Mutually located lines and planes. a sign of parallelism of a straight line and a plane
Removable element.
outgoing element.
- a) do not have common points;
Theorem.
Designation of cuts
GOST 2.305-2008 provides for the following requirements for section designation:
1. The position of the cutting plane is indicated in the drawing by a section line.
2. An open line should be used for the section line (thickness from S to 1.5S, line length 8-20 mm).
3. With a complex cut, strokes are also carried out at the intersections of the secant planes with each other.
4. Arrows indicating the direction of view should be placed on the initial and final strokes, the arrows should be applied at a distance of 2-3 mm from the outer end of the stroke.
5. The dimensions of the arrows must correspond to those shown in Figure 14.
6. The start and end strokes must not cross the outline of the corresponding image.
7. At the beginning and end of the section line, and, if necessary, at the intersection of the cutting planes, put the same capital letter of the Russian alphabet. The letters are applied near the arrows indicating the direction of view, and at the intersections from the side of the outer corner (Figure 24).
Figure 24 - Examples of section designation
8. The cut must be marked with an inscription of the type "A-A" (always two letters separated by a dash).
9. When the cutting plane coincides with the plane of symmetry of the object as a whole, and the corresponding images are located on the same sheet in direct projection connection and are not separated by any other images, the position of the cutting plane is not marked for horizontal, frontal and profile sections, and the incision is not accompanied by an inscription.
10. Frontal and profile sections, as a rule, are given a position corresponding to that adopted for a given subject in the main image of the drawing.
11. Horizontal, frontal and profile sections can be located in the place of the corresponding main views.
12. It is allowed to place the cut at any place in the drawing field, as well as with rotation with the addition of a conventional graphic symbol - the “Rotated” icon (Figure 25).
Figure 25 - Conditional graphic designation - icon "Rotated"
The designation of sections is similar designation of sections and consists of traces of a secant plane and an arrow indicating the direction of view, as well as a letter affixed on the outside of the arrow (Figure 1c, Figure 3). The removed section is not labeled and the cutting plane is not shown if the section line coincides with the axis of symmetry of the section, and the section itself is located on the continuation of the trace of the cutting plane or in the gap between parts of the view. For a symmetrical superimposed section, the cutting plane is also not shown. If the section is asymmetrical and located in a gap or is superimposed (Figure 2 b), the section line is drawn with arrows, but is not marked with letters.
The section is allowed to be rotated, providing the inscription above the section with the word "rotated". For several identical sections related to the same object, the section lines are designated by the same letter and draw one section. In cases where the section is obtained consisting of separate parts, cuts should be used.
General line
A straight line in general position (Fig. 2.2) is called a straight line that is not parallel to any of these projection planes. Any segment of such a straight line is projected in a given system of projection planes distortedly. The angles of inclination of this straight line to the projection planes are also distorted.
Rice. 2.2.
Direct private provision
Direct lines of particular position include straight lines parallel to one or two projection planes.
Any line (straight or curve) parallel to the projection plane is called a level line. In engineering graphics, there are three main level lines: horizontal, frontal and profile lines.
Rice. 2.3-a
A horizontal line is any line parallel to the horizontal plane of projections (Fig. 2.3-a). The frontal projection of the horizontal is always perpendicular to the communication lines. Any segment of the horizontal onto the horizontal projection plane is projected in true value. The true value is projected onto this plane and the angle of inclination of the horizontal (straight line) to the frontal projection plane. As an example, in Fig. 2.Z-a, a visual image and a complex drawing of a horizontal line are given h, inclined to the plane P 2 at an angle b . 
Rice. 2.3-b
The frontal is called a line parallel to the frontal projection plane (Fig. 2.3-b). The horizontal projection of the frontal is always perpendicular to the communication lines. Any segment of the frontal onto the frontal projection plane is projected in true size. The true value is projected onto this plane and the angle of inclination of the frontal (straight) to the horizontal projection plane (angle a). 
Rice. 2.3-in
A profile line is a line parallel to the profile plane of projections (Fig. 2.Z-c). The horizontal and frontal projections of the profile line are parallel to the communication lines of these projections. Any segment of the profile line (straight) is projected onto the profile plane in the true value. On the same plane are projected in true value and the angles of inclination of the profile straight line to the projection planes P 1 and P 2. When specifying a profile line in a complex drawing, it is necessary to specify two points of this line.
Level lines parallel to two projection planes will be perpendicular to the third projection plane. Such lines are called projecting. There are three main projecting lines: horizontal, frontal and profile projecting lines. 
Rice. 2.3-d 
Rice. 2.3-d 
Rice. 2.3rd
A horizontally projecting straight line (Fig. 2.3-d) is called a straight line perpendicular to the plane P one . Any segment of this line is projected onto the plane P P 1 - to the point.
A frontally projecting straight line (Fig. 2.Z-e) is called a straight line perpendicular to the plane P 2. Any segment of this line is projected onto the plane P 1 without distortion, but flat P 2 - to the point.
A profile projecting line (Fig. 2.Z-e) is called a straight line perpendicular to the plane P 3 , i.e. straight line parallel to projection planes P 1 and P 2. Any segment of this line is projected onto the plane P 1 and P 2 without distortion, but flat P 3 - to the point.
Main lines in the plane
Among the straight lines belonging to the plane, a special place is occupied by straight lines that occupy a particular position in space:
1. Horizontals h - straight lines lying in a given plane and parallel to the horizontal plane of projections (h / / P1) (Fig. 6.4).
Figure 6.4 Horizontal
2. Frontals f - straight lines located in the plane and parallel to the frontal plane of projections (f / / P2) (Fig. 6.5).
Figure 6.5 Frontal
3. Profile straight lines p - straight lines that are in a given plane and parallel to the profile plane of projections (p / / P3) (Fig. 6.6). It should be noted that traces of the plane can also be attributed to the main lines. The horizontal trace is the horizontal of the plane, the frontal is the front and the profile is the profile line of the plane.
Figure 6.6 Profile straight
4. The line of the largest slope and its horizontal projection form a linear angle j, which measures the dihedral angle made up by this plane and the horizontal plane of projections (Fig. 6.7). Obviously, if a line does not have two common points with a plane, then it is either parallel to the plane or intersects it.
Figure 6.7 The line of the largest slope
Kinematic way of surface formation. Setting the surface on the drawing.
In engineering graphics, a surface is considered as a set of successive positions of a line moving in space according to a certain law. In the process of surface formation, line 1 may remain unchanged or change its shape.
For clarity of the image of the surface on a complex drawing, it is advisable to set the law of displacement graphically in the form of a family of lines (a, b, c). The law of movement of line 1 can be specified by two (a and b) or one (a) lines and additional conditions specifying the law of movement 1.
The moving line 1 is called the generatrix, the fixed lines a, b, c are the guides.
We will consider the process of surface formation using the example shown in Fig. 3.1.
Here, line 1 is taken as a generatrix. The law of displacement of the generatrix is given by the guide a and the line b. This means that the generatrix 1 slides along the guide a, all the time remaining parallel to the straight line b.
This way of forming surfaces is called kinematic. With it, you can create and set various surfaces on the drawing. In particular, Figure 3.1 shows the most general case of a cylindrical surface.
Rice. 3.1.
Another way to form a surface and its image in the drawing is to set the surface by a set of points or lines belonging to it. In this case, points and lines are chosen so that they make it possible to determine the shape of the surface with a sufficient degree of accuracy and solve various problems on it.
The set of points or lines that define a surface is called its wireframe.
Depending on how the surface frame is specified, by points or lines, the frames are divided into point and linear.
Figure 3.2 shows a surface skeleton consisting of two orthogonally located families of lines a1, a2, a3, ..., an and b1, b2, b3, ..., bn.
Rice. 3.2.
Conic sections.
CONIC SECTIONS, plane curves, which are obtained by crossing a right circular cone with a plane that does not pass through its top (Fig. 1). From the point of view of analytical geometry, the conic section is the locus of points that satisfy a second-order equation. With the exception of the degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas, or parabolas.
Conic sections are often found in nature and technology. For example, the orbits of the planets revolving around the Sun are ellipses. A circle is a special case of an ellipse, in which the major axis is equal to the minor one. A parabolic mirror has the property that all incident rays parallel to its axis converge at one point (focus). This is used in most reflecting telescopes using parabolic mirrors, as well as in radar antennas and special microphones with parabolic reflectors. A beam of parallel rays emanate from a light source placed at the focus of a parabolic reflector. Therefore, parabolic mirrors are used in powerful spotlights and car headlights. A hyperbola is a graph of many important physical relationships, such as Boyle's law (which relates the pressure and volume of an ideal gas) and Ohm's law, which defines electric current as a function of resistance at constant voltage.
EARLY HISTORY
The discoverer of conic sections is supposedly Menechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menechmus used a parabola and an isosceles hyperbola to solve the problem of doubling a cube.
Treatises on conic sections written by Aristaeus and Euclid at the end of the 4th century. BC, were lost, but the materials from them were included in the famous Conic Sections of Apollonius of Perga (c. 260–170 BC), which have survived to our time. Apollonius abandoned the requirement that the secant plane of the generatrix of the cone be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We also owe to Apollonius the modern names of curves - ellipse, parabola and hyperbola.
In his constructions, Apollonius used a two-sheeted circular cone (as in Fig. 1), so for the first time it became clear that a hyperbola is a curve with two branches. Since the time of Apollonius, conic sections have been divided into three types, depending on the inclination of the cutting plane to the generatrix of the cone. An ellipse (Fig. 1, a) is formed when the cutting plane intersects all the generators of the cone at the points of one of its cavity; parabola (Fig. 1, b) - when the cutting plane is parallel to one of the tangent planes of the cone; hyperbola (Fig. 1, c) - when the cutting plane intersects both cavities of the cone.
CONSTRUCTION OF CONIC SECTIONS
While studying conic sections as intersections of planes and cones, ancient Greek mathematicians also considered them as trajectories of points on a plane. It was found that an ellipse can be defined as the locus of points, the sum of the distances from which to two given points is constant; parabola - as a locus of points equidistant from a given point and a given line; hyperbola - as a locus of points, the difference in distances from which to two given points is constant.
These definitions of conic sections as plane curves also suggest a way to construct them using a stretched thread.
Ellipse.
If the ends of a thread of a given length are fixed at points F1 and F2 (Fig. 2), then the curve described by the tip of a pencil sliding along a tightly stretched thread has the shape of an ellipse. The points F1 and F2 are called the foci of the ellipse, and the segments V1V2 and v1v2 between the intersection points of the ellipse with the coordinate axes are called the major and minor axes. If the points F1 and F2 coincide, then the ellipse turns into a circle.
rice. 2 Ellipsis
Hyperbola.
When constructing a hyperbola, point P, the point of a pencil, is fixed on a thread that slides freely along the pegs installed at points F1 and F2, as shown in Fig. 3a. The distances are chosen so that the segment PF2 is longer than the segment PF1 by a fixed amount, which is less than the distance F1F2. In this case, one end of the thread passes under the F1 peg and both ends of the thread pass over the F2 peg. (The tip of the pencil should not slide along the thread, so it must be secured by making a small loop on the thread and threading the tip into it.) We draw one branch of the hyperbola (PV1Q), making sure that the thread remains taut all the time, and pulling both ends thread down past the point F2, and when the point P is below the segment F1F2, holding the thread at both ends and carefully easing (i.e. releasing) it. We draw the second branch of the hyperbola (PўV2Qў), having previously changed the roles of the pins F1 and F2.
rice. 3 hyperbole
The branches of the hyperbola approach two straight lines that intersect between the branches. These lines, called the asymptotes of the hyperbola, are constructed as shown in Fig. 3b. The slopes of these lines are equal to ± (v1v2)/(V1V2), where v1v2 is the segment of the bisector of the angle between the asymptotes, perpendicular to the segment F1F2; the segment v1v2 is called the conjugate axis of the hyperbola, and the segment V1V2 is called its transverse axis. Thus, the asymptotes are the diagonals of a rectangle with sides passing through four points v1, v2, V1, V2 parallel to the axes. To build this rectangle, you need to specify the location of points v1 and v2. They are at the same distance, equal to
from the point of intersection of the axes O. This formula involves the construction of a right triangle with legs Ov1 and V2O and hypotenuse F2O.
If the asymptotes of the hyperbola are mutually perpendicular, then the hyperbola is called isosceles. Two hyperbolas having common asymptotes, but with rearranged transverse and conjugate axes, are called mutually conjugate.
Parabola.
The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola, apparently, was first established by Pappus (2nd half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called the director. The construction of a parabola using a stretched thread, based on the definition of Pappus, was proposed by Isidore of Miletus (6th century). Let us arrange the ruler so that its edge coincides with the directrix LLў (Fig. 4), and attach the leg AC of the drawing triangle ABC to this edge. We fix one end of the thread of length AB at the vertex B of the triangle, and the other at the focus of the parabola F. Pulling the thread with the tip of the pencil, press the tip at the variable point P to the free leg AB of the drawing triangle. As the triangle moves along the ruler, point P will describe the arc of a parabola with focus F and directrix LLў, since the total length of the thread is AB, the segment of the thread is adjacent to the free leg of the triangle, and therefore the remaining segment of the thread PF must be equal to the remaining parts of leg AB, i.e. PA. The point of intersection of the V parabola with the axis is called the vertex of the parabola, the straight line passing through F and V is called the axis of the parabola. If a straight line perpendicular to the axis is drawn through the focus, then the segment of this straight line cut off by the parabola is called the focal parameter. For an ellipse and a hyperbola, the focal parameter is defined similarly.
TICKETS ANSWERS: No. 1 (incomplete), 2 (incomplete), 3 (incomplete), 4, 5, 6, 7, 12, 13, 14 (incomplete), 16, 17, 18, 20, 21 , 22, 23, 26,
Removable element.
When making drawings, in some cases it becomes necessary to build an additional separate image of any part of the object that requires explanations regarding the shape, dimensions or other data. Such an image is called outgoing element. It is usually performed enlarged. A callout can be laid out as a view or as a section.
When constructing a remote element, the corresponding place in the main image is marked with a closed solid thin line, usually an oval or a circle, and is indicated by a capital letter of the Russian alphabet on the shelf of the leader line. The external element is recorded according to type A (5: 1). On fig. 191 shows an example of a remote element. It is placed as close as possible to the corresponding place on the image of the subject.
1. The method of rectangular (orthogonal) projection. Basic invariant properties of rectangular projection. Epure Monge.
Orthogonal (rectangular) projection is a special case of parallel projection, when all projecting rays are perpendicular to the projection plane. Orthogonal projections have all the properties of parallel projections, but with a rectangular projection, the projection of a segment, if it is not parallel to the projection plane, is always less than the segment itself (Fig. 58). This is explained by the fact that the segment itself in space is the hypotenuse of a right triangle, and its projection is the leg: A "B" \u003d ABcos a.
With rectangular projection, a right angle is projected in full size when both sides of it are parallel to the projection plane, and when only one of its sides is parallel to the projection plane, and the second side is not perpendicular to this projection plane.
Mutual arrangement of a straight line and a plane.
A straight line and a plane in space can:
- a) do not have common points;
- b) have exactly one common point;
- c) have at least two common points.
On fig. 30 shows all these possibilities.
In case a) the line b is parallel to the plane: b || .
In case b) the line l intersects the plane at one point O; l = O.
In case c) the line a belongs to the plane: a or a.
Theorem. If the line b is parallel to at least one line a belonging to the plane , then the line is parallel to the plane .
Suppose that the line m intersects the plane at the point Q. If m is perpendicular to each line of the plane passing through the point Q, then the line m is called perpendicular to the plane.
Tram rails illustrate the belonging of straight lines to the ground plane. Power lines are parallel to the ground plane, and tree trunks are examples of straight lines crossing the ground, some perpendicular to the ground plane, others not perpendicular (slanted).
In planimetry, the plane is one of the main figures, therefore, it is very important to have a clear idea of \u200b\u200bit. This article was created to cover this topic. First, the concept of a plane, its graphical representation, and the designations of planes are shown. Further, the plane is considered together with a point, a straight line or another plane, while options arise from the relative position in space. In the second, third and fourth paragraphs of the article, all variants of the mutual arrangement of two planes, a straight line and a plane, as well as a point and a plane, are analyzed, the main axioms and graphic illustrations are given. In conclusion, the main ways of specifying a plane in space are given.
Page navigation.
Plane - basic concepts, notation and image.
The simplest and most basic geometric figures in three-dimensional space are the point, the line, and the plane. We already have an idea of a point and a line in the plane. If we place a plane on which points and lines are depicted in three-dimensional space, then we will get points and lines in space. The idea of a plane in space allows you to get, for example, the surface of a table or wall. However, a table or wall has finite dimensions, and the plane extends beyond their boundaries to infinity.
Points and lines in space are denoted in the same way as on a plane - in capital and small Latin letters, respectively. For example, points A and Q, lines a and d. If two points are given that lie on a line, then the line can be denoted by two letters corresponding to these points. For example, the line AB or BA passes through points A and B. Planes are usually denoted by small Greek letters, for example, planes, or.
When solving problems, it becomes necessary to depict planes in the drawing. The plane is usually depicted as a parallelogram or an arbitrary simple closed area.
The plane is usually considered together with points, straight lines or other planes, and various variants of their mutual arrangement arise. We turn to their description.
Mutual arrangement of a plane and a point.
Let's start with an axiom: there are points in every plane. From it follows the first variant of the mutual arrangement of the plane and the point - the point may belong to the plane. In other words, a plane can pass through a point. To indicate the belonging of a point to any plane, the symbol "" is used. For example, if the plane passes through point A, then you can briefly write .
It should be understood that there are infinitely many points on a given plane in space.
The following axiom shows how many points in space must be marked in order for them to define a particular plane: through three points that do not lie on one straight line, a plane passes, and only one. If three points are known that lie in a plane, then the plane can be denoted by three letters corresponding to these points. For example, if the plane passes through points A, B and C, then it can be designated ABC.
Let us formulate one more axiom, which gives the second variant of the mutual arrangement of the plane and the point: there are at least four points that do not lie in the same plane. So, a point in space may not belong to the plane. Indeed, by virtue of the previous axiom, a plane passes through three points of space, and the fourth point may or may not lie on this plane. When shorthand, the symbol "" is used, which is equivalent to the phrase "does not belong."
For example, if point A does not lie in the plane, then a short notation is used.
Line and plane in space.
First, a line can lie in a plane. In this case, at least two points of this line lie in the plane. This is established by the axiom: if two points of a line lie in a plane, then all points of this line lie in the plane. For a short record of belonging to a certain line of a given plane, use the symbol "". For example, the entry means that the line a lies in the plane.
Second, the line can intersect the plane. In this case, the line and the plane have one single common point, which is called the point of intersection of the line and the plane. With a short record, the intersection is denoted by the symbol "". For example, the entry means that the line a intersects the plane at the point M. When a certain line intersects a plane, the concept of an angle between a line and a plane arises.
Separately, it is worth dwelling on a line that intersects a plane and is perpendicular to any line lying in this plane. Such a line is called perpendicular to the plane. For a short record of perpendicularity, the symbol "" is used. For a deeper study of the material, you can refer to the article perpendicularity of a straight line and a plane.
Of particular importance in solving problems related to the plane is the so-called normal vector of the plane. A normal vector of a plane is any non-zero vector lying on a line perpendicular to this plane.
Thirdly, a straight line can be parallel to a plane, that is, not have common points in it. When shorthand for parallelism, the symbol "" is used. For example, if the line a is parallel to the plane, then you can write . We recommend that you study this case in more detail by referring to the article parallelism of a straight line and a plane.
It should be said that a straight line lying in a plane divides this plane into two half-planes. The straight line in this case is called the boundary of the half-planes. Any two points of the same half-plane lie on the same side of the line, and two points of different half-planes lie on opposite sides of the boundary line.
Mutual arrangement of planes.
Two planes in space can coincide. In this case, they have at least three points in common.
Two planes in space can intersect. The intersection of two planes is a straight line, which is established by the axiom: if two planes have a common point, then they have a common straight line on which all common points of these planes lie.
In this case, the concept of the angle between intersecting planes arises. Of particular interest is the case when the angle between the planes is ninety degrees. Such planes are called perpendicular. We talked about them in the article perpendicularity of planes.
Finally, two planes in space can be parallel, that is, have no common points. We recommend that you read the article parallelism of planes to get a complete picture of this variant of the relative position of the planes.
Plane definition methods.
Now we list the main ways to set a specific plane in space.
First, a plane can be defined by fixing three points in space that do not lie on the same straight line. This method is based on the axiom: through any three points that do not lie on the same straight line, there is only one plane.
If a plane is fixed and given in three-dimensional space by specifying the coordinates of its three different points that do not lie on the same straight line, then we can write the equation of a plane passing through three given points.
The next two ways of specifying a plane are a consequence of the previous one. They are based on the consequences of the axiom about a plane passing through three points:
- a plane passes through a line and a point not lying on it, moreover, only one (see also the article equation of a plane passing through a line and a point);
- a single plane passes through two intersecting lines (we recommend that you familiarize yourself with the material of the article the equation of a plane passing through two intersecting lines).
The fourth way to define a plane in space is based on the definition of parallel lines. Recall that two lines in space are called parallel if they lie in the same plane and do not intersect. Thus, by specifying two parallel lines in space, we determine the only plane in which these lines lie.
If in three-dimensional space with respect to a rectangular coordinate system a plane is given in the indicated way, then we can compose an equation for a plane passing through two parallel lines.
In a high school course in geometry lessons, the following theorem is proved: a single plane passes through a fixed point in space, perpendicular to a given line. Thus, we can define a plane if we specify a point through which it passes and a line perpendicular to it.
If a rectangular coordinate system is fixed in three-dimensional space and a plane is given in the indicated way, then it is possible to compose an equation for a plane passing through a given point perpendicular to a given straight line.
Instead of a straight line perpendicular to a plane, one of the normal vectors of this plane can be specified. In this case, it is possible to write
The line may or may not belong to the plane. It belongs to the plane if at least two of its points lie on the plane. Figure 93 shows the plane Sum (axb). Straight l belongs to the plane Sum, since its points 1 and 2 belong to this plane.
If the line does not belong to the plane, it may be parallel to it or intersect it.
A line is parallel to a plane if it is parallel to another line in that plane. Figure 93 straight m || sum, since it is parallel to the line l belonging to this plane.
A straight line can intersect a plane at various angles and, in particular, be perpendicular to it. The construction of lines of intersection of a straight line with a plane is given in §61.
Figure 93 - A straight line belonging to a plane
A point in relation to a plane can be located as follows: to belong or not to belong to it. A point belongs to a plane if it is located on a line in that plane. Figure 94 shows a complex drawing of the Sum plane defined by two parallel lines l and P. The line is in the plane m. Point A lies in the plane Sum, since it lies on the line m. Dot AT does not belong to the plane, since its second projection does not lie on the corresponding projections of the line.
Figure 94 - Complex drawing of a plane defined by two parallel lines
Conical and cylindrical surfaces
Conical surfaces include surfaces formed by the displacement of a rectilinear generatrix l along a curved guide m. A feature of the formation of a conical surface is that in this case one point of the generatrix is always fixed. This point is the top of the conical surface (Figure 95, a). Conical surface definer includes vertex S and guide m, wherein l"~S; l"^ m.
Cylindrical surfaces include surfaces formed by a straight generatrix / moving along a curvilinear guide t parallel to the given direction S(Figure 95, b). A cylindrical surface can be considered as a special case of a conical surface with a vertex at infinity S.
The cylindrical surface determinant consists of a guide t and direction S, forming l, while l" || S; l" ^ m.
If the generators of a cylindrical surface are perpendicular to the plane of projections, then such a surface is called projecting. Figure 95, in a horizontally projecting cylindrical surface is shown.
On cylindrical and conical surfaces, given points are built using generators passing through them. Lines on surfaces, such as a line a to figure 95, in or horizontal h in figure 95, a, b, are built using individual points belonging to these lines.
Figure 95 - Conical and cylindrical surfaces
Torso surfaces
A torso surface is a surface formed by a rectilinear generatrix l, touching during its motion in all its positions a certain spatial curve t, called return edge(Figure 96). The return edge completely defines the torso and is the geometric part of the surface definer. The algorithmic part is the indication of the tangency of the generators to the cusp edge.
A conical surface is a special case of a torso with a return edge t degenerated into a point S- top of a conical surface. A cylindrical surface is a special case of a torso, whose cusp edge is a point at infinity.
Figure 96 - Torso surface
Faceted surfaces
Faceted surfaces include surfaces formed by the displacement of a rectilinear generatrix l along a broken line m. However, if one point S generatrix is motionless, a pyramidal surface is created (Figure 97) if the generatrix is parallel to a given direction when moving S, then a prismatic surface is created (Figure 98).
The elements of faceted surfaces are: vertex S(near the prismatic surface it is at infinity), face (part of the plane bounded by one section of the guide m and the extreme positions of the generatrix relative to it l) and an edge (line of intersection of adjacent faces).
Pyramid surface determinant includes vertex S, through which generators and guides pass: l" ~ S; l^ t.
Prismatic surface determinant, except guide t, contains direction S, to which all generators are parallel l surfaces: l||S; l^ t.
Figure 97 - Pyramidal surface
Figure 98 - Prismatic surface
Closed faceted surfaces formed by a certain number (at least four) of faces are called polyhedra. Among the polyhedra, a group of regular polyhedra is distinguished, in which all faces are regular and congruent polygons, and the polyhedral angles at the vertices are convex and contain the same number of faces. For example: hexahedron - cube (Figure 99, a), tetrahedron - regular quadrangle (Figure 99, 6) octahedron - polyhedron (Figure 99, in). Crystals have the shape of various polyhedra.
Figure 99 - Polyhedra
Pyramid- a polyhedron, at the base of which lies an arbitrary polygon, and the side faces are triangles with a common vertex S.
In the complex drawing, the pyramid is defined by the projections of its vertices and edges, taking into account their visibility. The visibility of an edge is determined using competing points (Figure 100).
Figure 100 - Determining the visibility of an edge using competing points
Prism- a polyhedron whose base is two identical and mutually parallel polygons, and the side faces are parallelograms. If the edges of the prism are perpendicular to the plane of the base, such a prism is called a straight line. If the ribs of a prism are perpendicular to any projection plane, then its side surface is called projecting. Figure 101 shows a complex drawing of a straight quadrangular prism with a horizontally projecting surface.
Figure 101 - Complex drawing of a straight quadrangular prism with a horizontally projecting surface
When working with a complex drawing of a polyhedron, you have to build lines on its surface, and since a line is a collection of points, you must be able to build points on the surface.
Any point on a faceted surface can be constructed using a generatrix passing through this point. In the figure 100 in the face ACS point built M with the help of generator S-5.
Helical surfaces
The helical surfaces are those created during the helical motion of a rectilinear generatrix. Ruled helical surfaces are called helicoids.
A straight helicoid is formed by the motion of a rectilinear generatrix i along two guides: a helix t and its axes i; while generating l crosses the helical axis at a right angle (Figure 102, a). A straight helicoid is used to create spiral staircases, screws, as well as power threads, in machine tools.
An inclined helicoid is formed by the movement of the generatrix along the helical guide t and its axes i so that the generator l crosses the axis i at a constant angle φ other than a right angle, i.e. in any position, the generatrix l parallel to one of the generatrices of the guide cone with an angle at the apex equal to 2φ (Figure 102, b). Inclined helicoids limit the surfaces of the threads.
Figure 102 - Helicoids
Surfaces of revolution
Surfaces of revolution include surfaces formed by the rotation of a line l around a straight line i representing the axis of rotation. They can be ruled, such as a cone or cylinder of revolution, and non-linear or curvilinear, such as a sphere. The determinant of the surface of revolution includes the generatrix l and axis i . Each point of the generatrix during rotation describes a circle, the plane of which is perpendicular to the axis of rotation. Such circles of the surface of revolution are called parallels. The largest of the parallels is called equator. Equator.defines the horizontal outline of the surface if i _|_ P 1 . In this case, the parallels are the horizontals of this surface.
The curves of the surface of revolution, formed as a result of the intersection of the surface with planes passing through the axis of revolution, are called meridians. All meridians of one surface are congruent. The frontal meridian is called the main meridian; it defines the frontal outline of the surface of revolution. The profile meridian determines the profile outline of the surface of revolution.
It is most convenient to build a point on curved surfaces of revolution using surface parallels. Figure 103 dot M built on the parallel h 4 .
Figure 103 - Building a point on a curved surface
Surfaces of revolution have found the widest application in engineering. They limit the surfaces of most engineering parts.
A conical surface of revolution is formed by the rotation of a straight line i around the straight line intersecting with it - the axis i(Figure 104, a). Dot M on the surface is built using a generatrix l and parallels h. This surface is also called a cone of revolution or a right circular cone.
A cylindrical surface of revolution is formed by the rotation of a straight line l around a parallel axis i(Figure 104, b). This surface is also called a cylinder or a right circular cylinder.
A sphere is formed by rotating a circle around its diameter (Figure 104, in). Point A on the surface of the sphere belongs to the prime meridian f, dot AT- equator h, a dot M built on an auxiliary parallel h".
Figure 104 - Formation of surfaces of revolution
A torus is formed by rotating a circle or its arc around an axis lying in the plane of the circle. If the axis is located within the circle formed, then such a torus is called closed (Figure 105, a). If the axis of rotation is outside the circle, then such a torus is called open (Figure 105, b). An open torus is also called a ring.
Figure 105 - Formation of a torus
The surfaces of revolution can also be formed by other curves of the second order. Ellipsoid of revolution (Figure 106, a) formed by the rotation of an ellipse around one of its axes; paraboloid of revolution (Figure 106, b) - rotation of the parabola around its axis; one-sheet hyperboloid of revolution (Figure 106, in) is formed by the rotation of the hyperbola around the imaginary axis, and two-sheeted (Figure 106, G) - rotation of the hyperbola around the real axis.
Figure 106 - Formation of surfaces of revolution by curves of the second order
In the general case, the surfaces are depicted as not limited in the direction of propagation of the generating lines (see Figures 97, 98). To solve specific problems and obtain geometric shapes, they are limited to cutting planes. For example, to obtain a circular cylinder, it is necessary to limit the section of the cylindrical surface with cut planes (see Figure 104, b). As a result, we get its upper and lower bases. If the cut planes are perpendicular to the axis of rotation, the cylinder will be straight; if not, the cylinder will be inclined.
To get a circular cone (see figure 104, a), you need to cut along the vertex and beyond. If the cut plane of the base of the cylinder is perpendicular to the axis of rotation, the cone will be straight; if not, it will be inclined. If both cut planes do not pass through the vertex, the cone will be truncated.
Using the cut plane, you can get a prism and a pyramid. For example, a hexagonal pyramid will be straight if all its edges have the same slope to the cut plane. In other cases, it will be oblique. If it is done With with the help of trim planes and none of them passes through the top - the pyramid is truncated.
A prism (see Figure 101) can be obtained by limiting a portion of the prismatic surface to two cut planes. If the cut plane is perpendicular to the edges, for example, an octagonal prism, it is straight, if not perpendicular, it is inclined.
By choosing the appropriate position of the cutting planes, it is possible to obtain various shapes of geometric figures depending on the conditions of the problem being solved.
Location
Feature: if a line not lying in a given plane is parallel to some line lying in this plane, then it is parallel to the given plane.
1. if a plane passes through a given line parallel to another plane and intersects this plane, then the line of intersection of the planes is parallel to the given line.
2. if one of the 2 lines is parallel to the given one, then the other line is either also parallel to the given plane, or lies in this plane.
RELATIONSHIP OF THE PLANES. PARALLEL PLANES
Location
1. planes have at least 1 common point, i.e. intersect in a straight line
2. the planes do not intersect, i.e. do not have 1 common point, in which case they are called parallel.
sign
if 2 intersecting lines of 1 plane are respectively parallel to 2 lines of another plane, then these planes are parallel.
Holy
1. if 2 parallel planes are crossed by 3, then the lines of their intersection are parallel
2. segments of parallel lines enclosed between parallel planes are equal.
PERPENDICULARITY OF A LINE AND A PLANE. SIGN OF PERPENDICULARITY OF A LINE AND A PLANE.
Direct naz perpendicular if they intersect<90.
Lemma: if 1 of 2 parallel lines is perpendicular to the 3rd line, then the other line is also perpendicular to this line.
A straight line is perpendicular to a plane, if it is perpendicular to any line in that plane.
Theorem: if 1 of 2 parallel lines is perpendicular to a plane, then the other line is also perpendicular to that plane.
Theorem: if 2 lines are perpendicular to a plane, then they are parallel.
sign
If a line is perpendicular to 2 intersecting lines lying in a plane, then it is perpendicular to that plane.
PERPENDICULAR AND SLANT
Let's construct a plane and m.A, not belonging to the plane. Their t.A draw a straight line, perpendicular to the plane. The point of intersection of a straight line with a plane is designated H. The segment AN is a perpendicular drawn from point A to the plane. T.N - the base of the perpendicular. Let us take in the plane t.M, which does not coincide with H. The segment AM is an oblique line drawn from point A to the plane. M - the base of the inclined. Segment MN - projection of the inclined onto the plane. Perpendicular AH - distance from point A to the plane. Any distance is a part of a perpendicular.
Theorem about 3 perpendiculars:
A straight line drawn in a plane through the base of an inclined plane perpendicular to its projection onto this plane is also perpendicular to the inclined one itself.
ANGLE BETWEEN A RIGHT AND A PLANE
The angle between the line and the plane is the angle between this line and its projection on the plane.
DIHEDRAL ANGLE. ANGLE BETWEEN PLANES
dihedral angle naz the figure formed by a straight line and 2 half-planes with a common boundary a does not belong to the same plane.
border a- dihedral edge. Half planes - faces of a dihedral angle. To measure dihedral angle. You need to build a linear angle inside it. We mark some point on the edge of the dihedral angle and draw a ray from this point in each face, perpendicular to the edge. The angle formed by these rays linear gl of the dihedral angle. There can be infinitely many of them inside the dihedral angle. They all have the same size.
PERPENDICULARITY OF TWO PLANES
Two intersecting planes perpendicular, if the angle between them is 90.
Feature:
If 1 of 2 planes passes through a line perpendicular to another plane, then such planes are perpendicular.
POLYHEDRALS
Polyhedron- a surface composed of polygons and limiting some geometric body. Facets are the polygons that make up the polyhedra. Ribs- the sides of the edges. Peaks- the ends of the ribs. Polyhedron diagonal back a segment connecting 2 vertices that do not belong to 1 face. A plane on both sides of which there are points of a polyhedron, called . cutting plane. The common part of the polyhedron and the secant area is called section of a polyhedron. Polyhedra are convex and concave. Naz polyhedron convex, if it is located on one side of the plane of each of its faces (tetrahedron, parallelepiped, octahedron). In a convex polyhedron, the sum of all plane angles at each of its vertices is less than 360.
PRISM
A polyhedron composed of 2 equal polygons located in parallel planes and n - parallelograms called prism.
Polygons A1A2..A(p) and B1B2..B(p) - prism bases. А1А2В2В1…- parallelograms, A(p)A1B1B(p) – side edges. Segments A1B1, A2B2..A(p)B(p) – side ribs. Depending on the polygon underlying the prism, the prism naz p-coal. A perpendicular drawn from any point of one base to the plane of another base is called height. If the side edges of the prism are perpendicular to the base, then the prism - straight, and if not perpendicular - then inclined. The height of a straight prism is equal to the length of its lateral edge. Direct prismanaz correct, if its base is regular polygons, all side faces are equal rectangles.
PARALLEPIPED
ABCD//A1B1C1D1, AA1//BB1//SS1//DD1, AA1=BB1=SS1=DD1 (according to the property of parallel planes)
The parallelepiped consists of 6 parallelograms. Parallelograms naz faces. ABSD and A1V1S1D1 - bases, the remaining faces are called side. Points A B C D A1 B1 C1 D1 - tops. Segments connecting vertices ribs. AA1, BB1, SS1, DD1 - side ribs.
Diagonal of a parallelepiped back a segment connecting 2 vertices that do not belong to 1 face.
Saints
1. Opposite faces of a parallelepiped are parallel and equal. 2. The diagonals of the parallelepiped intersect at one point and bisect this point.
PYRAMID
Consider a polygon A1A2..A(n), a point P not lying in the plane of this polygon. Let's connect the point P with the vertices of the polygon and get n triangles: PA1A2, PA2A3….RA(p)A1.
A polyhedron composed of an n-gon and n-triangles over the pyramid. Polygon - base. Triangles - side edges. R - top of the pyramid. Segments А1Р, А2Р..А(p)Р – side ribs. Depending on the polygon lying at the base, the pyramid is called p-coal. The height of the pyramid back a perpendicular drawn from the vertex to the plane of the base. Pyramid called correct, if its base is a regular polygon and the height is at the center of the base. Apothem is the height of the lateral face of a regular pyramid.
TRUNCATED PYRAMID
Consider the pyramid PA1A2A3A(n). draw a cutting plane parallel to the base. This plane divides our pyramid into 2 parts: the upper one is a pyramid similar to this one, the lower one is a truncated pyramid. The side surface consists of a trapezium. Lateral ribs connect the tops of the bases.
Theorem: the area of the lateral surface of a regular truncated pyramid is equal to the product of half the sum of the perimeters of the bases and the apothem.
REGULAR POLYTOPES
A convex polyhedron is called regular, if all its faces are equal regular polygons and the same number of edges converge at each of its vertices. An example of a regular polyhedron is a cube. All its faces are equal squares, and 3 edges converge at each vertex.
regular tetrahedron composed of 4 equilateral triangles. Each vertex is a vertex of 3 triangles. The sum of the flat angles at each vertex is 180.
Regular octahedron Consist of 8 equilateral triangles. Each vertex is a vertex of 4 triangles. Sum of plane angles at each vertex =240
Regular icosahedron Consist of 20 equilateral triangles. Each vertex is a vertex 5 triangle. The sum of flat angles at each vertex is 300.
Cube composed of 6 squares. Each vertex is a vertex of 3 squares. The sum of flat angles at each vertex =270.
Regular dodecahedron Consist of 12 regular pentagons. Each vertex is a vertex of 3 regular pentagons. The sum of flat angles at each vertex = 324.
There are no other types of regular polyhedra.
CYLINDER
A body bounded by a cylindrical surface and two circles with boundaries L and L1 called cylinder. Circles L and L1 back the bases of the cylinder. Segments MM1, AA1 - generators. Forming the composition of the cylindrical or lateral surface of the cylinder. Straight line, connecting the centers of the bases O and O1 naz axis of the cylinder. Generating length - cylinder height. The base radius (r) is the radius of the cylinder.
Cylinder sections
Axial passes through the axis and base diameter
Perpendicular to axis
A cylinder is a body of revolution. It is obtained by rotating a rectangle around 1 of the sides.
CONE
Let us consider a circle (o;r) and a straight line OP perpendicular to the plane of this circle. Through each point of the circle L and t.P we draw segments, there are infinitely many of them. They form a conical surface and generators.
R- vertex, OR - conical surface axis.
Body bounded by a conical surface and a circle with boundary L naz cone. A circle - the base of the cone. Vertex of a conical surface is the apex of the cone. Forming a conical surface - forming a cone. Conical surface - lateral surface of the cone. RO - cone axis. Distance from R to O - cone height. A cone is a body of revolution. It is obtained by rotating a right triangle around the leg.
Cone section
Axial section
Section perpendicular to the axis
SPHERE AND BALL
sphere called a surface consisting of all points in space located at a given distance from a given point. This point is the center of the sphere. This distance is sphere radius.
A line segment connecting two points on a sphere and passing through its center naz the diameter of the sphere.
A body bounded by a sphere ball. Center, radius and diameter of the sphere center, radius and diameter of the sphere.
Sphere and ball are bodies of revolution. Sphere is obtained by rotating a semicircle around the diameter, and ball obtained by rotating a semicircle around the diameter.
in a rectangular coordinate system, the equation of a sphere of radius R with center C(x(0), y(0), Z(0) has the form (x-x(0))(2)+(y-y(0))(2 )+(z-z(0))(2)= R(2)
The mutual position of a straight line and a plane is determined by the number of common points :
1) if a line has two common points with a plane, then it belongs to this plane,
2) if a line has one common point with a plane, then the line intersects the plane,
3) if the point of intersection of a line with a plane is removed to infinity, then the line and the plane are parallel.
Problems in which the relative position of various geometric shapes relative to each other is determined are called positional problems.
The straight line belonging to the plane was considered earlier.
Line parallel to plane, if it is parallel to some straight line lying in this plane. To construct such a straight line, it is necessary to specify any straight line in the plane and draw the required one parallel to it.
Rice. 1.53 Fig. 1.54 Fig.1.55
Let through the dot BUT(Fig. 1.53) it is necessary to draw a straight line AB, parallel to the plane Q, given by a triangle CDF. To do this, through the frontal projection of the point a / points BUT make a frontal projection a / in / desired line parallel to the frontal projection of any line lying in the plane R, e.g. straight CD (a / in /!!s / d /). Through a horizontal projection a points BUT parallel sd make a horizontal projection av desired line AB (av11 sd). Straight AB parallel to the plane R, given by a triangle CDF.
Of all the possible positions of a line intersecting a plane, we note the case when the line is perpendicular to the plane. Consider the properties of projections of such a line.
Rice. 1.56 Fig. 1.57
The line is perpendicular to the plane(a special case of the intersection of a straight line with a plane) if it is perpendicular to any straight line lying in the plane. To construct projections of a perpendicular to a plane in a general position, this is not enough without transforming the projections. Therefore, an additional condition is introduced: a line is perpendicular to a plane if it is perpendicular to two intersecting principal lines(to build projections, the right angle projection condition is used). In this case: the horizontal and frontal projections of the perpendicular are perpendicular, respectively, to the horizontal projection of the horizontal and the frontal projection of the frontal of a given plane in general position (Fig. 1.54). When a plane is specified by traces, the projections of the perpendicular are perpendicular, respectively, to the frontal - to the frontal trace, horizontal - to the horizontal trace of the plane (Fig. 1.55).
Intersection of a straight line with a projecting plane. Consider a straight line that intersects a plane when the plane is in a particular position.
A plane perpendicular to the projection plane (the projection plane) is projected onto it as a straight line. On this line (the projection of the plane) there must be a corresponding projection of the point at which some line intersects this plane (Fig. 1.56).
In Figure 1.56, the frontal projection of the point To line intersection AB with a triangle CDE is determined at the intersection of their frontal projections, because triangle CDE projected onto the frontal plane as a straight line. We find the horizontal projection of the point of intersection of the line with the plane (it lies on the horizontal projection of the line). Using the method of competing points, we determine the visibility of the line AB relative to the plane of the triangle CDE on the horizontal projection plane.
Figure 1.59 shows a horizontally projecting plane P and a straight line in general position AB. Because plane R is perpendicular to the horizontal plane of projections, then everything that is in it is projected onto the horizontal plane of projections on its trace, including the point of its intersection with the line AB. Therefore, in the complex drawing we have a horizontal projection of the point of intersection of the line with the plane R. According to the belonging of the point to the straight line, we find the frontal projection of the point of intersection of the straight line AB with a plane R. Determine the visibility of the line on the frontal projection plane.
Rice. 1.58 Fig. 1.59
Figure 1.58 shows a comprehensive drawing of the construction of projections of the point of intersection of the line AB with horizontal level plane G. Frontal plane trace G is its frontal projection. Frontal projection of the point of intersection of the plane G with a straight line AB are determined at the intersection of the frontal projection of the straight line and the frontal trace of the plane. Having a frontal projection of the point of intersection, we find the horizontal projection of the point of intersection of the line AB with plane G.
Figure 1.57 shows a plane in general position, given by a triangle CDE and front projection line AB? intersecting the plane at a point K. Frontal projection of a point - k / matches the points a / and b/ . To construct a horizontal projection of the intersection point, draw through the point K in plane CDE straight line (eg. 1-2 ). Let's construct its frontal projection, and then horizontal. Dot K is the point of intersection of the lines AB and 1-2. That is the point K simultaneously belongs to the line AB and the plane of the triangle and, therefore, is the point of their intersection.
The intersection of two planes. A straight line of intersection of two planes is defined by two points, each of which belongs to both planes, or by one point, which belongs to two planes, and the known direction of the line. In both cases, the task is to find a point common to two planes.
Intersection of projecting planes. Two planes can be parallel to each other or intersect. Consider the cases of mutual intersection of planes.
A straight line obtained at the mutual intersection of two planes is completely determined by two points, each of which belongs to both planes, therefore, it is necessary and sufficient to find these two points belonging to the line of intersection of two given planes.
Therefore, in the general case, to construct a line of intersection of two planes, it is necessary to find any two points, each of which belongs to both planes. These points determine the line of intersection of the planes. To find each of these two points, you usually have to perform special constructions. But if at least one of the intersecting planes is perpendicular (or parallel) to any projection plane, then the construction of the projection of the line of their intersection is simplified.
Rice. 1.60 Fig. 1.61
If the planes are given by traces, then it is natural to look for the points that define the line of intersection of the planes at the points of intersection of the traces of the planes of the same name in pairs: the line passing through these points is common to both planes, i.e. their line of intersection.
Consider special cases of the location of one (or both) of the intersecting planes.
The complex drawing (Fig. 1.60) shows horizontally projecting planes P and Q. Then the horizontal projection of their intersection line degenerates into a point, and the frontal projection into a straight line perpendicular to the axis ox.
The complex drawing (Fig. 1.61) shows the planes of private position: the plane R perpendicular to the horizontal projection plane (horizontal projection plane) and the plane Q- horizontal level plane. In this case, the horizontal projection of their line of intersection will coincide with the horizontal trace of the plane R, and the frontal - with a frontal trace of the plane Q.
In the case of specifying planes by traces, it is easy to establish that these planes intersect: if at least one pair of traces of the same name intersect, then the planes intersect each other.
The foregoing applies to planes defined by intersecting traces. If both planes have traces parallel to each other on the horizontal and frontal planes, then these planes can be parallel or intersect. The mutual position of such planes can be judged by constructing a third projection (third trace). If the traces of both planes on the third projection are also parallel, then the planes are parallel to each other. If the traces on the third plane intersect, then the planes given in space intersect.
The complex drawing (Fig. 1.62) shows front-projecting planes defined by a triangle ABC and DEF. The projection of the line of intersection on the frontal projection plane is a point, i.e. since the triangles are perpendicular to the frontal projection plane, their line of intersection is also perpendicular to the frontal projection plane. Therefore, the horizontal projection of the line of intersection of triangles ( 12 ) is perpendicular to the axis ox. The visibility of the elements of the triangles on the horizontal projection plane is determined using competing points (3,4).
On the complex drawing (Fig. 1.63), two planes are set: one of which is a triangle ABC general position, the other - a triangle DEF perpendicular to the frontal projection plane, i.e. located in a private position (front-projecting). Frontal projection of the line of intersection of triangles ( 1 / 2 / ) is found based on common points that simultaneously belong to both triangles (everything that is in the front-projecting triangle DEF on the frontal projection will result in a line - its projection onto the frontal plane, including the line of its intersection with the triangle ABC. According to the belonging of the points of intersection to the sides of the triangle ABC, we find the horizontal projection of the line of intersection of the triangles. Using the method of competing points, we determine the visibility of triangle elements on the horizontal plane of projections.
Rice. 1.63 Fig. 1.64
Figure 1.64 shows a complex drawing of two planes defined by a triangle in general position ABC and horizontally projecting plane R, given by traces. Since the plane R- horizontally projecting, then everything that is in it, including the line of its intersection with the plane of the triangle ABC, on the horizontal projection will coincide with its
horizontal track. The frontal projection of the line of intersection of these planes is found from the condition that the points of the element belong to (sides) of the plane in general position.
In the case of specifying planes in general position not by traces, then to obtain the line of intersection of the planes, the point of meeting of the side of one triangle with the plane of another triangle is sequentially found. If planes in general position are not given by triangles, then the line of intersection of such planes can be found by introducing two auxiliary secant planes in turn - projecting (for specifying planes by triangles) or level for all other cases.
Intersection of a line in general position with a plane in general position. Previously, cases of intersection of planes were considered, when one of them was projecting. Based on this, we can find the point of intersection of a line in general position with a plane in general position by introducing an additional projecting mediator plane.
Before considering the intersection of planes in general position, consider the intersection of a line in general position with a plane in general position.
To find the meeting point of a line in general position with a plane in general position, it is necessary:
1) enclose a straight line in an auxiliary projecting plane,
2) find the line of intersection of the given and auxiliary planes,
determine a common point belonging simultaneously to two planes (this is their line of intersection) and a straight line.
Rice. 1.65 Fig. 1.66
Rice. 1.67 Fig. 1.68
The complex drawing (Fig. 1.65) shows a triangle CDE general position and direct AB general position. To find the point of intersection of a line with a plane, we conclude the line AB Q. Let's find the line of intersection ( 12 ) intermediary plane Q and given plane CDE. When constructing a horizontal projection of the intersection line, there is a common point To, simultaneously belonging to two planes and a given line AB. From the belonging of a point to a straight line, we find the frontal projection of the point of intersection of a straight line with a given plane. The visibility of the elements of a straight line on the projection planes is determined using competing points.
Figure 1.66 shows an example of finding the meeting point of a straight line AB, which is a horizontal line (the line is parallel to the horizontal plane of projections) and the plane R, in general position, given by traces. To find the point of their intersection, the line AB lies in the horizontally projecting plane Q. Then proceed as in the above example.
To find the meeting point of a horizontally projecting line AB with a plane in general position (Fig. 1.67), through the meeting point of a straight line with a plane (its horizontal projection coincides with the horizontal projection of the straight line itself) we draw a horizontal line (i.e. we bind the point of intersection of a straight line with a plane to a plane R). Having found the frontal projection of the drawn horizontal in the plane R, mark the frontal projection of the meeting point of the line AB with plane R.
To find the line of intersection of planes in general position, given by traces, it is enough to mark two common points that simultaneously belong to both planes. Such points are the points of intersection of their traces (Fig. 1.68).
To find the line of intersection of planes in general position, given by two triangles (Fig. 1.69), we sequentially find the point
meeting of the side of one triangle with the plane of another triangle. Taking any two sides from any triangle, enclosing them in mediators projecting planes, two points are found that simultaneously belong to both triangles - the line of their intersection.
Figure 1.69 shows a complex drawing of triangles ABC and DEF general position. To find the line of intersection of these planes:
1. We conclude the side sun triangle ABC into the frontal projection plane S(the choice of planes is completely arbitrary).
2. Find the line of intersection of the plane S and plane DEF – 12 .
3. We mark the horizontal projection of the meeting point (common point of two triangles) To from intersection 12 and sun and find its frontal projection on the frontal projection of the line Sun.
4. We draw the second auxiliary projecting plane Q across the side D.F. triangle DEF.
5. Find the line of intersection of the plane Q and triangle ABC - 3 4.
6. Mark the horizontal projection of the point L, which is the meeting point of the party D.F. with triangle plane ABC and find its frontal projection.
7. We connect the same-named projections of points To and L. to L- line of intersection of planes in general position, given by triangles ABC and DEF.
8. Using the method of competing points, we determine the visibility of the elements of triangles on the projection planes.
Since the above is also valid for the main lines of parallel planes, we can say that planes are parallel if their traces of the same name are parallel(Fig. 1.71).
Figure 1.72 shows the construction of a plane parallel to the given one and passing through the point BUT. In the first case, through the point BUT a straight line (front) is drawn parallel to a given plane G. Thus, a plane is drawn R containing a line parallel to a given plane G and parallel to it. In the second case, through the point BUT a plane is drawn, given by the main lines from the condition of parallelism of these lines to a given plane G.
Mutually perpendicular planes.If one plane contains
at least one line perpendicular to another plane, then such
planes are perpendicular. Figure 1.73 mutually perpendicular planes are shown. Figure 1.74 shows the construction of a plane perpendicular to the one given through the point BUT, using the condition of perpendicularity of a straight line (in this case, the main lines) to the plane.
In the first case, through the point BUT a frontal is drawn perpendicular to the plane R, its horizontal trace is constructed and a horizontal trace of the plane is drawn through it Q , perpendicular to the horizontal trace of the plane R. Through the resulting vanishing point QX a frontal trace of the plane is drawn Q perpendicular to the front trace of the plane R.
In the second case, horizontal lines are drawn in the plane of the triangle BE and frontal bf and through a given point BUT we set the plane by intersecting straight lines (main lines) perpendicular to the plane of the triangle. To do this, draw through the point BUT horizontal and frontal. The horizontal projection of the horizontal of the desired plane ( N) we draw perpendicular to the horizontal projection of the horizontal of the triangle, the frontal projection of the front of the new plane ( M) is perpendicular to the frontal projection of the front of the triangle.
