane P 2.

Therefore, the highest and lowest point the intersection lines M(M 1, M 2) and N(N 1, N 2) are obtained at the intersection of the outline generators. All other points of the intersection line are found using auxiliary spheres, drawn from the point of intersection of the axes of the cone and cylinder O(O 1, O 2). Sphere smallest radius is a sphere inscribed in the surface of one of the intersecting bodies. Such a sphere must intersect with the surface of another body. In order to determine which of the intersecting figures fits into the smallest sphere and the intersection points of the O(O 2) axes, we lower the perpendiculars onto the outline generatrices of the figures; whichever perpendicular is larger will be the radius of the smallest sphere (R min =O 2 K 2). The sphere Ф(Ф 2) drawn from the center O(O 2) inscribed in the surface of the cone touches the surface of the cone along the circle m(m 2,m 1) and intersects with the surface of the cylinder along the circle n(n 2). Both of these circles are projected onto P 2 in the form of straight segments K 2 K` 2 and A 2 A` 2. Since the constructed circles belong to the same sphere Ф, they intersect at two points E(E 1, E 2) and F(F 1, F 2), which are common to the surfaces of the cone and cylinder, and, therefore, are located at the line of their intersection.

Arbitrary points 1, 2, 3, 4 are defined using a concentric sphere ( 2) with a radius arbitrarily slightly larger than the radius of the inscribed sphere. After all the points are found in two projections, they are connected by a smooth line on P 2, and on P 1, taking into account visibility.

If two intersecting surfaces are figures of revolution and have a common plane of symmetry, but the axes of these planes do not intersect, then in this case it is applied eccentric sphere method. In this method, the points of the line of intersection between two surfaces are determined using spheres drawn from different centers.

Let's look at the application of this method using an example.

Example:

Construct a line of intersection between the surfaces of the cone and torus (Fig. 66).

First we determine the reference points. The common plane of symmetry of both bodies is parallel to the P2 plane. That's why highest point the intersection line M(M 1, M 2) is obtained at the intersection of the outline generators. The base plane of both figures also coincides and is parallel to P 1. On P 1, both bases are projected from drawing the plane ( 2) in the form of circles and their intersection gives the two lower points of the intersection line E(E 1, E 2) and F(F 1, F 2). To determine arbitrary points 1, 2 draw through the axis of the torus an auxiliary frontally projecting plane, which will intersect the torus in a circle with the center A(A 2); this circle is projected onto P 2 in the form of a segment B 2 B` 2. from the center of this circle (A 2) a perpendicular is drawn to the segment B 2 B` 2. Intersecting with the axis of the cone, it determines the center of the sphere O(O 2). From the center O(O 2) an auxiliary sphere Ф(Ф 2) is drawn of such a radius that it intersects the torus along the circle ВВ`(В 2 В` 2). This sphere intersects the cone along the circle CC`(C 2 C` 2). Both found circles will intersect at two points 1(1 2 ,1 1) and 2(2 2 ,2 1), located on the line of intersection of the surfaces of the cone and torus.

Points 3 and 4 are determined using the auxiliary sphere ( 2) from the center O`(O` 2), found by a similar construction using the auxiliary plane Q(Q 2). After all the points have been found, in two projections they are connected by a smooth curved line on P 2 and P 1. Finally, the mutual visibility of the cone and torus is determined.

IN In some cases, the curve that is obtained by intersecting surfaces of revolution splits into two flat curves, i.e. to second order curves. The conditions under which the intersection line splits into two plane curves are specified in three theorems:

Theorem 1. If two surfaces of revolution (second order) intersect along one plane curve, then they intersect along another plane curve (Fig. 67a, b).

Theorem 2. If two surfaces of revolution touch at two points (Fig. 68 N and M), then the line of their intersection splits into two flat curves. The planes of these curves intersect along a straight line (Fig. 68 MN) connecting the points of contact of the surfaces.

Theorem 3.(G. Monge’s theorem) If surfaces of revolution of the second order are inscribed or described around the third surface of revolution of the second order (sphere), then as a result of their intersection two plane curves of the second order are formed (Fig. 69).

Development of surfaces.

The development is called flat figure, obtained by combining the developable surface with the plane.

Surfaces that can be aligned with a plane without breaks or folds are called developable.

Let's look at different types of scans:

a) Precision developments (faceted surfaces, cone and cylinder) (Fig. 70).

b) Approximate (curved developable surfaces). The curved surface is replaced with a faceted surface. The accuracy of the scan depends on the size of the sections of the faceted surface, and therefore on their number (Fig. 71). To obtain the desired surface from an approximate development, it is enough to bend the thin sheet on which the development is drawn.

c) Approximately - conditional developments (non-developable curved surfaces).

Theoretically, non-developable surfaces cannot have developments. A development is obtained conditionally if this surface is replaced by such simple developable surfaces as cylinders and cones. The latter, in turn, are replaced by multifaceted surfaces, which are deployed.

There are several ways to construct surface developments:

Triangle method (triangulation). This method is used to construct developments of faceted surfaces and all ruled surfaces. The curved ruled surface is replaced by an inscribed faceted surface (Fig. 70, 71).

Normal section method(Fig. 72).

Rolling method.

Auxiliary cylinder and cone method(for constructing conditionally approximate scans).

Let's look at several examples of constructing surface developments:

Example 1. Construct a development of the surface of the pyramid (Fig. 70). Since the side faces of the pyramid are triangles, the construction of its development comes down to the construction of the natural values ​​of these triangles and the natural values ​​of the base. The natural dimensions of the ribs are determined by the method of plane-parallel movement. The development of a pyramid is a series of faces and a base attached to each other.

Example 2. Construct a development of the lateral surface of a truncated cone (Fig. 71). We replace the surface of the cone with an octagonal pyramid inscribed in the cone. The natural size of the generatrices is determined by the method of plane-parallel movement.

This construction can be performed on the original drawing by moving all the generatrices and segments on them to the position of the outermost generatrix, which is parallel to P 2. We replace the arcs of the base of the cone with a series of chords and construct the development similarly to the development of a pyramid (a series of triangles). Then we connect the resulting points with a smooth curved line.

Example 3. Build a sweep inclined prism(Fig. 72). To determine the distance between the ribs of the prism, it is necessary to construct the natural value of the normal section by plane P(P 2), perpendicular to the side ribs. The actual size of a normal section is determined by replacing projection planes or plane-parallel movement. On a development, the figure of a normal section is a straight line, the length of which is equal to the sum of the sides of the normal section. The actual sizes of the ribs AA`, BB`, CC`, DD` are removed from P 2, because the edges of this prism are parallel to P 2, then their actual size is read at P 2. If the edges of the prism are straight lines of general position, then it is necessary to first determine their natural size, then the nature of the normal section and build a development according to the recommendation described above.