A ruled surface is a surface that is formed by the movement of a straight line (generator) in space. Depending on the law of motion of the generatrix of the straight line, three types of ruled surfaces are distinguished.

1.5.4.1. Ruled surfaces with three guides are formed by movement rectilinear generatrix along three guides a ,b And c (curve or straight), which uniquely determine the movement of the generatrix l (Fig. 1.55). So, by selecting on the guide a any point A , it will be possible to draw through this point infinite set rectilinear generatrices of a conical surface with a vertex at the point A and crossing the guide c . From Fig. 1.55 it is clear that through the point A , taken on the guide a ,one and only one straight line passes through, intersecting two other guides b And c .

In the described way through points belonging to the guide a ,you can construct any number of rectilinear generators that will highlight one single ruled surface in space.

Since the position of rectilinear generatrices is uniquely determined by the shape and position in space of the guides a ,b And c , then the determinant ruled surface of the type under consideration is written as:

Ф(a,b,c) –ruled surface.

An example of a ruled surface with three guides is a single-strip hyperboloid, in which the guides are three arbitrarily intersecting lines a ,b And c (Fig. 1.56).

Ruled surfaces are often defined fewer guides. In these cases, the absence of missing guides is supplemented with conditions that ensure the given nature of the motion of the generatrix.

1.5.4.2. To obtain ruled surfaces with two guides, set additional condition maintaining parallelism of the generatrix of any plane, called the plane of parallelism, or maintaining given angle the inclination of the generatrix relative to any plane or axis of rotation (for helicoids). Such surfaces are called surfaces with a plane of parallelism. These include:

- cylindroid l along two curved guides a And b, Σ (Fig. 1.57)

– cylindroid.

In the complex drawing (Fig. 1.5)7 a point was constructed using a surface frame A , which belongs to the cylinderoid. Dot A built on the principle of line belonging With , which in turn belongs to the surface of the cylindroid F :

Usually, for the convenience of constructing generatrices of ruled surfaces, one of the projection planes is taken as the plane of parallelism, then the generators will be the corresponding level lines;

- conoid is formed by the movement of a rectilinear generatrix l along two guides, one of which is a curved line a , and the other is straight b, and in all its positions the generatrix is ​​parallel to a certain plane of parallelism Σ . The surface determinant has the form:

– conoid.

If the conoid has a straight guide b is perpendicular to the plane of parallelism, then the conoid is called direct. In Fig. 1.58 shows a straight conoid with a plane of parallelism P 1 , whose generators are horizontal;

- oblique plane is formed by the movement of a rectilinear generatrix l along two crossing rectilinear guides a And b, and in all its positions the generatrix is ​​parallel to a certain plane of parallelism Σ . The surface determinant has the form:

– oblique plane.

If the guides a And b If there are not intersecting straight lines, but intersecting or parallel ones, then the oblique plane will degenerate into an ordinary plane to which the directrixes belong a And b .

In Fig. 1.59 shows an oblique plane, the guides of which are straight lines a And b, and the plane of parallelism is the horizontal plane of projections P 1 , therefore, the generators of the oblique plane are horizontals.

Since in the section of an oblique plane one can obtain, in addition to rectilinear generatrices and guides, also a hyperbola and a parabola, this surface is also called hyperbolic paraboloid. A parabola is a horizontal outline of an oblique plane shown in Fig. 1.59.

1.5.4.3. There are three types of ruled surfaces with one guide:

- conical surface general form is formed by the movement of a rectilinear generatrix l along some curved line m (guide) and having fixed point S (top) (Fig. 1.60). The surface determinant has the form:

Ф(m,S) –conical surface;

- cylindrical surface is formed as a result of the movement of a rectilinear generatrix l along some curved line m (guide) and having a constant direction s (Fig. 1.61). The surface determinant has the form:

Ф(m,s) – cylindrical surface.

If the guide is broken line, then we obtain special cases of conical and cylindrical surfaces – pyramidal and prismatic surfaces;

- torso is formed by the movement of a rectilinear generatrix l , tangent in all its positions to some spatial curve m , called the return edge. The return rib is the guide of the torso, which completely defines the surface (Fig. 1.62). In this regard, the surface determinant contains only one element:

Ф(m) –torso.

Conical and cylindrical surfaces can be considered as special cases of the torso surface when its cusp edge degenerates into a point (finite or at infinity).

Ruled surfaces with one guide are among the unfolding surfaces. All other ruled surface curves are classified as non-deployable, they are also called oblique.

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 the 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 cylindricals 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 cylinder, 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

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Chapter 8. SURFACES

§ 45. Formation of surfaces

A surface is a set of sequential positions of lines moving in space. This line can be straight or curved and is called generatrix surfaces. If the generatrix is ​​a curve, it may have a constant or variable view. The generatrix moves along guides, representing lines of a different direction than the generators. The guide lines set the law of movement for the generators. When moving the generatrix along the guides, a frame surface (Fig. 84), which is a set of several successive positions of the generatrices and guides. Examining the frame, one can be convinced that the generators l and guides T can be swapped, but the surface remains the same.

Any surface can be obtained in various ways. Yes, straight circular cylinder(Fig. 85) can be created by rotating the generatrix l around the z-axis parallel to it. The same cylinder is formed

by moving the circle t s centered at a point ABOUT, sliding along the axis i. Any curve k, lying on the surface of the cylinder, forms this surface during its rotation around the axis /".

In practice, of all possible ways surface formation, choose the simplest one.

Depending on the shape of the generatrix, all surfaces can be divided into ruled, which have a generative straight line, and non-ruled, which have a forming curved line.

In ruled surfaces, there are developable surfaces, which can be combined with all their points with the plane without breaks and folds, and non-developable surfaces, which cannot be combined with the plane without breaks and folds.

Developable surfaces include the surfaces of all polyhedra, cylindrical, conical and torso surfaces. All other surfaces are non-developable. Non-ruled surfaces can have a generatrix of a constant shape (surfaces of revolution and tubular surfaces) and a generatrix of variable shape (channel and frame surfaces).

To define surfaces, select a set of independent geometric conditions that uniquely defines a given surface in space. This set of conditions is called the surface determinant. The determinant consists of two parts: geometric, which includes the main geometric elements and the relationship between them, and the algorithmic one, which contains the sequence and nature of the operations of transition from the basic constant elements and quantities to the variable elements of the surface, i.e. the law for constructing individual points and lines of a given surface.

A surface in a complex drawing is specified by projections of the geometric part of its determinant, indicating the method of constructing its generators. In a drawing of a surface, for any point in space the question of whether it belongs to a given surface is unambiguously resolved. Graphic task elements of the surface determinant ensures the reversibility of the drawing, but does not make it visual. For clarity, they resort to constructing projections of a fairly dense frame of generatrices and to constructing outline lines of the surface (Fig. 86). When projecting surface Q onto the projection plane, the projecting rays touch this surface at points forming a certain line on it l which is called contour line. Projection contour line called essay surfaces. In a complex drawing, any surface has: P 1- horizontal outline, on P 2 - frontal outline, on P 3 - profile outline of the surface. The sketch includes, in addition to projections of the contour line, also projections of the cut lines.

Of the significant variety of surfaces in the course of engineering graphics, all developable surfaces will be considered, which include faceted, conical, cylindrical, torso surfaces, some surfaces of rotation and helical ones.

The simplest surface widely used in engineering graphics, is a plane, which is a surface formed by moving a rectilinear generatrix (Fig. 87) along two parallel or intersecting straight lines m 1 And m2.

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12. Self-test questions

SELF-TEST QUESTIONS

14. What lines are characteristic of a surface of revolution and what is their role in constructing images of the surface?

46. ​​Image of the plane in the drawing

§ 46. Image of the plane in the drawing

The plane in the drawing can be specified in various ways: by three points that do not lie on the same straight line Q(A, B, C) (Fig. 88, a);

line and a point not on the same line Q(aA; A does not belong to a)(Fig. 88, b);

two intersecting lines Q(a || b)(Fig. 88, c);

two parallel lines Q(a^ b)(Fig. 88, d);

any flat figure, for example, a triangle Q(ABC)(Fig. 88, d).

The planes specified in the drawing by one of these methods are not limited to the projections of the elements that define it.

Considering complex drawing plane, one can be convinced that each of the named methods of specifying it allows the possibility of transition from one of them to another.

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47. The location of the plane relative to the projection planes. Mutual position two planes

§ 47. The location of the plane relative to the projection planes. The relative position of the two planes

Based on their location relative to the projection planes, the planes are divided into planes general and particular provisions.

To planes general position These include planes that are not parallel and not perpendicular to any of the projection planes. In a complex drawing (see Fig. 88), the projections of the elements that define the plane, as a rule, occupy a general position.

Partial position planes include planes parallel or perpendicular to one of the projection planes.

In turn, the planes of particular position are divided into projecting planes and planes level. Projecting planes include planes perpendicular to one of the projection planes. All projecting planes will be denoted by the letter E. Projecting planes can be perpendicular P 1, P 2 or P 3. Depending on this they distinguish horizontally projecting plane when Sum_|_ P 1; front projecting plane when Sum_|_ P 2;profile projecting plane when Sum_|_ P 3;

The projecting plane is different in that its projection onto the projection plane perpendicular to it is always depicted in the form of a straight line and figures lying in the projecting plane. The projection of a plane expressed in a straight line completely determines the position of the plane relative to the projection planes. For example, in Fig. 89, A a complex drawing of plane I is given, defined by two parallel straight lines. From the figure it is clear that I (a \\ b) is a horizontally projecting plane and is located at an angle P to frontal plane projections and at an angle with the frontal plane of projections.

In Fig. 89, b a comprehensive drawing of the plane Sum is shown, making an angle a with the horizontal plane of projections and an angle y with the frontal plane of projections. This can be written like this: ABC ~ A 2 ~ Sum 2 B2~Sum 2, C 2 ~ Sum 2.

The presence of a degenerate projection makes it possible to specify projecting planes in a complex drawing with only one projection. In Fig. 89, V through the point A a profile projecting plane (Sum_|_P 3) is drawn at an angle a to P 1.

All images located in given plane, onto planes not perpendicular to it are projected with distortion.

Level planes include planes parallel to one of the projection planes. They can be considered doubly projecting

planes, since in their complex drawing two projections look like a straight line, located at right angles to the communication line, and the third projection gives an image of all the elements lying in this plane in full size. Level planes are usually designated: G- horizontal plane of the level; F - frontal plane of the level; U - profile

level plane. In Fig. 90, A a comprehensive drawing of the horizontal level plane is given (G || P 1); in Fig. 90, b a comprehensive drawing of the frontal level plane is shown (F || P 2), F e ABC, A 2 B 2 C 2- true size of the triangle ABC; in Fig. 90, V shows a complex drawing of a profile projecting plane (U || P 3, u aA; A~ A).

Level planes differ in that on the projection plane perpendicular to them, they are projected into a straight line on which points, lines and figures located in the level plane are located. These lines are degenerate projections of a given plane. On a projection plane parallel to a given plane, all images of this plane are projected without distortion, i.e. in natural size.

Two planes in space can be parallel or intersect. Planes will be parallel if one of them is defined by intersecting lines, parallel to intersecting ones,

giving a second plane; in Fig. 91 shows parallel planes: Sum (ahb) and Sum 2 (cxd), and a || c, ab || d.

If the planes intersect, then the line of their intersection is straight. Planes perpendicular to each other represent the case of their intersection, when the angle between the planes is 90°.

The construction of lines of intersection of planes is discussed in §62.

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48. Special lines in a plane

§ 48. Special lines in a plane

Special lines in a plane include lines parallel to the projection plane. They are called level lines.

A line belonging to a plane and parallel horizontal plane projections are called horizontal plane (Fig. 92, A). The construction of a horizontal line always begins with its frontal projection: h(A 1 1)~ Q(ABC);h 2 ~ A 2 ;h 2 _|_ A 2 A l ;h 2 ^ B 2 C 2 = l 2,l 2 l 1 || A 2 A 1 .

A line belonging to the plane and parallel to the frontal plane of projections is called frontal plane (Fig. 92, b). The construction of the frontal begins with a horizontal projection: f(F 1 1) ~ ^(DFE); F 1 ~f 1 , f 1 ,_|_F 1 F 2 ; f1^D 1 E 1 =l 1 ; l 1 l 2 || F 1 F 2 ;

l 1 l 2 ^D 2 E 2 =l 2 ^F 2 =l 2 .

By considering singular lines in planes of particular position, one can be convinced that the corresponding level lines in this case will also be projecting.

In Fig. 92, V horizontal shown h frontally projecting plane Sum. IN in this case it will also be a frontal projecting line, i.e. h e Sum; Sum _|_ P 2 .

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49. The relative position of a point, a line and a plane

§ 49. The relative position of a point, a line and a plane

A straight line may or may not belong to a plane. It belongs to a plane if at least two of its points lie on the plane. In Fig. 93 shows the Sum plane (axb). Straight l belongs to the Sum plane, since its points 1 and 2 belong to this plane.

If a line does not belong to the plane, it can be parallel to it or intersect it.

A straight line is parallel to a plane if it is parallel to another straight line

Rice. 94

mine, lying in this plane. In Fig. 93 straight m || Sum, since it is parallel to the line l belonging to this plane.

A straight line can intersect a plane at different angles and, in particular, be perpendicular to it. The construction of lines of intersection of a straight line and a plane is given in §61.

A point in relation to the plane can be located in the following way: belong to it or not belong to it. A point belongs to a plane if it is located on a straight line located in this plane. In Fig. 94 shows a complex drawing of the Sum plane defined by two parallel lines l And p. There is a line in the plane m. Point A lies in the Sum plane, since it lies on the line m. Dot IN does not belong to the plane, since its second projection does not lie on the corresponding projections of the line.

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50. Conical and cylindrical surfaces

§ 50. Conical and cylindrical surfaces

Conical surfaces include surfaces formed by the movement of a rectilinear generatrix l along a curved guide m . The peculiarity of the formation of a conical surface is that

Rice. 96

in this case, one point of the generatrix is ​​always motionless. This point is the vertex of the conical surface (Fig. 95, A). The determinant of a conical surface includes the vertex S and guide m, at the same time l"~S; l"^ m.

Cylindrical surfaces are those formed by a straight generatrix / moving along a curved guide T parallel to the given direction S(Fig. 95, b). A cylindrical surface can be considered as special case conical surface with vertex at infinity S.

The determinant of a cylindrical surface consists of a guide T and directions S forming l, while l" || S; l"^ m.

If the generators of a cylindrical surface are perpendicular to the projection plane, then such a surface is called projecting. In Fig. 95, V a horizontally projecting cylindrical surface is shown.

On cylindrical and conical surfaces, given points are constructed using generatrices passing through them. Lines on surfaces, such as a line A in Fig. 95, V or horizontal h in Fig. 95, a, b, are constructed using individual points belonging to these lines.

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51. Torso surfaces

§ 51. Torso surfaces

A torso surface is a surface formed by a rectilinear generatrix l, touching during its movement in all its positions some spatial curve T, called return edge(Fig. 96). The return edge completely defines the torso and is a geometric part of the surface determinant. The algorithmic part is the indication of the tangency of the generators to the cusp edge.

Conical surface is a special case of a torso, which has a return edge T degenerated into a point S- the top of the conical surface. A cylindrical surface is a special case of a torso, whose return edge is a point at infinity.

52. Faceted surfaces

§ 52. Faceted surfaces

Faceted surfaces include surfaces formed by the movement of a rectilinear generatrix l along a broken guide m. Moreover, if one point S the generatrix is ​​motionless, a pyramidal surface is created (Fig. 97), if the generatrix is ​​parallel to a given direction when moving S, then a prismatic surface is created (Fig. 98).

The elements of faceted surfaces are: vertex S(near a prismatic surface it is at infinity), face (part of the plane limited by one section of the guide m and extreme

relative to it by the positions of the generatrix l) and the edge (the line of intersection of adjacent faces).

The determinant of a pyramidal surface includes the vertex S, through which the generators and guides pass: l" ~ S;

l^ T.

Determinant of a prismatic surface other than a guide T, contains direction S, to which all generating l surfaces are parallel: l||S; l^ T.

Closed faceted surfaces formed by a certain number (at least four) of faces are called polyhedra. From among the polyhedra, a group of regular polyhedra is distinguished, in which all faces are regular and congruent polygons, and the polyhedral angles at the vertices are convex and contain same number faces. For example: hexahedron - cube (Fig. 99, A), tetrahedron - regular quadrilateral(Fig. 99, 6) octahedron - polyhedron (Fig. 99, 6) V). Crystals have the shape of various polyhedra.

Pyramid- a polyhedron whose base is an arbitrary polygon, and side faces- triangles with common top S.

In a complex drawing, a pyramid is defined by projections of its vertices and edges, taking into account their visibility. The visibility of an edge is determined using competing points (Fig. 100).

Prism- a polyhedron whose base is two identical and mutually parallel polygons, and the side faces are parallelograms. If the edges of the prism are perpendicular to the plane of the base, such a prism is called a straight one. If the edges of a prism are perpendicular to any projection plane, then lateral surface it is called projecting. In Fig. 101 given comprehensive drawing straight quadrangular prism with a horizontally projecting surface.

Rice. 100

When working with a complex drawing of a polyhedron, you have to build lines on its surface, and since a line is a collection of points, you need to be able to build points on the surface.

Any point on a faceted surface can be constructed using a generatrix passing through this point. In Fig. 100 on the verge ACS point built M using generatrix S-5.

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53. Helical surfaces

§ 53. Helical surfaces

Helical surfaces include surfaces created by screw motion rectilinear generatrix. Ruled helical surfaces are called helicoids.

A straight helicoid is formed by the movement of a rectilinear generatrix i along two guides: a helix T and its axis i; in this case, the generatrix l intersects the screw axis at a right angle (Fig. 102, a). Straight helicoid is used to create spiral staircases, augers, as well as power threads in machine tools.

An inclined helicoid is formed by moving the generatrix along a screw guide T and its axis i so that the generatrix l intersects the axis i at a constant angle φ, different from a right angle, i.e., in any position, the generatrix l is parallel to one of the generatrices of the guide cone with an apex angle equal to 2φ (Fig. 102, b). Inclined helicoids limit the surfaces of the threads.

Rice. 102 Ruled helical surfaces - helicoids.

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54. Surfaces of revolution

§ 54. Surfaces of rotation

Surfaces of revolution include surfaces formed by rotating line l around straight line i, which represents the axis of rotation. They can be linear, such as a cone or cylinder of revolution, and non-linear or curved, such as a sphere. The determinant of the surface of revolution includes the generatrix l and the axis i. A curved surface of rotation is formed when a person rotates

During rotation, each point of the generatrix describes a circle, the plane of which is perpendicular to the axis of rotation. Such circles of the surface of revolution are called parallels. The largest of the parallels is called equator. Equator determines the horizontal outline of the surface if i _|_ P 1 . In this case, the parallels are the horizontal lines h this surface.

Curves of a surface of revolution resulting from the intersection of the surface by planes passing through the axis of rotation are called meridians. All meridians of one surface are congruent. The frontal meridian is called the main meridian; it determines the frontal outline of the surface of revolution. The profile meridian determines the profile outline of the surface of rotation.

It is most convenient to construct a point on curved surfaces of revolution using surface parallels. In Fig. 103 point M built on parallel h4.

Surfaces of revolution have found the widest application in technology. They limit the surfaces of most engineering parts.

A conical surface of revolution is formed by rotating a straight line i around the straight line intersecting with it - axis i (Fig. 104, a). Dot M on the surface constructed using the generatrix l and parallel h. This surface is also called a cone of revolution or a right circular cone.

A cylindrical surface of revolution is formed by rotating a straight line l around an axis i parallel to it (Fig. 104, b). This surface is also called a cylinder or a right circular cylinder.

A sphere is formed by rotating a circle around its diameter (Fig. 104, c). Point A on the surface of the sphere belongs to the main

meridian f, dot IN- equator h, a point M built on an auxiliary parallel h".

A torus is formed by rotating a circle or its arc around an axis lying in the plane of the circle. If the axis is located within the resulting circle, then such a torus is called closed (Fig. 105, a). If the axis of rotation is outside the circle, then such a torus is called open (Fig. 105, b). An open torus is also called a ring.

Surfaces of revolution can also be formed by other second-order curves. Ellipsoid of revolution (Fig. 106, A) formed by rotating an ellipse around one of its axes; paraboloid of revolution (Fig. 106, b) - by rotating the parabola around its axis; A one-sheet hyperboloid of revolution (Fig. 106, c) is formed by rotating a hyperbola around an imaginary axis, and a two-sheet (Fig. 106, d) is formed by rotating a hyperbola around a real axis.

In the general case, surfaces are depicted as not limited in the direction of propagation of the generating lines (see Fig. 97, 98). To solve specific tasks and receiving geometric shapes limited to the cutting planes. For example, to obtain a circular cylinder, it is necessary to limit a section of the cylindrical surface to the cutting planes (see Fig. 104, b). As a result, we get its upper and lower bases. If the cutting planes are perpendicular to the axis of rotation, the cylinder will be straight; if not, the cylinder will be inclined.

To get circular cone(see Fig. 104, a), it is necessary to cut along the top and beyond. If the cutting plane of the base of the cylinder is perpendicular to the axis of rotation, the cone will be straight; if not, it will be inclined. If both cutting planes do not pass through the vertex, the cone will be truncated.

Using the cut plane, you can get a prism and a pyramid. For example, a hexagonal pyramid will be straight if all its edges have the same slope to the cutting plane. In other cases it will be slanted. If it is completed With using cutting planes and none of them passes through the vertex - the pyramid is truncated.

A prism (see Fig. 101) can be obtained by limiting a section of the prismatic surface to two cutting planes. If the cutting plane is perpendicular to the edges of, for example, an octagonal prism, it is straight; if not perpendicular, it is inclined.

If the line does not belong to the surface, then they intersect. The simplest case is the intersection of a straight line with a surface. The problem is solved by enclosing this line in some projecting plane and constructing the natural size of the section, from which it is easy to determine the entry and exit points of the line. Problems of this type are considered in § 63.

A point may or may not belong to the surface. A point belongs to a surface if it lies on a line located on this surface. In Fig. 104, V dot M belongs spherical surface, since it is on the line of the circle /z" lying on this surface. Points A And IN also belong to the spherical surface, since they are located on the lines of outline circles belonging to the spherical surface. Examples of a point belonging to a surface can also be given in the case of a conical surface (point M in Fig. 104, A), surface of the torus (point M in Fig. 105) and surfaces of more complex shape(dot M in Fig. 103).

The problem of determining whether a point belongs to a surface is solved in the following way. If projections of surface elements and points are given, it is necessary on one of the projection planes through given point draw a line belonging to the surface and construct a projection of this line on one projection plane. If the second projection passes through the second projection of the point, the point belongs to the surface; if it does not pass, it does not belong.

This problem can be considered using the example of Fig. 104, A. In a complex drawing, a conical surface is specified by outline lines. The point is also given M horizontal and frontal projections. Through the horizontal projection of a point we draw a horizontal projection h 1 circle belonging to a conical surface. Having built frontal projection h 2 of this circle, we make sure that it passes through the frontal projection of the point. This confirms that the point belongs to a conical surface.

This problem can be solved in another way. With the same initial data through the frontal projection M 1 points we carry out a projection of one of the generatrices f Having constructed a horizontal projection h generatrix, we make sure that it has passed through the horizontal projection M 1 points M, and this allows us to conclude that the point M belongs to a conical surface.

The principles of constructing points and lines on surfaces form the basis for constructing lines of intersection, cuts, cuts, penetrations, etc., which determines the construction of complex geometric bodies, and ultimately - parts, components, machines, buildings, structures.

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 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 a cylindrical surface can be considered as a special case of a conical surface.

When specifying 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.

When forming a ruled surface using a parallelism plane, the generators must be parallel. this plane, therefore they intersect with it at improper points, the set of which defines the improper line; this straight line should be considered as the third guide of the ruled surface, i.e. the plane of parallelism is, as it were, a proper representative of the improper straight line. The formation of a ruled surface using a plane of parallelism is a special case general method forming a ruled surface with two guides.

The determinant for the group of Catalan surfaces has the form

Ф(g; d 1, d 2, γ);

To specify the surface of this group on the Monge diagram, it is enough to indicate the projections of the guides d 1 and d 2 and the position of the parallelism plane γ (Table 5, Fig. 140 ... 142).

* Named after the Belgian mathematician Katalan, who studied the properties of these surfaces.

Table 5. Ruled surfaces with two guides and a plane of parallelism. Group B II; Ф(g; d 1, d 2, γ);

1. Surface of a straight cylinder (see Table 5, Fig. 140). The surface of a straight cylinder is formed when the guides d 1 and d 2 are smooth curved lines, and one of them must belong to the plane, perpendicular to the plane parallelism.

To determine the projections of the rectilinear generatrices of the surface of a right cylindroid, it is enough to draw straight lines parallel to the plane of parallelism. In Fig. 143 shows the construction of the generatrix g j.

First, we draw g" j -, determine the points M" and N", from them we find M" and N". (MN) we draw parallel to the plane of parallelism γ; for this it is enough that (M"N") || h 0γ.

The surface of a straight cylinder is used in engineering practice In particular, it is used in the manufacture of large-diameter air ducts.

2. Surface of a straight conoid (see Table 5, Fig. 141). The only difference between the surface of a conoid and a cylindroid is that one of the guide lines of the conoid is straight. Therefore, to specify the surface of the conoid on the Monge diagram, it is necessary to indicate the projections: curve ᵭ 2 (one guide), straight line d 1 (second guide) and plane of parallelism γ. E1if a rectilinear guide is perpendicular to the plane of parallelism, then we will be dealing with a special case of a surface, which is called straight conoid.

To obtain a projection drawing (Monge diagram) that is visual, you should indicate the projections of not one, but a number of rectilinear generatrices of this surface. To do this, we draw several straight lines parallel to the plane of parallelism γ and intersecting the guides d 1 and d 2. In Fig. 144 shows the construction of an arbitrary generator g j. In order for the straight line g j to be parallel to the plane of parallelism γ, it is necessary that it be parallel to the straight line belonging to the plane γ. Since the plane γ is horizontally projecting, then the horizontal projections of all lines belonging to this plane coincide with horizontal trail plane h 0γ . Therefore, we begin the construction of a particular generatrix of the conoid surface

from carrying out its horizontal projection g" j , and g" j || h 0γ (based on the invariant property 2r (see § 6) orthogonal projection] . We mark the points M" and N", at which the horizontal projection of the generatrix g" j intersects the horizontal projections of the guides d" 1 and d" 2, using M" and N" we find the points M" and N", which determine the frontal projection of the straight line g" j .

The surface of a straight conoid is used in hydraulic engineering to form the surface of bridge pier abutments.

3. The surface of a hyperbolic paraboloid is an oblique plane (see Table 5, Fig. 142). A hyperbolic paraboloid can be obtained by sliding a straight line along two intersecting rectilinear guides, while the generatrix remains parallel at all times. planes of parallelism. A hyperbolic paraboloid has two planes of parallelism, corresponding to two families of rectilinear guides. If the planes of parallelism are perpendicular to each other, then the hyperbolic paraboloid is called a straight paraboloid. In engineering practice, a hyperbolic paraboloid is often called oblique plane.

To specify an oblique plane in the drawing, it is enough to indicate the projections of two intersecting straight lines d 1 and d 2 and the position of the parallelism plane γ. To obtain a projection drawing that is visual, the projections of several rectilinear generatrices are usually indicated, for this:

1) on the guides d 1 and d 2 segments |AB| and |CD| ;

2) divide the projections of the segments |AB| and |CD| on arbitrary number equal parts (in Fig. 145 the projections of the division points are indicated 1", ... , 6"; 1", ... , 6" and 1" 1, ... , 6" 1; 1" 1, ... , 6" 1

3) the projections of the division points of the same name are connected by straight lines.

When defining an oblique plane in this way, we did not use planes of parallelism. If you need to determine their position, then it is enough to arbitrary point Draw lines e and f, parallel to lines d 2 and d 1, respectively. Second plane parallel

lism (for a family of guides g 1 and g 2) is determined by intersecting straight lines l and m (l || g 1, m || g 2).

The oblique plane is widely used in engineering and construction practice for forming the surfaces of slopes of iron and iron embankments. highways, embankments hydraulic structures at the junction of slopes having different angles of inclination.

4. Flatness. If the directing lines d 1 and d 2 intersect or are parallel, then when the rectilinear generatrix g moves along them, a plane is obtained. The image of the plane on Monge's survey and various options for its location in relation to the projection planes were discussed in detail in § 8 of Chapter. I.