Surface in geometry is called border separating a geometric body (cylinder, cone, ball, etc.) from outer space . In the drawings (diagrams), only points and lines (straight lines or curves) are depicted. Therefore, the surface can be depicted only when it is projected onto a line or a set of lines.

The surface can be specified using a model (shoe last, mannequin, etc.), using an equation, kinematically - as a trace of a line moving in space, etc. In descriptive geometry, a kinematic method of surface formation is adopted. It can be said that surface – it is a continuous set of successive positions of a straight or curved line moving in space . A line that forms a surface as it moves is called generatrix .

2.4.1. Defining a surface using a determinant. In order to set a surface, it is sufficient to set the generatrix of the surface and determine the law by which it moves in space. The laws of motion of generators can be specified in different ways:

1) The generatrix moves by crossing some fixed line, which is called guide .

2) The generatrix moves by crossing two or three guide lines.

3) The generatrix moves parallel to itself or parallel to some plane, which is called plane of parallelism and etc.

The generatrix, together with the geometric figures that determine its movement, as well as the law of its movement, constitute determinant surfaces. We can say that the determinant of the surface is a set of independent parameters that uniquely define the surface.

The determinant consists of two parts:

1) geometric part - figures (points, lines, surfaces) movable and fixed, with the help of which a surface is formed.

2) Algorithmic part - the rule of motion (the law of motion) of the generatrix in relation to the fixed figures of the determinant.

In some cases, the generatrix can be deformed during its movement, which is also specified in the algorithmic part of the determinant. The basis for the compilation of the determinant is the analysis of the method of formation of the surface and its main properties. Each surface can be defined by different determinants.

For example, consider the determinant of an arbitrary cylindrical surface (Fig. 2.34). The determinant record looks like this:

F(l, a) - cylindrical surface

(geometric part) (algorithmic part)

This entry is given in conjunction with the drawing. In the notation of the geometric part with the letter F the surface is denoted by the letter l- generatrix, letter A- guide. The shape and position in space of the generatrix and guide are determined by the drawing.

In the record of the algorithmic part, the name of the surface is given. For a surface with a given name, it is well known what movement the l, forming a surface F. But it is also possible to write down in detail the nature of the motion of the generatrix. In our case, the generator l moves parallel to itself and crosses the guide all the time A. The determinant completely determines the surface, since with its help it is possible to construct its projections.

On fig. 2.35 A a complex drawing of the determinant of a cylindrical surface is specified F(l, a) and projection A 2 points A belonging to the surface. It is necessary to build a horizontal projection A 1 points A.

Knowing the algorithmic part of the determinant, we perform the following constructions (Fig. 2.35, b):

1) Through A 2 parallel l 2 draw and find the frontal projection AT 2 points of intersection with a 2(stage 1). The steps are indicated by arrows.

2) Using the projection link on a 1 find IN 1(stage 2).

3) Through a point IN 1 run in parallel l 1(stage 3).

4) We build on using the communication line A 1(stage 4).

2.4.2. Surface wireframe. If we build a certain number of generators according to the method described in the determinant algorithm, then we get frame or net surfaces (Fig. 2.36).

Shown in fig. 2.36 A the frame is called one-parameter, because it consists of lines belonging to the same family. This is a discrete frame, it consists of a finite number of lines.

One can also imagine a continuous frame of generators. A continuous wireframe is a set of lines that fill the surface so that only one wireframe line passes through each point of the surface.

On the same surface, depending on the determinant, one can imagine other frameworks. If in the determinant of a cylindrical surface the generatrix and guide are interchanged and we assume that the curve A will be a generatrix that moves parallel to itself and intersects the guide all the time l, then another one-parameter frame will be obtained (Fig. 2.36, b).

If two frames are built on the surface, then a two-parameter frame will be obtained (Fig. 2.36, V). Two wireframe lines pass through each point of the surface defined by a two-parameter skeleton.

2.4.3. Specifying a surface that does not have a determinant. There are irregular surfaces, which include a mannequin, a shoe last, car bodies, aircraft fuselages, hulls of sea and river vessels, relief earth's surface etc. Such surfaces are called graphic and are given by a discrete framework. Most often, the lines of this frame are flat curves parallel to any projection plane. If the planes of the frame lines are parallel to the horizontal projection plane, then such lines are called horizontal.

2.4.4. Surface outline. The line of intersection of the projecting surface, enveloping the given surface, with the projection plane is called the outline of the surface . On fig. 2.37 shows the projection of the sphere T to the plane P 1. A set of horizontally projecting rays tangent to the surface of the sphere form an envelope of a horizontally projecting cylindrical surface F. Intersection line F And P 1 represents a horizontal outline of the surface - a circle a 1.

The outline line of a surface is the line along which the enveloping projecting surface touches the given surface. In our case, the outline line will be the great circle of the sphere A(equator).

The images of the surfaces given by the determinant are not always visual. The images of surfaces are more visual with the help of sketches. The outline of a surface almost always includes its determinant. When constructing projections of a point lying on a surface depicted by a sketch, it is necessary to first select the projections of the determinant, and then, using the determinant algorithm, construct the projections of the point.

On fig. 2.38 A the surface of an inclined elliptical cylinder is given by a determinant, and in fig. 2.38 b sketch. A horizontal outline is a line consisting of segments of straight lines and curves. ; the frontal outline is a parallelogram.

The generators of the horizontal outline and and the generators of the frontal outline and do not coincide with each other. From the projections of the essay, one can single out the geometric part of the determinant, which will consist of an ellipse and some generatrix, for example.

2.4.5. Plane projections. A plane can be considered as a special case of a surface. Plane Σ can be formed due to the movement of a rectilinear generatrix l parallel to itself, while the generatrix intersects all points of the directing line A(Fig. 2.39). The plane determinant in this case has the form: Σ (A, l).

It is known from geometry that planes are completely determined:

1) Three dots A, IN And WITH, not lying on one straight line (Fig. 2.40, A).

2) Straight A and dot A outside of it (Fig. 2.40, b).

3) Two parallel lines A And b(Fig. 2.40, V).

4) Two intersecting lines A And b(Fig. 2.40, G).

Specifying a plane with intersecting lines A And b(Fig. 2.40, G) can be considered as a universal way to define a plane, since all the others can be reduced to it. So, for example, if the plane is given by three points A, IN And WITH(Fig. 2.40, A), then by connecting the dots A With IN And IN With WITH, we get intersecting lines AB And sun.

2.4.6. Types of planes according to their location in space. According to the location relative to the projection planes, the plane can be divided into three types:

1) plane general position - planes that are not parallel and not perpendicular to the projection planes;

2) plane projecting - planes perpendicular to any projection plane;

3) plane level - planes parallel to one of the projection planes and perpendicular to the other two.

Consider some of the features of each of the listed types of planes.

Planes in general position. On fig. 2.40 planes of general position are shown. It is typical for these planes that the elements defining them (points, straight lines, etc.) do not merge into a straight line on any projection, i.e. do not lie on the same line.

On fig. 2.41 given plane Σ () and one projection A 2 points A belonging to the plane Σ . We will assume that A- guide, b- generatrix of the plane Σ . Keeping in mind that all generators are parallel to each other and all intersect with the guide, we will perform the following constructions:

1) Through a point A 2 let us carry out the projection of the generatrix m2║b 2 and build a point K 2 intersections m2 With a 2(stage 1).

2) On the communication line and on a 1 find K 1(stage 2).

3) Through K 1 carry out m 1║b 1(stage 3).

4) Using the communication line on m 1 find A 1(stage 4).

In this construction, the generator m 1, lying in the plane Σ , was built on a point and a known direction. However, when constructing a point lying in a plane, one can use not only the generatrix lying in the plane. On fig. 2.42 horizontal projection of a point A constructed using an arbitrary line. At the same time, the following constructions were carried out:

1) Through a given projection A 2 draw an arbitrary line m2 and considering that m lies in the plane Σ (), mark the points of its intersection K 2 And M 2 With a 2 And b 2(stage 1).

2) Building K 1 And M 1 on a 1 And b 1 using communication lines (stage 2).

3) Connect K 1 And M 1 and get m 1(stage 3).

4) On m 1 with the help of a communication line we find A 1(stage 4).

Obviously, in order to construct a point in a plane, it is necessary to draw a line in this plane and then take a point on the line. Wherein A line is in a plane if it passes through two points in the plane.

Projecting planes. There are three types of projecting planes:

1) Horizontal projection , perpendicular P 1.

2) front projecting , perpendicular P 2.

3) Profile projecting , perpendicular P 3.

When depicting projecting planes, one must keep in mind that the projection of the same name of such a plane always degenerates into a straight line, as was shown earlier. This line is called main projection or next projecting plane; this projection is also called degenerate . In order to distinguish the projecting plane from a straight line, the main projection of the projecting plane in the drawing is often depicted with a thickened end.

On fig. 2.43, A a visual image of an arbitrary horizontally projecting plane is shown Σ (A║b) and its main projection Σ 1. A comprehensive drawing of this plane is shown in Fig. 2.43, b. All points lying in the plane are projected onto the main projection of the plane.

Frontal projection plane T(With d) is shown in Fig. 2.44 A, profile-projecting plane G (e f) - in fig. 2.44 b and profile-projecting plane R (A ║ b) - in fig. 2.44 V.

Due to the projective property quoting planes can be defined by one of their principal projections (thereafter, a degenerate projection). On fig. 2.45 the front-projecting plane is set Σ .

It is known from stereometry that planes are perpendicular if one of them passes through a perpendicular to the other. Therefore, in each projecting plane, it is possible to construct a projecting line of the same name. On fig. 2.43, b in plane Σ (A║b) a horizontally projecting straight line is constructed With. On fig. 2.44 A in plane T (With d) a frontally projecting straight line is constructed f .

In planes G (e f) (Fig. 2.44, b) And R (A ║ b) (Fig. 2.44, V) there are lines, perpendicular P 3. Therefore, these planes are profile-projecting. Thus, profile-projecting planes can only be specified by projections onto P 1 And P 2.

The question of whether a point and a line belong to a projecting plane is solved more simply than in a plane in general position. The projection of a point or line is always in the main projection of a plane degenerated into a line. So, in Fig. 2.46, A point projections are shown A, and in fig. 2.46 b - straight A belonging respectively to the horizontally projecting plane Σ and front-projecting plane T.

Level planes. There are three types of level planes:

1)Horizontal plane, parallel P 1 and perpendicular P 2 And P 3.

2)Frontal plane, parallel P 2 and perpendicular P 1 And P 3.

3)Profile plane, parallel P 3 and perpendicular P 1 And P 2.

Level planes can be called doubly projecting , since each of them is perpendicular to two projection planes.

From the projecting property it follows that the level planes are projected into lines, each on two projection planes. On fig. 2.47 is a visual representation of the horizontal level plane Σ . characteristic feature drawings of level planes is the parallelism of the main (degenerate) projection of the plane of one of the axes of the drawing. On fig. 2.47 Σ ║ P 1 And Σ P 2, Σ P 3. Let's prove that Σ 2 ║ x 12.

It is known that If two parallel planes are intersected by a third plane, then parallel lines are formed. When crossing P 2 And P 1 an axis is formed x 12. When crossing P 2 With Σ its main projection is formed Σ 2. In the same way, it is proved that Σ 3 ║ 3.

horizontal plane G (A b) is shown in Fig. 2.48 A, frontal plane T (A ║ b) - in fig. 2.48 b, profile plane Ω (∆ ABC) - in fig. 2.48 V.

2.4.7. Examples for incidence . Consider several problems on the mutual belonging of a point and a straight line.

1) Through a point A draw a general plane Σ (A b), Where A ║ P 1 And b ║ P 2(Fig. 2.49, A).

Solution: through a point A(A 1, A 2) we carry out horizontal projections A ║ P 1 and frontals b ║ P 2. Other options are also possible. Yes, through the dot A one can draw a horizontal or frontal and intersect it with a line in general position. It is also possible through the dot A draw two straight lines in general position. However, in this case, it is necessary to check for the absence of profile-projecting lines in the resulting plane, the presence of which indicates the receipt of a profile-projecting plane.

2) Conclude a straight line A(a 1, a 2) of general position in the horizontally projecting plane Σ , setting it as its main projection Σ 1 (Fig. 2.49, b).

Solution: carry out the main projection Σ 1 coinciding with the horizontal projection a 1.

3) Construct a horizontal projection of a straight line b in general position intersecting with a straight line A so that both lines belong to the horizontally projecting plane T(Fig. 2.49, V).

Solution: carry out a frontal projection of a straight line b so that b 2 was not parallel or perpendicular x 12, and the horizontal projection b 1 coincided with a 1. Main projection T 1 plane T in this case coincides with the horizontal projections of the intersecting lines A And b.

4) Cross the line A direct private position d so that both lines are enclosed in a horizontally projecting plane G(Fig. 2.49, G).

Solution: direct A intersect the horizontally projecting line anywhere d. Main projection G 1 horizontally projecting plane G coincides with horizontal projections a 1 And d1 direct.

5) Conclude a straight line A into the profile-projecting plane Ψ (Fig. 2.50, A).

Solution: in the simplest case, we intersect the line A profile-projecting line b P 3. Two intersecting lines A And b form a profile-projecting plane Ψ , because if there is a perpendicular to another plane in a plane, then these planes are perpendicular to each other.

6) Through the dot A draw a horizontal projection plane Σ (fig.2.50, b).

Solution: through a point A 1 arbitrary, but not perpendicular or parallel x 12 carry out the main projection Σ 1 plane Σ.

7) Through the dot IN draw a horizontal level plane T(Fig. 2.50, V).

Solution: through a point AT 2 carry out the main projection T 2 plane T parallel x 12.

2.4.8. Parallelism of a line and a plane . A line is known to be parallel to a plane if it is parallel to any line in that plane. Let, for example, through the point M it is necessary to make a direct d general position parallel to the plane given as a triangle - Σ (ABC) (Fig. 2.51).

Solution : in plane Σ (ABC) we draw an arbitrary straight line in general position ED(E 1 D 1,E 2 D 2). Further through the point M 1 make a horizontal projection d 1 ║ E 1 D 1 and front projection d 2 ║E 2 D 2 straight d.

If through a dot TO need to be horizontal b parallel to the plane Σ (ABC), then the constructions should be performed in the following sequence:

1) We build a frontal projection A 2 D 2 horizontal AD parallel to axis x 12.

2) In the projection connection we find the horizontal projection A 1 D 1.

3) Through points K 1 And K 2 make projections b 1 ║ A 1 D 1 And b 2 ║ A 2 D 2 desired horizontal b. It should be noted that it is not at all necessary to draw a horizontal line through a point A, which we recommend to the reader to verify.

2.4.9. parallel planes. To construct parallel planes, we use the sign of their parallelism, known from stereometry: "Planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of the second plane."

Let it be required through a dot TO(Fig. 2.52) draw a plane Σ (A b) parallel to the plane T (With d). To solve a problem through a point TO carry out A ║ With so that a 1║ from 1 And a 2║ since 2, And b ║ d, to b 1 ║ d1 And b 2 ║ d2.

On fig. 2.53 the problem is considered when direct A And b enclose in a pair of parallel planes. The condition of the problem is given in fig. 2.53, A. To solve it, we take on straight lines A And b arbitrary points TO And M(Fig. 2.53, b). Further through the point TO draw a straight line With ║ b, and through the point M direct d ║ A. As a result, we obtain parallel planes Σ (A With) And T (b d), because two intersecting lines A And With plane Σ are respectively parallel to two intersecting lines b And d plane T.

2.4.10. Construction of plane projections when replacing projection planes. In order to build projections of the plane when replacing the projection plane, the plane must be defined by three points. When constructing, each point that defines a plane is transformed similarly to that considered earlier when replacing projection planes. On fig. 2.54 shows the transformation of the plane with an arbitrary replacement of the projection plane P 2 on P 4.

The most complex position of a plane in space is the generic plane, the simpler one is the projecting plane, and the simplest is the level plane. When solving problems, the plane is usually placed from a more complex position to a simpler one. Thus, a series of plane transformations has the form: a generic plane → a projecting plane → a level plane.

Let's make the first transformation. Let a plane be given in general position Σ (ABC) (Fig. 2.55), and it must be converted to front-projecting. A projecting plane always contains a projecting line. Any straight line can be transformed into a projecting one by replacing the projection planes: a general position line - using two transformations, a level line - using one transformation.

To solve the problem, we perform the first transformation. For this:

1) In plane Σ (ABC) build a horizontal AE (A 2 E 2, A 1 E 1).

2) We put AE to the projecting position by replacing P 2 on P 4, and x 14 A 1 E 1.

3) Project the triangle onto a new plane P 4. At the same time, in the system P 1–P 4 triangle ABC- projecting. His new frontal projection A 4 B 4 C 4 represents a straight line.

Let's do the second transformation. In system P 1–P 4(Fig. 2.53) Σ (ABC) is a front-projecting plane, and it must be converted to a level plane. Any level plane is parallel to one projection plane and perpendicular to the other two. In this case Σ (ABC) P 4. Therefore, if we replace P 1 on P 5, putting P 5 ║ Σ (ABC), then in the system P 4–P 5 plane Σ (ABC) becomes a level plane.

Let's make constructions. For this:

1) Let's draw an axis x 45 ║ Σ 4.

2) In the system P 4–P 5 build projections of points A 5, AT 5 And From 5. Triangle projection A 5 B 5 C 5 represents its natural size, since the plane Σ (ABC) ║ P 5. When transforming a generic plane into a level position, two successive transformations were performed. First, one projection plane was replaced, then another.

2.4.11. Surface classification. We will classify surfaces according to two criteria:

In the form of the generatrix:

1) Planes, polyhedral surfaces and ruled curved surfaces have a rectilinear generatrix.

2) Curvilinear generatrix, unchanging and changing, - all other curved surfaces.

According to the developability of the surface to the plane:

1) Deployable.

2) Non-deployable.

Deployment is such an isometric deformation of the surface, in which it can be combined with the plane.

Isometric surface deformation is called bending. When bending, line segments located on the surface do not change their length. If a surface can be aligned with a plane without wrinkles or breaks, then it deployable . Most surfaces are not compatible with a plane without folds and breaks and are called non-deployable .

Developable are polyhedral surfaces and part of the ruled ones - cylindrical, conical and torso. There is no need to talk about the deployability of the plane - it can be combined with any plane.

Consider the features of constructing images of certain types of surfaces.

2.4.12. Polyhedral surfaces and polyhedra . It is considered to be , What A polyhedral surface is a surface formed by parts (by compartments) intersecting planes.

The surface of a polyhedral angle is a surface whose edges and faces intersect at one point.(top) . If you intersect the surface of a polyhedral angle with a plane, then a geometric figure is formed - pyramid.

The surface of a polyhedral angle can be obtained by moving a generatrix that always passes through the vertex of the angle and at the same time slides along the guiding polygon.

If the vertex of a polyhedral angle is taken to infinity, then the edges of the surface become parallel, and a prismatic surface .

If we limit the prismatic surface to two parallel flat bases, then a geometric figure is formed - prism .

A polyhedral surface definer includes a guide polygon, a vertex for a polyhedral angle, and some edge for a prismatic surface.

On fig. 2.56 shows the surface of a polyhedral angle F (ABCD, S) in a spatial image with a guiding quadrilateral ABCD and top S. On fig. 2.56 A the determinant of the surface is given. On fig. 2.56 b surface frame is constructed.

+

On fig. 2.57 A prismatic surface shown F (ABC, l) in a spatial image with a guiding triangle ABC and generating l; in fig. 2.57 b prism is shown.

The determinant of a pyramid can be its base and top. The determinant of a prism is its base and one side edge or one vertex of the other base.

When depicting polyhedra, they try to arrange them so that on the projections their edges and faces are projected as far as possible without distortion or with the least distortion.

From the whole variety of polyhedral surfaces, as an example, consider the sequence of constructing only regular trihedral straight prisms and pyramids.

Straight trihedral regular prism. On fig. 2.58 A given the graphic task of the prism F (ABC, ) by its determinant. In order to get a complex drawing of a prism (Fig. 2.58, b), it is necessary to complete two horizontally projecting edges IN And WITH and three horizontal edges of the upper base , and .

Let us analyze the elements of the lateral surface of the prism.

Lateral ribs are horizontally projecting straight lines. The edges of the bases are horizontals, of which the edges AC and - profile-projecting straight lines.

The side faces are horizontally projecting planes, of which the face is the frontal plane. The bases are horizontal planes. On a horizontal projection, both bases and their edges are projected in full size. On the frontal projection, the side edges and the rear front face are projected in full size.

Consider examples of incidence. Let the projection K 2 points TO. Find K 1, assuming that the point lies on the visible face of the prism (Fig. 2.58, b).

On the frontal projection, the faces and are visible, the face is not visible. Therefore, we consider that the point TO lies on the visible face, and its horizontal projection K 1 falls on the degenerate projection of the face (the projecting trace of the face).

Let the projection M 1 points M. Find M 2, assuming that the point lies on the apparent base of the prism.

Tangent planes are widely used in solving various positional problems on the surface.

1. The construction of tangent planes to surfaces is the basis of the theory of shadows. When constructing shadows, tangent planes to surfaces are built either passing through a point lying on the surface, or parallel to a given direction.

2. Tangent planes to the surfaces of the cone and cylinder, parallel to a given direction, are used to determine the line of intersection of these bodies with the plane of general position, closest and farthest from the planes of projections of the points of the curve, without constructing these curves (see Bubennshchiv § 68).

3. Tangent planes are used in the construction of contiguous one-lane hyperboloids of revolution in the design of hyperbolic gears. In gears with crossed shafts. (see Tambourines § 68)

4. Tangent planes are also used in the construction of the outlines of surfaces (outlines).

Let's consider this problem in more detail.

As you know, the outline of the surface (body) is obtained as a projection of the contour line onto the back projection plane (for example, P 1) (see Fig. 7.5). Recall that the contour line is the line along which the set of planes P, perpendicular to the plane P 1, touch the given body T (Fig. 10.13). The envelope of this family of tangent planes will be some cylindrical ray surface Ф, also perpendicular to П 1 .

Figure 10.13

The contour line m divides the body into two parts, one of which is visible on given plane projections P 1 , and the other invisible. At any point on the contour line, both surfaces - the body and the cylindrical ray - have a common tangent plane P. The line of intersection m 1 of the ray cylindrical surface Ф with the plane P 1 and is body outline. If we assume that the cylindrical ray surface consists of light rays touching an opaque body, then the outline of the body is a line that limits the shadow of the body on the plane P 1. This line on the projection planes is also called line of sight.

Figure 10.13 shows that the outline of the ball of the P 1 plane will be the projection of the equator m (m 1), which is projected onto the P 2 plane in the form of a straight line parallel to the OX axis. The outline of the ball on the plane P 2 will be the projection of its main meridian.

In figure 10.14 there will be a rectangle (principal meridian). The outline on the plane P 1 is determined by two tangent ray planes perpendicular to the plane P 1 . These planes touch the cylinder along the two extreme generatrixes AB and CD, the projections of which on the plane P 2 coincide. Horizontal projections A 1 B 1 and C 1 D 1 together with the outer surfaces (projections of the circles of the bases) determine the outline of the cylinder on the plane P 1 .

Figure 10.14

In the general case, to construct an outline of a body on the P 1 plane, you must first construct a projection of the contour line on the P 2 plane, along which the body is wrapped by a cylindrical ray surface, and then project it onto the P 1 plane.

The construction of a contour line is easiest to implement using inscribed spheres.

Example 8. Construct on a horizontal projection an outline of a cone whose axis i is parallel to the plane P 2 and inclined to the plane P 1. (Fig. 10.15)

Solution. It is not difficult to see that the contour of the cone on the plane P 2 , limited by the main meridian m, completely determines the shape of the surface of the cone.

Figure 10.15

And to build a horizontal outline from any point C (C 2) lying on the i-axis, we draw a sphere touching the cone along the circle k (k 2). Its frontal projection is a straight line perpendicular to the axis (i 2), as coaxial bodies.

We draw the equator q 2 through the center of the sphere and find the point A 2 its intersection with the circle k 2 . By connecting the points S 2 and A 2 we get the contour line. Projecting point A 2 onto the horizontal projection of the equator, we get two points A 1, which together with the top S1 and set the horizontal outline of the contour n 1 . Note that the frontal projection n 2 of the horizontal outline does not coincide with the projection of the axis i 2 .

Example 9. Build on a horizontal projection P 1 Sketch of the details of rotation, the I axis of which is parallel to the plane P 2 and inclined to the plane P 1. The part surface consists of a cone of revolution (S, k) and a torus, the generatrix of which is an arc of a circle with a radius R centered on a point ABOUT. (Fig. 10.16)

Figure 10.16

Solution:

1. The outline of the frontal projection - this is the main meridian - completely sets the shape of the part.

2. The outline of the horizontal projection is made up of the ellipse of the upper base, the spatial curve and the outline of the cone.

3. We build an ellipse along two axes - a small one 1 1 2 1 and a large one 1 2 2 2 .

4. We build the outline of the cone according to example 8 (Fig. 10.15).

6. To construct a contour line on the surface of the torus, we inscribe a number of spheres into it. The centers of the spheres C 2 lie at the points of intersection of the axis of rotation i 2 with the radius R drawn from the point O 2 to the meridian. The spheres touch the torus along the parallels k 2 .

7. The planes tangent to the torus are also tangent to the auxiliary spheres at the points A 2 of the intersection of the equators q2 spheres with parallels k 2 .

8. Horizontal projections A 1 of these points are determined at the intersection of communication lines with the horizontal projection of the equator q1.

9. A number of points are found by similar constructions (for example, B 2). The set of points form a contour space curve l 2 .

10. The horizontal projection l 1 will give the outline of the torus.

11. So, the outline of the detail is a composite flat curve from the outlines of the contour n 1, the torus l 1 and the ellipse.

Each surface of one of its sides can be directed towards the observer, and then this side will be visible. Otherwise, the side of the surface will not be visible from the viewpoint. It may happen that only part of the side of the surface is visible. In this case, a line can be drawn on the surface separating the visible and invisible pure surfaces. A sketch line is a line on a surface that separates the visible part of the surface or face from its invisible part.

Rice. 9.5.1. Surface Sketch Line Projections

Rice. 9.5.2. Projections of a grid of polygons and outline lines

On fig. 9.5.1 shows the lines of the outline of the surface. On fig. 9.5.2 shows the outline lines together with the surface grid.

When crossing the line of outline, the surface normal changes direction with respect to the line of sight. At the points of the outline line, the surface normal is orthogonal to the line of sight. In the general case, there can be several outline lines near the surface. Each line of the outline is a spatial curve. It is either closed or ends at the edges of the surface. For different directions of view, there is a set of outline lines, therefore, when the surface is rotated, the outline lines must be built anew.

parallel projections.

For some surfaces, for example, a sphere, a cylinder, a cone, outline lines are built quite simply. Let us consider the general case of constructing lines of the outline of a surface.

Let it be required to find the outline lines of the surface described by the radius vector. Each point of the outline line for a parallel projection onto the plane (9.2.1) must satisfy the equation

where is the normal to the surface for which the sketch line is being constructed. For a surface described by the radius vector, the normal is also a function of the parameters and . The scalar equation (9.5.1) contains two required parameters u, v. If you set one of the parameters, then the other can be found from equation (9.5.1), i.e. one of the parameters is a function of the other. For equality of parameters, they can be represented as functions of some common parameter

The result of solving equation (9.5.1) is a two-dimensional line

on the surface This line is the outline of the surface.

We will construct a sketch line from an ordered set of points that satisfy equation (9.5.1). By points we mean a pair of surface parameters, which are the coordinates of two-dimensional points on the parametric plane. Having separate points of the outline line, located in their order and at a certain distance from each other, it is always possible to find any other point of the line. For example, to find a point lying between two given neighboring points of the outline line, we draw a plane perpendicular to the segment connecting neighboring points and find a common point for the surface and the plane by solving three scalar intersection equations together with equation (9.5.1). The position of the plane on the segment can be set by the line parameter. By the extreme points of the segment, the zero approximation for the desired point is determined. Thus, the set of individual two-dimensional points of the surface outline line serves as a zero approximation of this line, by which one of the numerical methods can always find the exact position of the point. The algorithm for constructing surface outline lines can be divided into two stages.

At the first stage, we find at least one point on each line of the outline. To do this, walking along the surface and examining the sign of the scalar product at neighboring points, we find pairs of points on the surface at which the sign changes. Taking as a zero approximation the average values ​​of the parameters of these points, one of the numerical methods will find the parameters of the point of the outline line. Let, for example, when moving from a point to a point close to it, change sign. Then, setting with the help of the iterative process of Newton's method

or iterative process

find the parameters of one of the points of the outline line. The derivatives of the normal are determined by the Weingarten formulas (1.7.26), (1.7.28). In this way, we get a set of points for the outline lines. The points from the set obtained at the first stage are not related to each other in any way and may belong to different outline lines. It is only important that at least one point is present from each outline line in the set.

At the second stage, we take any point from the existing set and, moving from it with some step, first in one direction and then in the other, we find point by point the desired set of points of the outline line. The direction of movement gives the vector

where - partial derivatives of the normal - partial derivatives of the radius vector of the surface with respect to the parameters .

The sign in front of the term coincides with the sign of the scalar product The motion step is calculated in accordance with the curvatures of the surfaces at the current point by formula (9.4.7) or by formula (9.4.8). If

then by formula (9.4.7) we give an increment to the parameter u and by formula (9.5.4) we find the parameter v of the surface corresponding to it. Otherwise, by the formula (9.4.8) we will give an increment to the parameter and and by the formula (9.5.5) we will find the parameter corresponding to it and the surface. We will finish moving along the curve when we reach the edge of one of the surfaces or when the line closes (the new point will be at the distance of the current step from the starting point).

In the process of movement, we will check whether points from the set obtained at the first stage lie near the route. To do this, along the route, we will calculate the distance from the current point of the outline curve to each point from the set obtained at the first stage. If the calculated distance to any point of the set is commensurate with the current step of movement, then this point will be removed from the set as more unnecessary. So we get a set of individual points of one line of the essay. In this case, the set of points obtained at the first stage will not contain any points of this line. If there are more points left in the set, then the given surface has at least one more outline line.

Rice. 9.5.3. body outline lines

Rice. 9.5.4. Body of rotation

We find the set of its points by taking any point from the set and repeating the second stage of construction. We will complete the construction of lines when there is not a single point left in the set. Using the described method, we will construct the outline lines of all the faces of the model.

The outline lines of the faces are the outline lines of their surfaces. The outline line of the body will be visible if it is not covered by a face that is closer to the viewpoint. On fig. 9.5.3 shows the outline line of the body of revolution shown in fig. 9.5.4. The outline line projection may have breaks and cusps, but the outline line itself is smooth.

Breakpoints in the projection occur where the tangent line of the outline is collinear to the vector

To construct the projection of the outline line, we will build its polygon, the projection of which will be taken as the projection of the outline line.

central projections.

The outline lines in the central projections satisfy the equation

(9.5.7)

where - surface normal - radius-vector of the observation point. The outline line for the central projection differs from the outline line for the parallel projection, although the algorithms for their construction are similar. Instead of a constant vector in (9.5.7) there is a vector whose direction depends on the projected point. The sketch line for the central projection also represents a certain curve on the surface, described by the dependencies (9.5.3), and is a spatial curve. This line must be projected onto the plane according to the rules for constructing the central projection of a spatial line.

On fig. 9.5.5 shows a parallel projection of the outline lines of the torus, and in fig. 9.5.6 for comparison, the central projection of the outline lines of the torus is shown. As you can see, these projections are different.

Rice. 9.5.5. Parallel projection of torus outline lines

Rice. 9.5.6. Central projection of torus outline lines

The algorithm for constructing outline lines for the central projection of a surface described by a radius vector differs from the algorithm for constructing outline lines for a parallel projection of this surface in that at the first stage we will look for surface points at which the scalar product changes sign. To determine these points, instead of formulas (9.5.4) and (9.5.5), one should use the formulas

and formulas

respectively. Otherwise, the algorithm for constructing outline lines for the central surface projection does not differ from the algorithm for constructing outline lines for a parallel projection.

On fig. 354 shows a straight circular cone, the axis of which is parallel to the square. π 2 and inclined to the square. π 1 The outline of its frontal projection is given: it is an isosceles triangle S"D"E". It is required to construct an outline of the horizontal projection.

The desired outline is made up of a part of an ellipse and two straight lines tangent to it. Indeed, the cone in its given position is projected onto the square. π 1 using the surface of an elliptical cylinder, the generators of which pass through the points of the circumference of the base of the cone, and using two planes tangent to the surface of the cone.

An ellipse on a horizontal projection can be built along its two axes: a small D "E" and a large one, equal in size to D "E" (the diameter of the circumference of the base of the cone). The lines S "B" and S "F" are obtained by drawing tangents to the ellipse from the point S ". The construction of these lines consists in finding the projections of those generators of the cone along which the cone and the planes mentioned above come into contact. For this, a sphere inscribed in cone Since the plane projecting onto π 1 simultaneously touches the cone and the sphere, it is possible to draw a tangent from the point S "to the circle - the projection of the equator of the sphere - and take this tangent as the projection of the desired generatrix. The construction can be started by finding point A "- the frontal projection of one of the points of the desired generatrix. Point A" is obtained at the intersection of the frontal projections: 1) the circle of contact between the cone and the sphere (line M "N") and 2) the equator of the sphere (line K "L "). Now you can find the projection A "on the horizontal projection of the equator and through the points S" and A "draw a line - the horizontal projection of the desired generatrix. Point B is also determined on this line, the horizontal projection of which (point B") is the point of contact of the line with the ellipse.

With the construction of outlines of the projections of the cone of revolution, we meet, for example, in this case: given the projections of the top of the cone (S", S "), the direction of its axis (SK), the dimensions of the height and diameter of the base; construct projections of the cone. On fig. 355 this is done using additional projection planes.

So, to build a frontal projection, sq. π 3 perpendicular to π 2 and parallel to the straight line SK, which determines the direction of the cone axis. On the projection S""K"" the segment S""C"" is plotted, equal to the given height of the cone. At point C"" a perpendicular to S""C"" is drawn, and a segment C""B"" is drawn on it, equal to the radius of the base of the cone. By points C"" and B"" the points C" and B" are obtained, and thus the minor semi-axis C"B" of the ellipse-frontal projection of the base of the cone is obtained. The segment C"A" equal to C""B"" represents the semi-major axis of this ellipse. Having the axes of the ellipse, it is possible to construct it as shown in Fig. 147.

To construct a horizontal projection, the projection plane π 4 is introduced, which is perpendicular to π 1 and parallel to SK. The construction progress is similar to that described for the frontal projection.

How to build projection sketches? On fig. 356 shows a different one than in fig. 354, the method of drawing a tangent to an ellipse - without a sphere inscribed in a cone.

First, with a radius equal to the minor semiaxis of the ellipse, an arc was drawn from its center (in Fig. 356 this is a quarter of a circle). The point 2 of the intersection of this arc with a circle of diameter S"C" is determined. A straight line is drawn from point 2 parallel to the major axis of the ellipse; this

the line intersects the ellipse at points K "1 and K 2. Now it remains to draw straight lines S "K" 1 and S "K" 2, they are tangent to the ellipse and are included in the outline of the frontal projection of the cone.

On fig. 357 shows a body of revolution with an inclined axis parallel to the square. π 2. This body is bounded by a combined surface consisting of two cylinders, the surface of a circular ring and two planes. The sketch of the frontal projection of this body is its main meridian.

The outline of the horizontal projection of the upper cylindrical part of a given body is made up of an ellipse and two straight lines tangent to it. The straight line A"B" is a horizontal projection of the generatrix of the cylinder, along which the plane projecting onto π 1 touches the surface of the cylinder. The same applies to the sketch of the projection of the lower cylinder (in Fig. 357 this sketch is not shown in full).

Let's move on to the more difficult part of the essay - the intermediate one. We must construct a horizontal projection of that spatial curve line, at the points of which the projecting lines pass, tangent to the surface of the circular ring and perpendicular to the square. pi 1 . The frontal projection of each point of such a curve is constructed in the same way as was done for point A "in Fig. 354, using inscribed spheres. The horizontal projections of the points are determined on the projection of the equator of the corresponding sphere. For example, point D 1 (D" 1 , D" 1).

Points K "1 and K" 2 are obtained from the point K "1 (aka K" 2) on the equator of the sphere with center O, and this point K "1 (K" 2) is obtained by drawing a communication line tangent to the constructed curve B "D" 1 C".

So, the curve B"D" 1 K" 1 contains frontal projections of points whose horizontal projections B", D" 1 , K" 1 are included in the outline of the horizontal projection of the body under consideration.

Questions to §§ 53-54

1. What is called a plane tangent to a curved surface at a given point on this surface?
2. What is called an ordinary (or regular) point on a surface?
3. How to construct a plane tangent to a curved surface at some point on it?
4. What is the surface normal?
5. How to construct a plane tangent to the sphere at some point on the sphere?
6. When is a curved surface classified as convex?
7. Can a plane tangent to a curved surface at any point on that surface intersect the latter? Give an example of an intersection along two lines.
8. How are spheres inscribed in a surface of revolution, the axis of which is parallel to the square, used? π 2 , to construct an outline of the projection of this surface on the square. π 1 with respect to which the axis of the surface of revolution is inclined at an acute angle?
9. How to draw a tangent to an ellipse from a point lying on the continuation of its minor axis?
10. In what case will the outlines of the projections of the cylinder of revolution and the cone of revolution be exactly the same on the square. π 1 , and pl. p2?

Ministry of Education of the Russian Federation

Saratov State Technical University

SURFACES

Guidelines for completing task 2

for students of specialties
1706, 1705, 1201, 2503, 2506

Approved

editorial and publishing council

Saratov State

technical university

Saratov 2003

INTRODUCTION

In the practice of mechanical engineering, parts with cylindrical, conical, spherical, toric and helical surfaces are widespread. Technical forms of products are often a combination of surfaces of revolution with coinciding, intersecting and crossing axes. When making drawings of such products, it becomes necessary to depict lines of intersection of surfaces, also called transition lines.

A common way to construct intersection lines is to find the points of this line using some auxiliary cutting planes or surfaces, sometimes called "mediators".

In these guidelines, general and particular cases of constructing lines of intersection of two surfaces and methods for constructing sweeps of surfaces are considered.

1. MAIN PROVISIONS.

In descriptive geometry, a surface is considered as a set of successive positions of a line moving in space, called a generatrix.

If one of the surface lines is taken as a guide q and move along it according to a certain law the generatrix l, we obtain a family of surface generators that define the surface (Fig. 1).

To define a surface on a drawing, the concept of a surface determinant has been introduced.

A determinant is a set of conditions necessary and sufficient to uniquely define a surface.

The determinant consists of a geometric part containing geometric figures and the law of surface formation. For example, the geometric part of the shape determinant a(l,q) in Fig. 1 are generatrix l and guide q, the position of which is given in the drawing. Law of education: direct l, moving in space, always touches q while remaining parallel to the direction S. These conditions uniquely define a cylindrical surface. For any point in space, it is possible to solve the problem of belonging to its surface (AÎ a, inÏ a).

Geometric part of the determinant of a conical surface b(q,S) consists of guide q and peaks S(Fig. 2). The law of formation of a conical surface: generatrix straight line l q, always passes through the top S, forming a continuous set of straight lines of the conical surface.

Surfaces obtained by continuous movement are called kinematic. Such surfaces are accurate, regular, as opposed to irregular or random.

Surfaces formed by the movement of a straight line are called ruled, a curved line is called non-linear.

According to the law of motion of the generatrix, surfaces with translational movement of the generatrix are distinguished, with rotational motion of the generatrix - surfaces of revolution, with helical motion of the generatrix - helical surfaces.

Surfaces can be defined by a wireframe. A wireframe is a surface that is defined by a certain number of lines belonging to such a surface (Fig. 3).

Knowing the coordinates of the points of intersection of the lines, it is possible to draw a drawing of the wireframe surface.

1.2. Surfaces of revolution.

Among curved surfaces, surfaces of revolution are widespread. A surface of revolution is a surface obtained by rotating a generatrix around a fixed straight line - the axis of the surface.

A surface of revolution can be formed by rotating a curved line (sphere, torus, paraboloid, ellipsoid, hyperboloid, etc.) and by rotating a straight line (cylinder of revolution, cone of revolution, one-sheeted hyperboloid of revolution).

From the definition of a surface of revolution it follows that the geometric part of the determinant a(i,l) surfaces of revolution a must consist of an axis of rotation i and generating l. Surface formation law, rotation l around I allows you to build a continuous set of successive positions of the generatrix of the surface of revolution.

Of the many lines that can be drawn on surfaces of revolution, parallels (equator) and meridians (principal meridian) occupy a special position. The use of these lines greatly simplifies the solution of positional problems. Let's take a look at these lines.

Each point of the generatrix l(Fig. 4) describes around the axis i a circle lying in a plane perpendicular to the axis of rotation. This circle can be represented as a line of intersection of the surface by some plane (b) perpendicular to the axis of the surface of revolution. Such circles are called parallels. (R). The largest of the parallels is called the equator, the smallest - the throat.

Rice. 5 Fig. 6

On fig. 5 parallel RA points A equator, parallel RV points R surface throat.

If the axis of the surface i is perpendicular to the plane of projections, then the parallel is projected onto this plane by a circle in the true value (P1A), and on the projection plane parallel to the axis - a straight line (P2A) equal to the diameter of the parallel. In this case, the solution of positional problems is simplified. Linking any point of the surface (for example WITH) with a parallel, you can easily find the position of the projections of the parallel and a point on it. On fig. 5 by projection C2 points WITH belonging to the surface a, with the help of parallel Rs found horizontal projection C1.

The plane passing through the axis of rotation is called meridional. On fig. 4 is flat g. The line of intersection of the surface of revolution by the meridional plane is called the meridian of the surface. A meridian lying in a plane parallel to the plane of projections is called the main meridian ( m0 in fig. 4.5). In this position, the meridian is projected onto the plane P2 without distortion, but P1- straight parallel axis X12. For a cylinder and a cone, the meridians are straight lines.

Equator R2(Fig. 6) and the main meridians (m) delimit the surface into visible and invisible parts.

On fig. 6 surface equator a obtained as a result of a section of the surface by a plane d(P=a∩d), and the main meridian is a plane g(m=a∩g).

1.3. Surface outline.

The projecting surface, enclosing the given one, intersects the projection plane along a line called the outline of the surface projection. In other words, a surface outline is a line that delimits the projection of a figure from the rest of the drawing space. To construct an essay, it is necessary to construct extreme boundary sketch generators. The outline generators lie in a plane parallel to the plane of projections.

Any meridian of the surface of revolution can be taken as its generatrix. The construction of the essay will be simplified if we take the main meridian as a generatrix, since the main meridian is a flat curve (straight line) parallel to the projection plane and projected onto it without distortion.

Example 1. Cylinder a a(i,l). Build an outline of the surface (Fig. 7).

With this arrangement of the axis i horizontal outline is a circle of radius R(R=i1l1). Pass through the axis i meridian plane b||P2. To build a frontal outline, we find the horizontal projections of the outlines of the generators that lie in the plane of the main meridian (l1',l1”) and determine the frontal projections from them l2' And l2”.

Frontal projection of the main meridian of the cylinder outline generators l2' And l2”. The rectangle is the frontal outline of the surface.

Example 2. Cone a given by the geometric part of the determinant a(i,l). Build an outline of the surface (Fig. 8).

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Out of position geometric shapes l, i in fig. 9 shows that the given surface is a one-sheeted hyperboloid of revolution. Each point of the generatrix (A, B, C etc. ) when rotating around an axis i describes a circle (parallel). At i ^ P1 to the plane P1 parallels are projected by circles with a radius equal to the true value of the radius of the parallel. Dot WITH on the generatrix l describes the smallest parallel, the parallel of the throat. This is the shortest distance between the axis of rotation and the generatrix l. For finding Rc draw a perpendicular from i To l1. i1C1=Rc is the throat radius of the surface.

The horizontal projection of the hyperboloid will be three concentric circles.

The frontal outline of the surface should have the outline of its main meridian.

Pass through the axis i main meridional plane b and construct the horizontal projections of the parallels of the points A, B, C. Parallels intersect with a plane b at points А′, В′, С′ belonging to the main meridian of the surface. A continuous set of these parallels form the skeleton of the surface, and the points of intersection with the plane b- prime meridian m0 surfaces. The main meridian can be constructed as a bypass of the intersection points of the parallels with the plane b. The figure shows the construction of a point WITH And D.

Example 4. Construct an outline of an inclined cylinder a(l,m). Cylinder generatrix l, moving along the guide m, remains parallel to itself. The outline of the surface is built in Fig. 10. Any point on the surface of the cylinder is determined by drawing a generatrix through it (“connecting” the point with the generatrix). On fig. 10a according to the frontal projection of the point A2 belonging to the surface, its horizontal projection is found A1.

1.4. Ruled surfaces, with a plane of parallelism.

Ruled surfaces with a plane of parallelism are formed by moving a rectilinear generatrix along two guides. In this case, the generatrix in all its positions retains the parallelism of some given plane, called the plane of parallelism.

Geometric part of the determinant a(m,n,b) such a surface a contains two guides and a plane of parallelism. Depending on the shape of the guides, these surfaces are divided into: cylindroids - both guide curves; conoids - one guide - a straight line, one - a curve; oblique plane - both guides are straight.

Example: build a surface wireframe a(m,n,b)(Fig. 10b).

In this case, the horizontal plane of projections is taken as the plane of parallelism. Generating line, cutting off a curve m and direct n, in any position remains parallel to the plane P1.

Any plane parallel to the plane of parallelism intersects these surfaces in a straight line. Hence, if it is required to construct any generatrix of the surface, it is necessary to cut the surface with a plane (for example b) parallel to the plane of parallelism, find the points of intersection of the guide lines of the surface with this plane (b∩n=1;b∩m=2; rice. 10b) and draw a straight line through these points.

To construct the conoid in Fig. 10b, you can do without auxiliary cutting planes, since the frontal projections of the generators must be parallel to the axis X12. The density of the frame lines on the frontal projection is set arbitrarily. We build horizontal projections of the given generators along the connection line using the membership property.

If you need to find the projection of a point A, given by the projection A2, it is necessary to cut the surface with a plane g passing through the point A and parallel to the plane of parallelism (in Fig. 10b g//P1), find the generatrix as the line of intersection of the plane g with surface a(a∩g=3, 4), according to the frontal projection 32, 42 find the horizontal 31, 41 and determine on it A1.

1.5. Construction of the meeting point of the line with the surface.

Find the meeting point of the curve l with surface a(P,S).

Solution 1. Enclose the curve l(Fig. 11) into the auxiliary projection surface b^P1. Projection b1 coincides with the projection l1. 2. We build the line of intersection A surfaces α with surface b′, (αÇ b=e). Horizontal projection of this line a1 known, it matches b1. Plan view a1 building a frontal projection a2(Fig. 1 We determine the desired point to the intersection of the curve l with surface a.. K=lÇ a there is a meeting point l And a. On the one side l And A belong b And lÇ a=k. With another AÌ a, hence ToÌ α , that is To there are meeting points l with surface α .

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1.6. Construction of a line of intersection of surfaces.

When solving the problem of constructing a line of intersection of one surface with another, the method of sections is used - the main method for solving positional problems. In this case, the given surfaces are cut by auxiliary planes or curved surfaces (for example, spheres).

Auxiliary cutting surfaces are sometimes called "mediators".

1.5.1. General case.

In the general case, to solve the problem of determining the line of intersection of two surfaces, you can specify a family of generators on one of the surfaces (Fig. 12), find the meeting point of these generators with the second surface using the algorithm for solving the problem in Fig. 11, and then outline the meeting points.

Using this method to construct lines of intersection of two curved surfaces, we can use auxiliary planes or curved surfaces as secant "intermediaries".

If possible, you should choose such auxiliary surfaces that, in intersection with the given ones, give simple lines for constructing lines (straight lines or circles).

1.5.2. The axes of the surfaces of revolution coincide
(coaxial surfaces).

On fig. 13 surfaces a And b given by a common axis i and main meridians m0m0'.

The main meridians intersect at a point A(B). Dot A(B) intersections of meridians during rotation around the axis will describe a parallel R, which will belong to both surfaces, therefore, will be their intersection line.

Thus, two coaxial surfaces of revolution intersect along parallels that describe the points of intersection of their meridians. On fig. 13 axes of surfaces are parallel P2. On the plane of projections to which the axes of the surfaces are parallel, the line of intersection R2 a straight line is projected, the position of which is determined by the intersection points of the main meridians A And IN.

1.5.3. Cutting plane method.

In the case when the axes of the surfaces of revolution are parallel, the simplest constructions are obtained by using cutting planes as mediators. In this case, the auxiliary cutting planes are chosen so that they intersect both surfaces in circles.

On fig. 14 outlines the projections of two surfaces of revolution α And b, their axes i And j are parallel. In this case, the use of cutting planes perpendicular to the axes of the surfaces gives a simple solution to the problem. The resulting intersection lines of the surfaces will be parallels, the frontal projections of which are straight lines equal to the diameter of the parallel, and the horizontal projections are life-size circles.

When constructing points of intersection lines, you must first find the reference and characteristic points. Reference points are those that lie on the main meridian (3) and the equator (4, 5). Finding these points is not associated with additional constructions and is based on the use of membership properties.

Given in Fig. 14 surfaces have a common plane of the main meridian, their axes ^ P1, the bases lie in the plane P1. The reference points of the line of intersection are point 3 of the intersection of the main meridians and points 4 and 5 of the intersection of the parallels of the bases of the surfaces. Using the membership properties, by the known projections 32, 41 and 51 we find 31, 42 and 52.

The remaining intersection points are found using auxiliary cutting planes. Let's dissect the surface α And b horizontal plane g. Because g^ axes i And j, then the surfaces α And b intersect the plane g, in parallel Ra And Rb. And since the axes i And j^P1, then these parallels are projected onto P1 circles Ra, Rb in true size, but P2 direct P2a, R2b equal to the diameter of the parallel.

The intersection points of parallels 1 and 2 are the desired ones. Indeed, on one side of the parallel Ra And Rb belong to the same plane g and intersect at points 2 and 1. On the other hand, Ra And Rb belong different surfaces α And b. Therefore, points 2 and 1 simultaneously belong to the surfaces A And b, that is, they are the points of the line of intersection of the surfaces. Horizontal projections 21 and 11 of these points are at the intersection P1a, P1b, and we construct the front ones using the membership property.

Repeating the indicated method, we obtain the required number of points. The secant planes are distributed evenly in the interval from the point of the highest rise of the curve 32 to the main figure.

The number of points of the intersection line, and hence the cutting planes, is determined by the required accuracy of graphic constructions. The projections of the intersection line are built as the contours of the projections of its points. On fig. 14 line at points 4, 1, 3, 2, 5.

The considered example of solving problems is called the method of cutting planes.

1.5.4. Sphere method.

This technique is used when the axes of the surfaces of revolution intersect. It is based on the one shown in Fig. 13 case of intersection of coaxial surfaces.

On fig. 15 shows a cone and a cylinder with intersecting axes i And j. Their axes are parallel to the plane P2. The plane of the main meridian is common for both surfaces.

) . The construction is simplified due to the fact that the plane of the main meridian is common. The circles along which the sphere intersects two surfaces simultaneously ( Ra, Rb Rb") is projected onto the plane P2 in the form of straight lines ( R2a, R2b, P2b") equal to the diameters of the parallels.

At the intersection of these circles, points (5, 6, 7, 8), (52, 62, 72, 82) are obtained, common to both surfaces and, therefore, belonging to the line of intersection. Really parallels Ra, Rb, Pb", on the one hand, belong to the same surface - the sphere and have common points (5, 6, 7, 8), on the other - belong to different surfaces A And b. That is, points 5, 6, 7, 8 belong to both surfaces or the line of intersection of the surfaces.

To get enough points to draw the desired intersection line, several spheres are drawn.

The radius of the largest sphere ( Rmax) is equal to the distance from the center O2 to the most distant point of intersection of the outline generatrix (in this case, points 32 and 42, Rmax= 0232=0242. In this case, both lines of intersection of surfaces with a sphere ( Ra And Rb) intersect each other at points 3 and 4 with a larger radius of the sphere there will be no intersection.

The radius of the smallest sphere ( Rmin) is equal to the distance from the center 02 to the most distant sketch generatrix ( Rmin=02A2). In this case, the sphere will touch the cone along the circle, and the cylinder will cross twice and give points 5, 6, 7, 8. With a smaller radius of the sphere, there will be no intersection with the cone.

Now it remains to draw curved lines of intersection of surfaces through points 1, 5, 4, 6, 1 and 2, 7, 3, 8, 2.

On fig. 15, all constructions are made on the same projection. Number of secant spheres, with radii ranging from Rmax before Rmin, depends on the required construction accuracy. The construction of a horizontal projection of the intersection line is performed along the frontal 1, 5, 4, 6, 1 and 2, 7, 3, 8, 2 using the membership property.

1.5.5. Applying the cutting plane method
in cases of ruled surfaces with a plane of parallelism.

Two surfaces are given by the geometric part of the determinant: a(l,i) And b(m,n, P1). It is necessary to build sketches of surfaces and find the line of their intersection (Fig. 16).

Solution: 1. We build an outline of the surface a, n of the geometric part of the determinant, it can be seen that the surface a- sphere. Its horizontal and frontal outlines are circles of radius R. 2. We build the frame of the ruled surface. Since the plane is parallel P1, then the frontal projections of the generators are parallel to the axis X12. Having set the frame of a certain plane of lines on the frontal projection (four lines in Fig. 16), we build horizontal projections of these generators. 3. To construct a line of intersection of surfaces, we use secant planes as intermediaries. The position of the secant planes must be chosen such that they intersect the given surfaces along lines that are easy to construct (straight lines or circles). This condition is satisfied by horizontal planes. The horizontal planes are parallel to the plane of parallelism of the conoid ( P1), so they will intersect the conoid in straight lines. Such planes intersect the sphere along parallels.

,A" sphere along a parallel Ra. Frontal projection of the parallel ( R2a) is a straight line equal to the diameter of the parallel, and the horizontal projection ( P1a) is a circle. On a horizontal projection at the intersection of the parallel P1a and generatrix 1, 11 "is determined by the projection of two points of the line of intersection of the surface A And b. By horizontal projections of points A1 And IN 1 we build their frontal projections. By repeating the operation, we get a series of points of the intersection line, the outline of which will give the line of intersection.

The equator and prime meridian of the sphere delimits the line into visible and non-visible parts.

1.6. Construction of sweeps.

A developed surface is a figure obtained by combining the developed surface with a plane.

Developable surfaces are surfaces that are aligned with the plane without breaks or folds.

Developable surfaces include faceted surfaces, and curvilinear surfaces include only cylindrical, conical and torso surfaces.

Developments are divided into exact (development of faceted surfaces), approximate (development of a cylinder, cone, torso) and conditional (development of a sphere and other non-developable surfaces).

1.6.1. Reamers of faceted surfaces.

Unfold the pyramid given by the projections in Fig.17.

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The rolling method is applicable if the edges of the prism are parallel to the plane of projections and the true value of the edges of one of the bases is known (Fig. 18).

Rolling out a figure represents the process of combining the faces of a prism with a plane, in which true view each face is obtained by rotating around its edge.

Points A, B, C during rolling move along arcs of circles, which are depicted on the P2 plane as straight lines perpendicular to the projections of the edges of the prism. The sweep vertices are built as follows: from point A2 with radius R1=A1B1 (true length AB) we make a notch on the line B2B0 perpendicular to B2B2¢. From the constructed point B0 with radius R2=B1C1 a notch is made on the straight line C2C0^C2C2¢. Then a notch from point C0 with radius R3=A1C1 on the straight line A2A0^A2A2¢. We get point A0. Points A2B0C0A0 are connected by straight lines. From the points A0B0C0 we draw lines parallel to the edges (A2 A2¢), put on them the true values ​​of the side edges А2A¢, B2B¢, C2C¢. We connect points A¢B¢C¢A¢ with line segments.

1.6.2. Development of curved surfaces.

Theoretically, it is possible to obtain an exact development, i.e., a development that exactly repeats the dimensions of the surface being developed. In practice, when making drawings, one has to put up with an approximate solution of the problem, assuming that individual elements of the surface are approximated by plane sections. Under such conditions, the implementation of approximate developments of a cylinder and a cone is reduced to the construction of developments of prisms and pyramids inscribed in them (or described).

Figure 19 shows an example of a cone sweep.

We inscribe a multifaceted pyramid into the cone. From point S we draw an arc with a radius equal to the true value of the generatrix of the cone (S212) and put aside chords 1121 on the arc; 2, replacing arcs 1121;2

To find any point on the development, it is necessary to draw a generatrix through a given point (A), find the place of this generatrix on the development (2B=21B1), determine the true value of the SA or AB segment and put it on the generatrix on the development. Any line on the surface consists of a continuous set of points. Having found the required number of points on the development using the method described for point A and by tracing these points, we will get a line on the development. When constructing developments of inclined cylindrical surfaces, the methods of normal section and rolling are applicable.

Any non-developable surface can also be approximated by a polyhedral surface with any given accuracy. But the development of such a surface will not be a continuous flat figure, since these surfaces do not develop without breaks and folds.

1.6.3. Construction of a plane, tangent
to the surface at that point.

To construct a tangent plane to the surface at a given point (point A in Fig. 20), it is necessary to draw two arbitrary curves a and b on the surface through point A, then at point A construct two tangents t and t¢ to curves a and b. The tangents will determine the position of the tangent plane a to the surface b.

Figure 21 shows a surface of revolution a. It is required to draw a tangent plane at point A belonging to a.

To solve the problem through point A, we draw a parallel a and build a tangent t to it at point A (t1;t2).

Let's take the meridian as the second curve passing through point A. It is not shown in Fig. 21. The solution will be simplified if the meridian together with point A is rotated around the axis until it coincides with the main meridian. In this case, point A will take position A¢. Then draw a tangent t¢¢ to the main meridian through point A¢ until it intersects with the axis at point B. Returning the meridian to its previous position, draw a tangent t¢ to this meridian through point A and a fixed point B on the axis of rotation (t1¢;t2 ¢). The tangents t and t¢ will define the tangent plane.

When drawing a tangent plane to a ruled surface, one of the tangents that define the tangent plane can be taken as the generatrix t of the surface (Fig. 22). As the second, one can take the tangent t¢ to the parallel (if it is a cylinder or a cone) or the tangent to any curve drawn through a given point of a conoid, cylindroid, oblique plane. It is easy to construct a curve by cutting the surface with a projecting plane passing through a given point.

2.1. Goal of the work:

Consolidate the program material in the sections "Surface" and "Developments" and gain skills in solving the problems of constructing essays, intersection lines and developments of surfaces.

2.2. Exercise:

The drawing contains two intersecting surfaces. Surfaces are given by coordinated projections of the geometric part of the determinant.

Necessary:

Using the coordinates of the geometric part of the determinant, apply the projections of the determinant on the drawing, connect the necessary points to obtain the geometric shapes of the determinant;

Build Essays given surfaces by projections of the geometric part of the determinant;

Construct a line of intersection of surfaces;

Build a development of one of the surfaces with drawing a line of intersection (as directed by the teacher);

Draw a tangent plane to one of the surfaces at the point indicated by the teacher;

Make a layout of intersecting surfaces.

The work is done first on A2 graph paper, then on Whatman paper in A2 format. The drawing must be drawn up in accordance with GOST ESKD. The main inscription is made according to the form 1.

When performing the work, lectures, practical training materials and recommended literature are used.

Task options are given in the appendix.

2.3. The order of the task.

The student receives a version of the assignment corresponding to the number on the list in the group journal, and works on the assignment for four weeks.

A week after receiving the assignment, the student presents to the teacher the constructions of the geometric part of the determinants and sketches of the given surfaces, made on A2 graph paper.

Two weeks later, a drawing is presented, supplemented by the construction of the line of intersection of the surfaces and the tangent plane.

During the third week, work on A4 graph paper ends with the construction of a development of one of the surfaces with drawing on it the line of intersection of the surfaces.

During the fourth week, a layout of intersecting surfaces is performed.

The completed work is presented to the teacher leading the practical lesson. According to the completed construction on graph paper, the assimilation of the studied material by the student is checked.

When solving the positional problem of constructing a line of intersection of surfaces, the section method is used. As "intermediaries" choose secant planes or spheres. Attention should be paid to the particular cases considered above (the method of cutting planes and the method of spheres), which give the simplest solution to the problem. If necessary, resort to a combination of these methods.

When performing a surface sweep, it is necessary to study the constructions performed by the normal section method and the rolling method, as well as the methods for constructing approximate and conditional sweeps and use the most rational way in the work.

When drawing a tangent plane to a surface at a given point, it is sufficient to draw two curved lines on the surface passing through a point, and draw tangents to these lines at a given point, remembering that a tangent to a flat curved line is projected by a tangent to its projection.

LITERATURE.

1. Vinitsky geometry. Moscow: Higher school, 1975.

2. Gordon geometry. Moscow: Nauka, 1975.

3. Surfaces. Methodical instructions. / Compiled, / Saratov, SGTU, 1990.

ASSIGNMENT OPTIONS

option	Designation of points	Point coordinates	verbal information
					1. Hyperbolic paraboloid Guide lines - AB and CD Parallelism plane - P2 2. Front projecting cylinder: Axis of rotation - I I¢ Generating - MN
					Top - S Base - AB 2. Truncated cone: Bottom base - CF 3. Upper base - DE
					Axis of rotation t ^ P1 Generating - CD 2. Hyperboloid: Axis of rotation i ^ P1 Generator - AB
					1. Surface of rotation: Rotation axis-KK¢ Generating - frontal arc (O - center of rotation OA - radius) 2. Cylinder: Rotation axis-MM¢ Generator - LL¢

					1. Cylinder: Axis of rotation - I I¢ Generator - EF 2. Pyramid: Tops of the pyramid - A, B, C, D
					1. Hyperbolic paraboloid Guide rails AB, CD Plane of parallelism. – P2 2. Hemisphere: Center - O Radius - OK
	A 1.5.6				1. Part of the sphere (from R to R¢) Center - O Radius - OR = OR¢ 2. Conoid: directing straight line - OA, BC-directing curve of projection of which: on P2- straight line, on P1-arc (center - O, radius - OB). P1-plane parallelism.
					1. Pyramid: Vertices - S, A, B, C. 2. Conoid: Guide straight - EF Guide curve - RR¢, projections of which: on P2-arc (O¢-center, O¢R =O¢R¢-radius), on the P1-arc (O - center, OR \u003d OR¢- radius), P1-plane of parallelism.
	A 1.5.7				1. Cylinder: Generating - CD 2. Conoid: Leading straight line - AB Guide circle belongs to the plane P1. O - center, OE - radius, P2 - plane of parallelism.
					1. Torus surface: Generating circle belongs to sq. P1. O - center, OS - radius. 2. Ruled surface: Generator - MM¢ Guide bow-KDM (O¢-center, O¢D-radius)
					1. Hyperboloid: Axis of rotation - I I¢ Generating - AB 2. Cylinder: Generating - NM Guide circle frontal (O-center, ON - radius).
	A 1.5.8 B 1.5.9				1. Cylinder: Generating - CD Axis of rotation t ^ P1 2. Hyperboloid: Axis of rotation i ^ P1 Generating - AB
	A 1.5.10				1. Cylinder: Axis of rotation - I I¢ Generating - AB Axis of rotation - TT¢ Generating circle belongs to the plane P1 (O - center, OS - radius)
	About 1.5.11				1. Hemisphere: (O-center, OK-radius) 2. Conoid: Guide straight - LM Guide circle belongs to sq. P1 (O - center, OK - radius) P2 - plane of parallelism

					1. Prism: ВВ¢ - edges. Rotation axis - I I¢ Generating arc of a circle (Center - O2,
					1. Hyperboloid: Rotation axis - I I¢ Generating - AB Rotation axis - OS Base Radius - OS
					1. Hyperbolic paraboloid Guides - AB and CD P1 - plane of parallelism Axis of rotation - SI Generating - SE
					1. Conoid: Guide straight - AB Guide circle belongs to sq. P1 Center - O, radius - OS P2 - plane of parallelism 2. Hemisphere: Center - O, radius - OS
					1. Cylinder: Guide circle belongs to sq. P2 (Center - O, radius - OA), Forming - OA Axis of rotation - CD Generating - CB
					1. Prism: ВВ¢- ribs Axis of rotation - EF Generating - ED
					1. Conoid: Guide straight - AB guide arc, belonging to P1- MN Center - O. Radius - OM P2 - plane of parallelism 2. Half cylinder: Generating - CD
					1. Conoid: Guide straight - AB guide arc, owned by P1-CD (center - O, radius - OS) E2F2 - plane traces concurrency 2. Cylinder: Rotation axis - I I¢ Generating - MN
					(Center - O, radius - OR) Axis of rotation - VK Generating - AB
					OS - axis of rotation, AS - generatrix Axis of rotation - CD Generating - NE
					1. Hemisphere: Radius - OS 2. Hyperboloid: Rotation axis - I I¢ Generating - AB