What is outline and contour of a surface. Specifies a surface in a complex drawing. Helical ruled surfaces

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

The desired outline is composed of a part of an ellipse and two lines tangent to it. In fact, the cone in its given position is projected onto the square. π 1 using the surface of an elliptical cylinder, whose generatrices 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 two of its axes: small D "E" and large, equal in size to D "E" (the diameter of the circumference of the base of the cone). Straight lines S "B" and S "F" are obtained if we draw tangent lines to the ellipse from point S ". Since the plane projecting on 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 sought 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 when the frontal projections intersect: 1) the circle of tangency of the cone and the sphere (straight line M "N") and 2) the equator of the sphere (straight 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 straight line - the horizontal projection of the desired generatrix. On this line, point B is also determined, the horizontal projection of which (point B") is the tangent point of the straight line with the ellipse.

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

So, to build a frontal projection, a square was introduced. π 3 perpendicular to π 2 and parallel to the straight line SK defining the direction of the cone axis. On the projection S "" K "" the segment S "" C "" is plotted equal to the specified height of the cone. At point C "" a perpendicular is drawn to S "" C "", and a segment C "" B "" is plotted on it, equal to the radius of the base of the cone. The points C "" and B "" are used to obtain points C "and B" and thus the semi-minor axis C "B" of the ellipse-frontal projection of the base of the cone is obtained. Segment C "A" equal to C "" B "" represents the major half-axis of this ellipse. Having the axes of the ellipse, you can build it as shown in Fig. 147.

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

How do you build projection sketches? In fig. 356 is shown different from Fig. 354, the method of drawing a tangent to an ellipse is without a sphere inscribed in a cone.

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

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

In fig. 357 shows a body of revolution with an inclined axis parallel to pl. π 2 This body is bounded by a combined surface consisting of two cylinders, the surface of a circular ring and two planes. An outline 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 composed of an ellipse and two lines tangent to it. 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 outline of the projection of the lower cylinder (in Fig. 357 this outline is not fully depicted).

We pass on to the more complex part of the essay - the intermediate one. We must construct a horizontal projection of that spatial curved line, at the points of which there are projection lines tangent to the surface of the circular ring and perpendicular to pl. π 1. The frontal projection of each point of such a curve is constructed in the same way as it was done for point A "in Fig. 354, - using inscribed spheres. The horizontal projections of points are determined on the projection of the equator of the corresponding sphere. This is how, for example, point D 1 (D" 1, D "1).

Points K "1 and K" 2 are obtained by point K "1 (aka K" 2) at 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, the horizontal projections of which B", D "1, K" 1 are included in the outline of the horizontal projection of the body in question.

Questions to §§ 53-54

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

Surface concept

SURFACES

In descriptive geometry, surfaces are considered as a set of successive positions of a certain line moving in space according to a certain law. This method of surface formation is called kinematic.

A line (curve or straight) moves in space according to a certain law and creates a surface. It is called a generatrix. During the formation of the surface, it can remain unchanged or change its shape. The law of displacement of the generatrix is ​​set in the form of a set of lines and indications of the nature of the displacement of the generatrix. These lines are called guidelines.

In addition to the kinematic method, the surface can be specified

· Analytically, that is, it is described by a mathematical expression;

· Wireframe method, which is used when defining complex surfaces; a surface wireframe is an ordered set of points or lines that belong to a surface.

To define a surface in a complex drawing, it is enough to have such surface elements on it that allow you to construct each of its points. The collection of these elements is called a surface determinant.

The surface identifier consists of two parts:

· Geometric part, including constant geometric elements (points, lines), which are involved in the formation of the surface;

· The algorithmic part, which sets the law of motion of the generator, the nature of the change in its shape.

In symbolic form, the determinant of the surface F can be written in the form: F (Г) [A], where Г is the geometric part of the determinant, A is the algorithmic part.

To distinguish a determinant near the surface, one should proceed from the kinematic method of its formation. But since many identical surfaces can be obtained in different ways, they will have different determinants. Below we will consider the most common surfaces in accordance with the classification criteria, pleasant in the course of descriptive geometry.

To define a surface in a complex drawing, it is enough to indicate the projections of not the entire set of points and lines belonging to the surface, but only geometric shapes included in its determinant. This way of defining the surface allows you to build projections of any of its points. Specifying a surface by projections of its determinant does not provide clarity, which makes it difficult to read the drawing. To increase clarity, if possible, sketch lines (sketches) of the surface are indicated in the drawing.

When any surface W is projected parallel to the projection plane S, then the projection lines tangent to the surface W , form a cylindrical surface (Figure 11.1). These projected straight lines touch the surface W at points forming some line m, which is called a contour line.

The projection of the contour line m onto the plane S - m / is called the outline of the surface. Surface outline separates the surface projection from the rest of the projection plane.

The surface contour line is used to determine the visibility of points relative to the projection plane. So, in fig. 11.1 the projections of points of the surface W located to the left of the contour m on the plane S will be visible. The projections of the rest of the surface points will be invisible.

Essays

When defining an object with curved edges for projection, in addition to defining a set of points, edges and faces of the projection object, it is necessary to define a set of outlines for its curved edges.

Curved surface sketches are lines on that curved surface that divide the surface into parts that are not visible and parts that are visible on the projection plane. In this case, we are talking about the projection of only the curved surface under consideration and does not take into account the possible shading of this surface by other foreground surfaces.

The parts into which the sketches are divided by a curved surface are called compartments.

The position of the sketches of curvilinear faces is determined by the projection parameters, therefore, the sketches should be determined after the transition to the species coordinate system is completed.

Determining the outline of a curved surface, in the general case, is a relatively difficult task. Therefore, as a rule, a given curved surface is approximated using one of the typical curved surfaces, which include:

Cylindrical surface;

Spherical surface;

Conical surface.

Consider finding sketches for these types of curved surfaces.

Finding outlines of a spherical surface illustrated in Fig. 6.6-7.

The figure uses the following designations:

О - the center of the sphere;

О п - projection of the center of the sphere;

GM is the main meridian of a given sphere;

Pl1 - plane passing through the center of the sphere, parallel to the projection plane;

X in, Y in, Z in - coordinate axes of the view coordinate system;

X p, Y p - coordinate axes on the projection plane.

To find the outline on the surface of the sphere, it is necessary to draw a plane (pl1 in Fig. 6.6‑7) through the center of the sphere, parallel to the projection plane. The line of intersection of this surface and the sphere, which has the shape of a circle, is called the main meridian (GM) of the spherical surface. This main meridian is the desired outline.

The projection of this outline will be a circle with the same radius. The center of this circle is the projection of the center of the original sphere onto the projection plane (O p in Fig. 6.7-1).

Rice.6.7 ‑ 1

For determining outline of a cylindrical surface, through the axis of the given cylinder o 1 o 2 (Fig. 6.7-2) plane Pl1 is drawn, perpendicular to the projection plane. Further, through the axis of the cylinder, plane Pl2 is drawn, perpendicular to the plane Pl1. Its intersections with the cylindrical surface form two straight lines o h 1 och 2 and o h 3 o h 4, which are outlines of the cylindrical surface. The projections of these sketches are straight lines o h 1p och 2p and o h 3p o h 4p, shown in Fig. 6.7-2.

Construction of essays conical surface illustrated in Fig. 6.7-3.

In the figure, the following designations are adopted:

O is the top of the cone;

OO 1 - axis of the cone;

X in, Y in, Z in - species coordinate system;

PP - projection plane;

X p, Y p, - coordinate system of the projection plane;

Лп - projection lines;

O 1 - the center of the sphere inscribed in the cone;

O 2 - a circle tangent to the inscribed sphere, having a center at point O 1, and the original conical surface;

O h 1, O h 1 - points lying on the outlines of the conical surface;

O h 1p, O h 1p are the points through which the lines corresponding to the projections of the outlines of the conical surface pass.

The conical surface has two outlines in the form of straight lines. Obviously, these lines pass through the vertices of the cone - point O. To define the outline unambiguously, therefore, it is necessary to find one point for each outline.

To build outlines of a conical surface, perform the following steps.

A sphere is inscribed into a given conical surface (for example, with a center at point O 1) and the tangent of this sphere with a conical surface is determined. In the case considered in the figure, the tangency line will have the shape of a circle with the center at point O 2 lying on the axis of the cone.

Obviously, of all points of a spherical surface, points belonging to outlines can only be points belonging to a tangent circle. On the other hand, these points must be located on the circumference of the main meridian of the inscribed sphere.

Therefore, the points of intersection of the circle of the main meridian of the inscribed sphere and the circle-tangent will be the required points. These points can be defined as the points of intersection of the tangent circle and the plane passing through the center of the inscribed sphere O 1, parallel to the projection plane. Such points in the figure are O h 1 and O h 2.

To construct the projections of sketches, it is enough to find the points O h 1p and O h 2p, which are the projections of the found points O h 1 and O h 2 on the projection plane, and, using these points and the point O n of the projection of the apex of the cone, construct two straight lines corresponding to the projections of the outlines of a given conical surface (see Fig. 6.7-3).

Rice. 3.15

Surfaces of revolution are widely used in all areas of technology. The surface of revolution is called the surface resulting from the rotation of some generating line 1 around a fixed line i- the axis of rotation of the surface (Figure 3.15). In the drawing, the surface of revolution is defined by its outline. Surface outline are the lines that delimit the area of ​​its projection. During rotation, each point of the generatrix describes a circle, the plane of which is perpendicular to the axis. Accordingly, the line of intersection of the surface of revolution with a plane perpendicular to the axis is a circle. Such circles are called parallels (Figure 3.15). The parallel of the largest radius is called the equator, the smallest - the throat. The plane passing through the axis of the surface of revolution is called meridian, the line of its intersection with the surface of revolution is called the meridian. A meridian lying in a plane parallel to the projection plane is called the main meridian. In the practice of drawing, the following surfaces of revolution are most often encountered: cylindrical, conical, spherical, torus.

Rice. 3.16

Cylindrical surface of revolution... As a guide a one should take a circle, and as a straight line b- axis i(Figure 3.16). Then we get that the generator l parallel to the axis i, revolves around the latter. If the axis of rotation is perpendicular to the horizontal plane of the projections, then on NS 1 cylindrical surface is projected into a circle, and on NS 3 - into a rectangle. The main meridian of the cylindrical surface is two parallel straight lines.

Fig 3.17

Conical surface of revolution we obtain by rotating the straight generatrix l around the axis i... In this case, the generator l crosses the axis i at the point S called the top of the cone (Figure 3.17). The main meridian of the conical surface is two intersecting straight lines. If we take a straight line segment as a generator, and the axis of the cone is perpendicular NS 1, then on NS 1 conical surface is projected into a circle, and on NS 2 - into a triangle.

Spherical surface is formed by rotating a circle around an axis passing through the center of the circle and lying in its plane (Figure 3.18). The equator and meridians of a spherical surface are equal circles. Therefore, with orthogonal projection onto any plane, a spherical surface is projected into circles.

Rice. 3.18 When a circle rotates around an axis lying in the plane of this circle, but not passing through its center, a surface is formed, called a torus (Figure 3.19).

Rice. 3.19

11. POSITIONAL PROBLEMS. ACCESSORIES OF A POINT, SURFACE LINE. MONGE'S THEOREM. Under positional means tasks, the solution of which allows you to get an answer about the belonging of an element (point) or a subset (line) to a set (surface). Positional also includes tasks to determine common elements belonging to various geometric shapes. The first group of tasks can be combined under the general title of a membership task. These, in particular, include tasks to determine: 1) the belonging of a point of the line; 2) the belonging of a point of the surface; 3) the belonging of the line of the surface. The second group includes problems of intersection. This group also contains three types of tasks: 1) for the intersection of a line with a line; 2) for the intersection of a surface with a surface; 3) for an intersection of a line with a surface. Surface point affiliation ... The main position in solving problems for this option of belonging is as follows : a point belongs to a surface if it belongs to any line of that surface... In this case, the lines should be chosen as the simplest ones to make it easier to build projections of such a line, then use the fact that the projections of a point lying on the surface should belong to the same projection of the line of this surface ... An example of the solution to this problem is shown in the figure.... There are two ways to solve it, since you can draw two simple lines belonging to a conical surface. In the first case, a straight line is drawn - the generator of the conical surface S1 so that it passes through any given projection of the point C. Thus, we assume that the point C belongs to the generator S1 of the conical surface, and therefore to the conical surface itself. In this case, the projections of the same name of point C should lie on the corresponding projections of this generatrix. Another simple line is a circle with a diameter of 1-2 (the radius of this circle is measured from the axis of the cone to the outline generatrix). This fact is also known from the school geometry course: when a circular cone intersects with a plane parallel to its base, or perpendicular to its axis, a circle will be obtained in the section. The second solution method allows you to find the missing projection of the point C, given by its frontal projection, belonging to the surface of the cone and coinciding in the drawing with the axis of rotation of the cone, without constructing a third projection. You should always keep in mind whether the point lying on the surface of the cone is visible or not visible (if it is not visible, the corresponding projection of the point will be enclosed in brackets). Obviously, in our problem, point C belongs to the surface, since the projections of the point belong to the projections of the same name, used to solve both the first and the second solution methods. Surface line affiliation. Basic position: a line belongs to a surface if all points of the line belong to a given surface... This means that in this case of belonging, the problem of belonging of a point to a surface must be solved several times. Monge's theorem: if two surfaces of the second order are described near the third or are inscribed in it, then the line of their intersection splits into two curves of the second order, the planes of which pass through a straight line connecting the points of intersection of the circle of tangency.

12.SECTIONS OF THE CONE OF ROTATION BY THE PROJECTING PLANES . When crossing surfaces bodies with projection planes, one section projection coincides with the projection of the projection plane. The cone can have five different shapes in cross section. Triangle- if the cutting plane intersects the cone through the vertex along two generatrices. Circle- if the plane intersects the cone parallel to the base (perpendicular to the axis). Ellipse- if the plane intersects all generators at a certain angle. Parabola- if the plane is parallel to one of the generatrices of the cone. Hyperbola- if the plane is parallel to the axis or two generatrices of the cone. Surface section by plane is a flat figure bounded by a closed line, all points of which belong to both the cutting plane and the surface. When a plane intersects a polyhedron in a section, a polygon is obtained with vertices located on the edges of the polyhedron. Example... Construct the projections of the line of intersection L of the surface of the right circular cone ω by the plane β. Solution... In the section, a parabola is obtained, the vertex of which is projected to point A (A ', A' '). Points A, D, E of the line of intersection are extreme. In fig. the construction of the sought line of intersection was carried out using the horizontal planes of the level αi, which intersect the surface of the cone ω along the parallels рi, and the plane β - along the segments of the frontally projecting straight lines. The intersection line L is fully visible on planes.

№13. Coaxial surfaces. Concentric spheres method.

When constructing a line of intersection of surfaces, the features of the intersection of coaxial surfaces of revolution allow using spheres coaxial with these surfaces as auxiliary intermediate surfaces. Coaxial surfaces of revolution include surfaces that have a common axis of revolution. In fig. 134 shows a coaxial cylinder and a sphere (Fig. 134, a), a coaxial cone and a sphere (Fig. 134, b) and a coaxial cylinder and a cone (Fig. 134, c)

Coaxial surfaces of revolution always intersect along circles whose planes are perpendicular to the axis of revolution. There are as many of these circles common to both surfaces as there are points of intersection of the outline lines of the surfaces. The surfaces in Fig. 134 intersect in circles created by points 1 and 2 of the intersection of their main meridians. The auxiliary sphere-mediator intersects each of the specified surfaces in a circle, at the intersection of which points are obtained that belong to the other surface, and therefore the intersection lines. If the axes of the surfaces intersect, then the auxiliary spheres are drawn from one center-point of intersection of the axes. In this case, the line of intersection of surfaces is constructed by the method of auxiliary concentric spheres. When constructing a line of intersection of surfaces to use the method of auxiliary concentric spheres, the following conditions must be met: 1) the intersection of surfaces of revolution; 2) the axes of the surfaces - intersecting straight lines - are parallel to one of the projection planes, that is, there is a common plane of symmetry; 3) the method cannot be used construction clipping planes, as they do not produce graphically simple lines on surfaces. Typically, the construction spheres method is used in conjunction with the construction clipping planes method. In fig. 135, a line of intersection of two conical surfaces of revolution with axes of revolution intersecting in the frontal plane of the level Ф (Ф1) is constructed. This means that the main meridians of these surfaces intersect and give in their intersection the points of visibility of the line of intersection relative to the plane P2 or the highest A and the lowest B points. At the intersection of the horizontal meridian h and parallel h ", lying in one auxiliary cutting plane Г (Г2), the points of visibility C and D of the intersection line relative to the plane P1 are determined. Ф, will intersect both surfaces along the hyperbolas, and the planes parallel to Г, will give in the intersection of the surfaces of the circle and the hyperbola.Auxiliary horizontal or frontal projection planes drawn through the vertex of one of the surfaces will intersect them along generators and ellipses. conditions allowing the use of auxiliary spheres for constructing points of the intersection line.The axes of the surfaces of revolution intersect at the point O (O1; O2), which is the center of the auxiliary spheres, the radius of the sphere changes within Rmin< R < Rmах- Радиус максимальной сферы определяется расстоянием от центра О наиболее удаленной точки В (Rmax = О2В2), а радиус минимальной сферы определяется как радиус сферы, касающейся одной поверхности (по окружности h2) и пересекающей другую (по окружности h3).Плоскости этих окружностей перпендикулярны осям вращения поверхностей. В пересечении этих окружностей получаем точки Е и F, принадлежащие линии пересечения поверхностей:

h22 ^ h32 = E2 (F2); Е2Е1 || A2A1; E2E1 ^ h21 = E1; F2F ^ h1 = F1 An intermediate sphere of radius R intersects the surfaces along the circles h4 and h5, at the intersection of which the points M and N are located: h42 ^ h52 = M2 (N2); M2M1 || A2A1, M2M1 ^ h41 = M1; N2N1 ^ h41 = N1 Connecting the projections of the same name of the constructed points, taking into account their visibility, we obtain the projections of the line of intersection of the surfaces.

No. 14. construction of a line of intersection of surfaces, if at least one of them is projecting. Keypoints of the line of intersection.

Before proceeding with the construction of the line of intersection of surfaces, it is necessary to carefully study the condition of the problem, i.e. which surfaces intersect. If one of the surfaces is projecting, then the solution of the problem is simplified, since on one of the projections, the intersection line coincides with the surface projection. And the task comes down to finding the second projection line. When solving the problem, it is necessary to note, first of all, "characteristic" points or "special" ones. It:

Points on extreme generatrices

Points dividing the line into visible and invisible parts

· Upper and lower points, etc. Next, you should wisely choose the method that we will use when constructing the line of intersection of surfaces. We will use two methods: 1. auxiliary cut planes. 2. auxiliary secant spheres. Projection surfaces include: 1) a cylinder if its axis is perpendicular to the projection plane; 2) a prism if the edges of the prism are perpendicular to the projection plane. The projection surface is projected into a line on the projection plane. All points and lines belonging to the lateral surface of the projection cylinder or the projection prism are projected into a line on the plane to which the axis of the cylinder or the edge of the prism is perpendicular. The line of intersection of surfaces belongs to both surfaces at the same time, and if one of these surfaces is projecting, then the following rule can be used to construct the line of intersection: if one of the intersecting surfaces is projection, then one projection of the line of intersection is in the drawing ready-made and coincides with the projection of the projection surface (the circle into which the cylinder is projected or the polygon into which the prism is projected). The second projection of the intersection line is constructed based on the condition that the points of this line belong to another non-projecting surface.

The considered features of the characteristic points make it easy to check the correctness of the construction of the line of intersection of surfaces, if it is constructed from arbitrarily selected points. In this case, ten points are enough to draw smooth projections of the intersection line. Any number of intermediate points can be plotted if necessary. The constructed points are connected with a smooth line, taking into account the peculiarities of their position and visibility. Let's formulate general rule construction of the line of intersection of surfaces: select the type of auxiliary surfaces; build lines of intersection of auxiliary surfaces with specified surfaces; find the intersection points of the constructed lines and connect them to each other. We choose the auxiliary cutting planes in such a way that, at the intersection with the given surfaces, geometrically simple lines (straight lines or circles) are obtained. Select auxiliary clipping planes. Most often, projection planes, in particular level planes, are chosen as auxiliary section planes. In this case, it is necessary to take into account the intersection lines obtained on the surface as a result of the intersection of the surface by the plane. So the cone is the most complex surface in terms of the number of lines obtained on it. Only planes passing through the apex of the cone or perpendicular to the axis of the cone intersect it, respectively, in a straight line and a circle (geometrically simplest lines). A plane running parallel to one generatrix intersects it in a parabola, a plane parallel to the axis of the cone intersects it along a hyperbola, and a plane that intersects all generators and inclined to the axis of the cone intersects it along an ellipse. On a sphere, when crossing it by a plane, a circle is always obtained, and if it is crossed by a level plane, then this circle is projected on the projection plane, respectively, into a straight line and a circle. So, as auxiliary planes, we select the horizontal planes of the level, which intersect both the cone and the sphere in circles (the simplest lines). Some special cases of intersecting surfaces In some cases, the location, shape or aspect ratio of curved surfaces is such that no complex constructions are required to depict the line of their intersection. These include the intersection of cylinders with parallel generatrices, cones with a common vertex, coaxial surfaces of revolution, surfaces of revolution described around one sphere.