What are the cases of the relative position of a straight line and a plane. The relative position of a straight line and a plane, two planes. Prism. Definition. Elements. Types of prisms
A straight line may or may not belong to a plane. It belongs to a plane if at least two of its points lie on the plane. Figure 93 shows the Sum plane (axb). Straight l belongs to the Sum plane, since its points 1 and 2 belong to this plane.
If a line does not belong to the plane, it can be parallel to it or intersect it.
A line is parallel to a plane if it is parallel to another line lying in that plane. In Figure 93 there is a straight line m || Sum, since it is parallel to the line l belonging to this plane.
A straight line can intersect a plane at different angles and, in particular, be perpendicular to it. The construction of lines of intersection of a straight line and a plane is given in §61.
Figure 93 - A straight line belonging to a plane
A point in relation to the plane can be located in the following way: belong to it or not belong to it. A point belongs to a plane if it is located on a straight line located in this plane. Figure 94 shows a complex drawing of the Sum plane defined by two parallel lines l And P. There is a line in the plane m. Point A lies in the Sum plane, since it lies on the line m. Dot IN does not belong to the plane, since its second projection does not lie on the corresponding projections of the line.
Figure 94 - Complex drawing of a plane defined by two parallel lines
Conical and cylindrical surfaces
Conical surfaces include surfaces formed by the movement of a rectilinear generatrix l along a curved guide m. The peculiarity of the formation of a conical surface is that in this case one point of the generatrix is always motionless. This point is the vertex of the conical surface (Figure 95, A). The determinant of a conical surface includes the vertex S and guide m, wherein l"~S; l"^ m.
Cylindrical surfaces are those formed by a straight generatrix / moving along a curved guide T parallel to the given direction S(Figure 95, b). A cylindrical surface can be considered as a special case of a conical surface with a vertex at infinity S.
The determinant of a cylindrical surface consists of a guide T and directions S forming l, while l" || S; l"^m.
If the generators of a cylindrical surface are perpendicular to the projection plane, then such a surface is called projecting. In Figure 95, V a horizontally projecting cylindrical surface is shown.
On cylindrical and conical surfaces, given points are constructed using generatrices passing through them. Lines on surfaces, such as a line A to figure 95, V or horizontal h in figure 95, a, b, are constructed 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 movement in all its positions some spatial curve T, called return edge(Figure 96). The return edge completely defines the torso and is a geometric part of the surface determinant. The algorithmic part is the indication of the tangency of the generators to the cusp edge.
A conical surface is a special case of a torso, which has a return edge T degenerated into a point S- the top of the conical surface. A cylindrical surface is a special case of a torso, whose return edge is a point at infinity.
Figure 96 – Torso surface
Faceted surfaces
Faceted surfaces include surfaces formed by the movement of a rectilinear generatrix l along a broken guide m. Moreover, if one point S the generatrix is motionless, a pyramidal surface is created (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 a prismatic surface it is at infinity), face (part of the plane limited by one section of the guide m and the extreme positions of the generatrix relative to it l) and edge (line of intersection of adjacent faces).
The determinant of a pyramidal surface includes the vertex S, through which the generators and guides pass: l" ~ S; l^ T.
Determinant of a prismatic surface other than a guide T, contains direction S, to which all generators are parallel l surfaces: l||S; l^ t.
Figure 97 - Pyramid surface
Figure 98 - Prismatic surface
Closed faceted surfaces formed by a certain number (at least four) of faces are called polyhedra. From among the polyhedra, a group of regular polyhedra is distinguished, in which all faces are regular and congruent polygons, and the polyhedral angles at the vertices are convex and contain the same number of faces. For example: hexahedron - cube (Figure 99, A), tetrahedron - regular quadrilateral (Figure 99, 6) octahedron - polyhedron (Figure 99, V). Crystals have the shape of various polyhedra.
Figure 99 - Polyhedra
Pyramid- a polyhedron, the base of which is an arbitrary polygon, and the side faces are triangles with a common vertex S.
In a complex drawing, a pyramid is defined by projections of its vertices and edges, taking into account their visibility. The visibility of an edge is determined using competing points (Figure 100).
Figure 100 – Determining edge visibility 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 one. If the edges of a prism are perpendicular to any projection plane, then lateral surface it is called projecting. Figure 101 shows a comprehensive drawing of a right quadrangular prism with a horizontally projecting surface.
Figure 101 - Complex drawing of a right 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 need to be able to build points on the surface.
Any point on a faceted surface can be constructed using a generatrix passing through this point. In the figure there are 100 in the face ACS point built M using generatrix S-5.
Helical surfaces
Helical surfaces include surfaces created by the helical movement of a rectilinear generatrix. Ruled helical surfaces are called helicoids.
A straight helicoid is formed by the movement of a rectilinear generatrix i along two guides: helix T and its axes i; while forming l intersects the screw axis at a right angle (Figure 102, a). Straight helicoid is used to create spiral staircases, augers, as well as power threads in machine tools.
An inclined helicoid is formed by moving the generatrix along a screw guide T and its axes i so that the generator l crosses the axis i at a constant angle φ, different from a straight line, i.e. in any position the generatrix l parallel to one of the generatrices of the guide cone with an apex angle 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 rotating a line l around a straight line i , which is the axis of rotation. They can be linear, such as a cone or cylinder of revolution, and non-linear or curved, such as a sphere. The determinant of the surface of revolution includes the generatrix l and axis i . During rotation, each point of the generatrix describes a circle, the plane of which is perpendicular to the axis of rotation. Such circles of the surface of revolution are called parallels. The largest of the parallels is called equator. Equator determines the horizontal outline of the surface if i _|_ P 1 . In this case, the parallels are the horizontals of this surface.
Curves of a surface of revolution resulting from the intersection of the surface by planes passing through the axis of rotation are called meridians. All meridians of one surface are congruent. The frontal meridian is called the main meridian; it determines the frontal outline of the surface of rotation. The profile meridian determines the profile outline of the surface of rotation.
It is most convenient to construct a point on curved surfaces of revolution using surface parallels. There is 103 point in the figure M built on parallel h4.
Figure 103 – Constructing a point on a curved surface
Surfaces of revolution have found the widest application in technology. They limit the surfaces of most engineering parts.
A conical surface of revolution is formed by rotating a straight line i around the straight line intersecting with it - the axis i(Figure 104, A). Dot M on the surface is constructed 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 rotating a straight line l around an axis parallel to it 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, V). Point A on the surface of the sphere belongs to the prime meridian f, dot IN- equator h, a point 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 resulting circle, 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
Surfaces of revolution can also be formed by other second-order curves. Ellipsoid of rotation (Figure 106, A) formed by rotating an ellipse around one of its axes; paraboloid of revolution (Figure 106, b) - rotation of the parabola around its axis; single-sheet hyperboloid of revolution (Figure 106, V) is formed by rotating a hyperbola around an imaginary axis, and a two-sheet (Figure 106, G) - rotation of the hyperbola around the real axis.
Figure 106 – Formation of surfaces of revolution by second-order curves
In the general case, surfaces are depicted as not limited in the direction of propagation of the generating lines (see Figures 97, 98). For solutions specific tasks and receiving geometric shapes limited to the cutting planes. For example, to obtain a circular cylinder, it is necessary to limit a section of the cylindrical surface to the cutting planes (see Figure 104, b). As a result, we get its upper and lower bases. If the cutting planes are perpendicular to the axis of rotation, the cylinder will be straight; if not, the cylinder will be inclined.
To obtain a circular cone (see Figure 104, A), it is necessary to trim along the top and beyond. If the cutting plane of the base of the cylinder is perpendicular to the axis of rotation, the cone will be straight; if not, it will be inclined. If both cutting planes do not pass through the vertex, the cone will be truncated.
Using the cut plane, you can get a prism and a pyramid. For example, a hexagonal pyramid will be straight if all its edges have the same slope to the cutting plane. In other cases it will be slanted. If it is completed With using cutting planes and none of them passes through the vertex - the pyramid is truncated.
A prism (see Figure 101) can be obtained by limiting a section of the prismatic surface to two cutting planes. If the cutting plane is perpendicular to the edges of, for example, an octagonal prism, it is straight; if not perpendicular, it is inclined.
By choosing the appropriate position of the cutting planes, you can obtain different shapes of geometric figures depending on the conditions of the problem being solved.
Remote element.
remote element.
- a) have no common points;
Theorem.
Designation of cuts
GOST 2.305-2008 provides the following requirements for the designation of a section:
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. In case of a complex cut, strokes are also made at the intersection of the cutting planes with each other.
4. Arrows should be placed on the initial and final strokes indicating the direction of view; the arrows should be placed 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 starting and ending strokes should not intersect the contour of the corresponding image.
7. At the beginning and end of the section line, and, if necessary, at the intersection of the cutting planes, place the same capital letter of the Russian alphabet. The letters are placed near the arrows indicating the direction of view, and at the intersection points from the outer corner (Figure 24).
Figure 24 - Examples of section designation
8. The cut must be marked with an inscription like “AA” (always two letters separated by a dash).
9. When the secant 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, for horizontal, frontal and profile sections the position of the secant plane is not noted, and the incision is not accompanied by an inscription.
10. Frontal and profile sections, as a rule, are given a position corresponding to that accepted for a given item in the main image of the drawing.
11. Horizontal, frontal and profile sections can be located in place of the corresponding main views.
12. It is allowed to place the section anywhere in the drawing field, as well as with a rotation with the addition of a conventional graphic designation - the “Rotated” icon (Figure 25).
Figure 25 - Graphic symbol – “Rotated” icon
The designation of sections is similar designation of cuts and consists of traces of a secant plane and an arrow indicating the direction of view, as well as a letter placed on the outside of the arrow (Figure 1c, Figure 3). The offset 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 a gap between parts of the view. For a symmetrical superimposed section, the cutting plane is also not shown. If the section is asymmetrical and is 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 may be positioned with a rotation, providing the inscription above the section with the word “rotated”. For several identical sections related to one object, the section lines are designated with the same letter and one section is drawn. In cases where the section turns out to consist of separate parts, cuts should be used.
Straight general position
A straight line in general position (Fig. 2.2) is a straight line that is not parallel to any of the given projection planes. Any segment of such a straight line is projected distortedly in a given system of projection planes. The angles of inclination of this straight line to the projection planes are also projected distortedly.
Rice. 2.2.
Direct private provisions
Lines of particular position include 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
The horizontal 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 horizontal segment on the horizontal projection plane is projected to its true size. The true magnitude is projected onto this plane and the angle of inclination of the horizontal (straight line) to the frontal plane of projections. As an example, Fig. 2.3-a shows a visual image and a comprehensive horizontal drawing h, inclined to the plane P 2 at an angle b . 
Rice. 2.3-b
The frontal is the line parallel to the frontal plane of projections (Fig. 2.3-b). The horizontal projection of the front is always perpendicular to the communication lines. Any segment of the frontal onto the frontal plane of projections is projected to its true size. The true magnitude is projected onto this plane and the angle of inclination of the frontal (straight line) to the horizontal plane of projections (angle a). 
Rice. 2.3-v
A profile line is a line parallel to the profile plane of projections (Fig. 2.3-c). Horizontal and frontal projections of the profile line are parallel to the connection lines of these projections. Any segment of a profile line (straight line) is projected onto the profile plane to its true size. The angles of inclination of the profile straight line to the projection planes are projected onto the same plane in true magnitude. P 1 and P 2. When specifying a profile line in a complex drawing, you must 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 lines. There are three main projection lines: horizontal, frontal and profile projection lines. 
Rice. 2.3-g 
Rice. 2.3-d 
Rice. 2.3rd
A horizontally projecting straight line (Fig. 2.3-d) is a straight line perpendicular to the plane P 1 . Any segment of this line is projected onto the plane P P 1 - to the point.
The frontally projecting straight line (Fig. 2.H-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 on a plane P 2 - to the point.
A profile projecting straight line (Fig. 2.3-f) is a straight line perpendicular to the plane P 3, i.e. straight line parallel to the 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 on a plane 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. Fronts f - straight lines, located in the plane and parallel to the frontal plane of projections (f//P2) (Fig. 6.5).
Figure 6.5 Front
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 frontal 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 formed by this plane and the horizontal plane of projections (Fig. 6.7). Obviously, if a straight line does not have two common points with a plane, then it is either parallel to the plane or intersects it.
Figure 6.7 Line of greatest slope
Kinematic method of surface formation. Specifying a surface in a 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. During the formation of the surface, line 1 can remain unchanged or change its shape.
For clarity of the surface image in a complex drawing, it is advisable to specify the law of movement 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) line and additional conditions that clarify the law of movement 1.
The moving line 1 is called the generatrix, the fixed lines a, b, c are called the guides.
Let us consider the process of surface formation using the example shown in Fig. 3.1.
Here straight line 1 is taken as a generatrix. The law of movement of the generatrix is given by guide a and straight line b. This means that generatrix 1 slides along guide a, remaining parallel to straight line b all the time.
This method of surface formation is called kinematic. With its help, you can create and define various surfaces in the drawing. In particular, Fig. 3.1 shows the most general case of a cylindrical surface.
Rice. 3.1.
Another way to form a surface and depict it in a drawing is to specify the surface with 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 frame.
Depending on whether the surface frame is defined by points or lines, frames are divided into point and linear.
Figure 3.2 shows a surface frame consisting of two orthogonally located families of lines a1, a2, a3, ..., an and b1, b2, b3, ..., bn.
Rice. 3.2.
Conic sections.
CONIC SECTIONS, flat curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex (Fig. 1). From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. Except for 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 planets revolving around the Sun are shaped like ellipses. A circle is a special case of an ellipse in which the major axis is equal to the minor. 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 that use parabolic mirrors, as well as in radar antennas and special microphones with parabolic reflectors. A beam of parallel rays emanates from a light source placed at the focus of a parabolic reflector. That's why parabolic mirrors are used in high-power spotlights and car headlights. The hyperbola is a graph of many important physical relationships, such as Boyle's law (relating the pressure and volume of an ideal gas) and Ohm's law, which defines electric current as a function of resistance at a constant voltage.
EARLY HISTORY
The discoverer of conic sections is supposedly considered to be Menaechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menaechmus used a parabola and an equilateral 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 materials from them were included in the famous Conic Sections of Apollonius of Perga (c. 260–170 BC), which have survived to this day. Apollonius abandoned the requirement that the secant plane of the cone's generatrix be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We also owe the modern names of curves to Apollonius - ellipse, parabola and hyperbola.
In his constructions, Apollonius used a two-sheet 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. 1a) is formed when the cutting plane intersects all generatrices 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
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 given point and a given straight 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 method for constructing them using a stretched string.
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 point of a pencil sliding along a tightly stretched thread has the shape of an ellipse. Points F1 and F2 are called the focuses of the ellipse, and the segments V1V2 and v1v2 between the points of intersection of the ellipse with the coordinate axes are the major and minor axes. If points F1 and F2 coincide, then the ellipse turns into a circle.
rice. 2 Ellipsis
Hyperbola.
When constructing a hyperbola, point P, the tip of a pencil, is fixed on a thread, which slides freely along pegs installed at points F1 and F2, as shown in Fig. 3, a. The distances are selected so that segment PF2 is longer than segment PF1 by a fixed amount less than distance F1F2. In this case, one end of the thread passes under the pin F1 and both ends of the thread pass over the pin F2. (The point 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 point through it.) We draw one branch of the hyperbola (PV1Q), making sure that the thread remains taut at all times, and pulling both ends thread down past point F2, and when point P is below segment F1F2, holding the thread at both ends and carefully etching (i.e. releasing) it. We draw the second branch of the hyperbola (PўV2Qў), having previously swapped 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 asymptotes of the hyperbola, are constructed as shown in Fig. 3, b. The angular coefficients of these lines are equal to ± (v1v2)/(V1V2), where v1v2 is the bisector segment 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 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 construct this rectangle, you need to specify the location of points v1 and v2. They are at the same distance, equal
from the intersection point of the O axes. This formula assumes the construction right triangle with legs Ov1 and V2O and hypotenuse F2O.
If the asymptotes of a hyperbola are mutually perpendicular, then the hyperbola is called equilateral. Two hyperbolas that have 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 was apparently 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's position 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. Let's fasten 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. Having pulled the thread with the tip of a 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 equal to AB, the piece of thread is adjacent to the free leg of the triangle, and therefore the remaining piece of thread PF must be equal to the remaining parts of leg AB, i.e. PA. The point of intersection of V of the parabola with the axis is called the vertex of the parabola, the straight line passing through F and V is the axis of the parabola. If a straight line is drawn through the focus, perpendicular to the axis, 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 determined similarly.
ANSWERS TO TICKETS: No. 1 (not completely), 2 (not completely), 3 (not completely), 4, 5, 6, 7, 12, 13, 14 (not completely), 16, 17, 18, 20, 21 , 22, 23, 26,
Remote element.
When making drawings, in some cases it becomes necessary to construct an additional separate image of any part of an object that requires explanation regarding the shape, size or other data. This image is called remote element. It is usually performed enlarged. The detail can be laid out as a view or as a section.
When constructing a callout element, the corresponding place of the main image is marked with a closed solid thin line, usually an oval or a circle, and is designated with a capital letter of the Russian alphabet on the shelf of the leader line. A type A (5:1) entry is made for the remote element. In Fig. 191 shows an example of the implementation of a remote element. It is placed as close as possible to the corresponding place in the image of the object.
1. 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 rectangular projection, the projection of a segment, if it is not parallel to the projection plane, is always smaller 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 a leg: А "В" = 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.
The relative position of a straight line and a plane.
A straight line and a plane in space can:
- a) have no common points;
- b) have exactly one common point;
- c) have at least two common points.
In Fig. 30 depicts all these possibilities.
In case a) line b is parallel to the plane: b || .
In case b) straight line l intersects the plane at one point O; l = O.
In case c) straight line a belongs to the plane: a or a.
Theorem. If line b is parallel to at least one line a belonging to the plane, then the line is parallel to the plane.
Suppose that line m intersects the plane at point Q. If m is perpendicular to every line of the plane passing through point Q, then line m is said to be perpendicular to the plane.
Tram rails illustrate that straight lines belong to the plane of the earth. Power lines are parallel to the plane of the earth, and tree trunks are examples of straight lines crossing the surface of the earth, some perpendicular to the plane of the earth, others not perpendicular (oblique).
Location
Sign: if a line that does not lie in a given plane is parallel to some line that lies 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 a given one, then the other line is either also parallel to a given plane or lies in this plane.
MUTUAL POSITION OF PLANES. PARALLELITY OF 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 straight lines of 1 plane are respectively parallel to 2 straight lines of another plane, then these planes are parallel.
Holy
1. if 2 parallel planes are intersected 3, then the lines of their intersection are parallel
2. segments of parallel lines contained between parallel planes are equal.
PERPENDICULARITY OF STRAIGHT AND PLANE. SIGN OF PERPENDICULARITY OF STRAIGHT AND PLANE.
Direct names perpendicular, if they intersect under<90.
Lemma: If 1 of 2 parallel lines is perpendicular to the 3rd line, then the other line is perpendicular to this line.
A straight line is said to be perpendicular to a plane, if it is perpendicular to any line in this plane.
Theorem: If 1 of 2 parallel lines is perpendicular to a plane, then the other line is perpendicular to this 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 this plane.
PERPENDICULAR AND OBLIQUE
Let's construct a plane and so on, not belonging to the plane. Their t.A we will draw a straight line, perpendicular to the plane. The point of intersection of the straight line with the plane is designated H. The segment AN is a perpendicular drawn from point A to the plane. T.N – base of the perpendicular. Let us take in the plane t.M, which does not coincide with H. The segment AM is inclined, drawn from t.A to the plane. M – inclined base. Segment MH is a projection of an inclined plane onto a plane. Perpendicular AN - the distance from t.A to the plane. Any distance is part of a perpendicular.
Theorem of 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 itself.
ANGLE BETWEEN A STRAIGHT AND A PLANE
The angle between a straight line and A plane is the angle between this line and its projection on the plane.
DIHEDRAL ANGLE. ANGLE BETWEEN PLANES
Dihedral angle called a figure formed by a straight line and 2 half-planes with a common boundary a, not belonging to the same plane.
Boundary a – edge of a dihedral angle. Half planes – dihedral angle faces. To measure the dihedral angle. You need to construct a linear angle inside it. Let's mark some point on the edge of the dihedral angle and draw a ray from this point at each face, perpendicular to the edge. The angle formed by these rays is called linear dihedral angle. There can be an infinite number of them inside a dihedral angle. They all have the same size.
PERPENDICULARITY OF TWO PLANES
Two intersecting planes are called perpendicular, if the angle between them is 90.
Sign:
If 1 of 2 planes passes through a line perpendicular to another plane, then such planes are perpendicular.
POLYhedra
Polyhedron– a surface composed of polygons and bounding a certain geometric body. Edges– polygons from which polyhedra are made. Ribs– sides of faces. Peaks- ends of ribs. Diagonal of a polyhedron called 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 is called . cutting plane. The common part of the polyhedron and the secant area is called cross section of a polyhedron. Polyhedra can be convex or concave. The polyhedron is called 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 vertex is less than 360.
PRISM
A polyhedron composed of 2 equal polygons located in parallel planes and n - parallelograms is called prism.
Polygons A1A2..A(p) and B1B2..B(p) – prism base. А1А2В2В1…- parallelograms, A(p)A1B1B(p) – side edges. Segments A1B1, A2B2..A(p)B(p) – lateral ribs. Depending on the polygon underlying the prism, the prism called p-coal. A perpendicular drawn from any point of one base to the plane of another base is called height. If the lateral edges of the prism are perpendicular to the base, then the prism – straight, and if not perpendicular – it's slanted. The height of a straight prism is equal to the length of its side edge. Direct prism is correct, if its base is regular polygons, all side faces are equal rectangles.
PARALLEPIPED
ABCD//A1B1S1D1, AA1//BB1//CC1//DD1, AA1=BB1=CC1=DD1 (according to the nature of parallel planes)
A parallelepiped consists of 6 parallelograms. Parallelograms are called edges. ABCD and А1В1С1Д1 are the bases, the remaining faces are called lateral. Points A B C D A1 B1 C1 D1 – tops. Line segments connecting vertices - ribs AA1, BB1, SS1, DD1 – lateral ribs.
The diagonal of the parallelepiped is called a segment connecting 2 vertices that do not belong to 1 face.
Saints
1. The opposite faces of the parallelepiped are parallel and equal. 2. The diagonals of the parallelepiped intersect at one point and are bisected by this point.
PYRAMID
Consider the polygon A1A2..A(n), a point P that does not lie in the plane of this polygon. Let's connect point P with the vertices of the polygon and get n triangles: RA1A2, RA2A3....RA(p)A1.
Polyhedron composed of n-gon and n-triangles called a pyramid. Polygon – foundation. Triangles – side edges. R - top of the pyramid. Segments A1P, A2P..A(p)P – lateral ribs. Depending on the polygon lying at the base, the pyramid is called p-coal. Pyramid height called a perpendicular drawn from the top to the plane of the base. The pyramid is called correct, if its base contains a regular polygon and its height falls in the center of the base. Apothem– the height of the side face of a regular pyramid.
TRUNCATED PYRAMID
Consider the pyramid PA1A2A3A(n). Let's 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 lateral surface consists of a trapezoid. 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 POLYHEDES
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 the cube. All its faces are equal squares, and 3 edges meet at each vertex.
Regular tetrahedron composed of 4 equilateral triangles. Each vertex is the vertex of 3 triangles. The sum of the plane angles at each vertex is 180.
Regular octahedron composed of 8 equilateral triangles. Each vertex is the vertex of 4 triangles. Sum of plane angles at each vertex = 240
Regular icosahedron composed of 20 equilateral triangles. Each vertex is a vertex 5 triangle. The sum of plane angles at each vertex is 300.
Cube composed of 6 squares. Each vertex is the vertex of 3 squares. The sum of plane angles at each vertex = 270.
Regular dodecahedron composed of 12 regular pentagons. Each vertex is the vertex of 3 regular pentagons. The sum of plane 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 is called cylinder. Circles L and L1 are called the bases of the cylinder. Segments MM1, AA1 – formative. Forming a cylindrical or lateral surface of a cylinder. Straight line connecting the centers of the bases O and O1 axis of the cylinder. Generator length – cylinder height. Base radius (r) – radius of the cylinder.
Cylinder sections
Axial passes through the axis and diameter of the base
Perpendicular to axis
A cylinder is a body of revolution. It is obtained by rotating the rectangle around one of its sides.
CONE
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 etc. we will draw segments; there are infinitely many of them. They form a conical surface and are called formative.
R- vertex, OR – axis of conical surface.
A body bounded by a conical surface and a circle with boundary L called a cone. Circle - base of the cone. Top of the conical surface - the top of the cone. Forming a conical surface - forming a cone. Conical surface – lateral surface of the cone. RO – cone axis. Distance from P to O – cone height. A cone is a body of revolution. It is obtained by rotating a right triangle around a 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 center of the sphere. This distance is radius of the sphere.
A segment connecting 2 points of a sphere and passing through its center called the diameter of the sphere.
A body bounded by a sphere called ball. The center, radius and diameter of the sphere are called center, radius and diameter of the ball.
A sphere and a ball are bodies of rotation. 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)
Direct can belong to the plane, be her parallel or cross plane. A line belongs to a plane if two points belonging to the line and the plane have the same elevations. The corollary that follows from what has been said: a point belongs to a plane if it belongs to a line lying in this plane.
A line is parallel to a plane if it is parallel to a line lying in this plane.
A straight line intersecting a plane. To find the point of intersection of a straight line with a plane, it is necessary (Fig. 3.28):
1) draw an auxiliary plane through a given straight line m T;
2) build a line n intersection of a given plane Σ with an auxiliary plane T;
3) mark the intersection point R, given straight line m with the line of intersection n.
Consider the problem (Fig. 3.29). The straight line m is defined on the plan by a point A 6 and an inclination angle of 35°. An auxiliary vertical plane is drawn through this line T, which intersects the plane Σ along the line n (B 2 C 3). Thus, one moves from the relative position of a straight line and a plane to the relative position of two straight lines lying in the same vertical plane. This problem is solved by constructing profiles of these straight lines. Intersection of lines m And n on the profile determines the desired point R. Point elevation R determined by the vertical scale scale.
Straight line perpendicular to the plane. A straight line is perpendicular to a plane if it is perpendicular to any two intersecting lines of this plane. Figure 3.30 shows a straight line m, perpendicular to the plane Σ and intersecting it at point A. On the plan, the projection of the line m and the horizontal planes are mutually perpendicular (a right angle, one side of which is parallel to the projection plane, is projected without distortion. Both lines lie in the same vertical plane, therefore the positions of such lines are inverse in magnitude to each other: l m = l/l u. But l uΣ = lΣ, then l m = l/lΣ, that is, the position of the straight line m is inversely proportional to the position of the plane. The falls of a straight line and a plane are directed in different directions.
3.4. Projections with numerical marks. Surfaces
3.4.1.Polyhedra and curved surfaces. Topographic surface
In nature, many substances have a crystalline structure in the form of polyhedra. A polyhedron is a collection of flat polygons that do not lie in the same plane, where each side of one of them is also a side of the other. When depicting a polyhedron, it is enough to indicate the projections of its vertices, connecting them in a certain order with straight lines - projections of the edges. In this case, it is necessary to indicate visible and invisible edges in the drawing. In Fig. Figure 3.31 shows a prism and a pyramid, as well as finding the marks of points belonging to these surfaces.
A special group of convex polygons is the group of regular polygons in which all faces are equal regular polygons and all polygonal angles are equal. There are five types of regular polygons.
Tetrahedron- a regular quadrilateral, bounded by equilateral triangles, has 4 vertices and 6 edges (Fig. 3.32 a).
Hexahedron- regular hexagon (cube) - 8 vertices, 12 edges (Fig. 3.32b).
Octahedron- a regular octahedron, bounded by eight equilateral triangles - 6 vertices, 12 edges (Fig. 3.32c).
Dodecahedron- a regular dodecahedron, bounded by twelve regular pentagons, connected by three near each vertex.
It has 20 vertices and 30 edges (Fig. 3.32 d).
Icosahedron- a regular twenty-sided triangle, bounded by twenty equilateral triangles, connected by five near each vertex. 12 vertices and 30 edges (Fig. 3.32 d).
When constructing a point lying on the face of a polyhedron, it is necessary to draw a straight line belonging to this face and mark the projection of the point on its projection.
Conical surfaces are formed by moving a rectilinear generatrix along a curved guide so that in all positions the generatrix passes through a fixed point - the vertex of the surface. General conical surfaces on the plan are represented by a horizontal line and a vertex. In Fig. Figure 3.33 shows the location of a point mark on the surface of a conical surface.
A straight circular cone is represented by a series of concentric circles drawn at equal intervals (Fig. 3.34a). Elliptical cone with a circular base - a series of eccentric circles (Fig. 3.34 b)
Spherical surfaces. A spherical surface is classified as a surface of revolution. It is formed by rotating a circle around its diameter. On the plan, a spherical surface is defined by the center TO and the projection of one of its horizontal lines (the equator of the sphere) (Fig. 3.35).
Topographic surface. A topographic surface is classified as a geometrically irregular surface, since it does not have a geometric law of formation. To characterize a surface, determine the position of its characteristic points relative to the projection plane. In Fig. 3.3 b a gives an example of a section of a topographic surface, which shows the projections of its individual points. Although such a plan makes it possible to get an idea of the shape of the depicted surface, it is not very clear. To give the drawing greater clarity and thereby make it easier to read, projections of points with identical marks are connected by smooth curved lines, which are called horizontals (isolines) (Fig. 3.36 b).
The horizontal lines of a topographic surface are sometimes defined as the lines of intersection of this surface with horizontal planes spaced from each other at the same distance (Fig. 3.37). The difference in elevations between two adjacent horizontal lines is called the section height.
The smaller the difference in elevations between two adjacent horizontal lines, the more accurate the image of a topographic surface is. On plans, contour lines are closed within the drawing or outside it. On steeper slopes, the surface projections of the contour lines come closer together; on flat slopes, their projections diverge.
The shortest distance between the projections of two adjacent horizontal lines on the plan is called the lay. In Fig. 3.38 through point A several straight line segments are drawn on the topographic surface AND YOU And AD. They all have different angles of incidence. The segment has the greatest angle of incidence AC, the location of which is of minimal importance. Therefore, it will be a projection of the line of incidence of the surface at a given location.
In Fig. 3.39 shows an example of constructing a projection of the line of incidence through a given point A. From point A 100, as if from the center, draw an arc of a circle touching the nearest horizontal line at the point At 90. Dot At 90, horizontal h 90, will belong to the fall line. From point At 90 draw an arc tangent to the next horizontal line at the point From 80, etc. From the drawing it is clear that the line of incidence of the topographic surface is a broken line, each link of which is perpendicular to the horizontal, passing through the lower end of the link, which has a lower elevation.
3.4.2.Intersection of a conical surface with a plane
If a cutting plane passes through the vertex of a conical surface, then it intersects it along straight lines forming the surface. In all other cases, the section line will be a flat curve: a circle, an ellipse, etc. Let us consider the case of a conical surface intersecting a plane.
Example 1. Construct the projection of the intersection line of a circular cone Φ( h o , S 5) with a plane Ω parallel to the generatrix of the conical surface.
A conical surface with a given plane location intersects along a parabola. Having interpolated the generatrix t we build horizontal lines of a circular cone - concentric circles with a center S 5 . Then we determine the intersection points of the same horizontals of the plane and the cone (Fig. 3.40).
3.4.3. Intersection of a topographic surface with a plane and a straight line
The case of the intersection of a topographic surface with a plane is most often encountered in solving geological problems. In Fig. 3.41 gives an example of constructing the intersection of a topographic surface with the plane Σ. The curve I'm looking for m are determined by the intersection points of the same horizontal planes and the topographic surface.
In Fig. 3.42 gives an example of constructing a true view of a topographic surface with a vertical plane Σ. The required line m is determined by points A, B, C... intersection of the horizontals of the topographic surface with the cutting plane Σ. On the plan, the projection of the curve degenerates into a straight line coinciding with the projection of the plane: m≡ Σ. The profile of the curve m is constructed taking into account the location of the projections of its points on the plan, as well as their elevations.
3.4.4. Surface of equal slope
A surface of equal slope is a ruled surface, all straight lines of which make a constant angle with the horizontal plane. Such a surface can be obtained by moving a straight circular cone with an axis perpendicular to the plane of the plan, so that its top slides along a certain guide, and the axis remains vertical in any position.
In Fig. Figure 3.43 shows a surface of equal slope (i=1/2), the guide of which is a spatial curve A, B, C, D.
Graduation of the plane. As examples, consider the slope planes of the roadway.
Example 1. Longitudinal slope of the roadway i=0, slope of the embankment i n =1:1.5, (Fig. 3.44a). It is required to draw horizontal lines every 1 m. The solution comes down to the following. We draw the scale of the slope of the plane perpendicular to the edge of the roadway, mark points at a distance equal to an interval of 1.5 m taken from the linear scale, and determine marks 49, 48 and 47. Through the obtained points we draw the contours of the slope parallel to the edge of the road.
Example 2. Longitudinal slope of the road i≠0, slope of the embankment i n =1:1.5, (Fig. 3.44b). The plane of the roadway is graded. The slope of the roadway is graded as follows. At the point with the vertex 50.00 (or another point) we place the vertex of the cone, describe a circle with a radius equal to the interval of the embankment slope (in our example l= 1.5m). The elevation of this horizontal line of the cone will be one less than the elevation of the vertex, i.e. 49m. We draw a series of circles, we get horizontal marks 48, 47, tangent to which from the edge points with marks 49, 48, 47 we draw horizontals of the embankment slope.
Graduation of surfaces.
Example 3. If the longitudinal slope of the road is i = 0 and the slope of the embankment is i n = 1: 1.5, then the contour lines of the slopes are drawn through the points of the slope scale, the interval of which is equal to the interval of the embankment slopes (Fig. 3.45a). The distance between two projections of adjacent horizontal lines in the direction of the general norm (slope scale) is the same everywhere.
Example 4. If the longitudinal slope of the road is i≠0, and the slope of the embankment is i n =1:1.5, (Fig. 3.45b), then the contour lines are constructed in the same way, except that the slope contours are drawn not in straight lines, but in curves.
3.4.5. Determination of the excavation limit line
Since most soils are unable to maintain vertical walls, slopes (artificial structures) have to be built. The slope imparted by a slope depends on the soil.
In order to give a section of the earth's surface the appearance of a plane with a certain slope, you need to know the line of limits for excavation and excavation work. This line, limiting the planned area, is represented by the lines of intersection of the slopes of embankments and excavations with a given topographic surface.
Since every surface (including flat ones) is depicted using contours, the line of intersection of surfaces is constructed as a set of intersection points of contours with the same marks. Let's look at examples.
Example 1. In Fig. 3.46 shows an earthen structure in the shape of a truncated quadrangular pyramid, standing on a plane N. Upper base ABCD pyramid has a mark 4m and side sizes 2×2.5 m. The side faces (embankment slopes) have a slope of 2:1 and 1:1, the direction of which is shown by arrows.
It is necessary to construct a line of intersection of the slopes of the structure with the plane N and among themselves, as well as construct a longitudinal profile along the axis of symmetry.
First, a diagram of slopes, intervals and scales of deposits, and given slopes is constructed. Perpendicular to each side of the site, the scales of the slopes are drawn at specified intervals, after which the projections of the contour lines with the same marks of adjacent faces are the intersection lines of the slopes, which are projections of the side edges of this pyramid.
The lower base of the pyramid coincides with the zero horizontal slopes. If this earthen structure is crossed by a vertical plane Q, in cross-section you will get a broken line - the longitudinal profile of the structure.
Example 2. Construct a line of intersection of the pit slopes with a flat slope and with each other. Bottom ( ABCD) the pit is a rectangular area with an elevation of 10 m and dimensions of 3x4 m. The axis of the site makes an angle of 5° with the south-north line. The slopes of the excavations have the same slopes of 2:1 (Fig. 3.47).
The line of zero works is established according to the site plan. It is constructed at the points of intersection of the same-named projections of the horizontal lines of the surfaces under consideration. At the points of intersection of the contours of the slopes and the topographic surface with the same marks, the line of intersection of the slopes is found, which are projections of the side edges of a given pit.
In this case, the side slopes of the excavations are adjacent to the bottom of the pit. Line abcd– the desired intersection line. Aa, Bb, Cs, Dd– the edges of the pit, the lines of intersection of the slopes with each other.
4. Questions for self-control and tasks for independent work on the topic “Rectangular projections”
Dot
4.1.1. The essence of the projection method.
4.1.2. What is point projection?
4.1.3. What are projection planes called and designated?
4.1.4. What are projection connection lines in a drawing and how are they located in the drawing in relation to the projection axes?
4.1.5. How to construct the third (profile) projection of a point?
4.1.6. Construct three projections of points A, B, C on a three-picture drawing, write down their coordinates and fill out the table.
4.1.7. Construct the missing projection axes, x A =25, y A =20. Construct a profile projection of point A.
4.1.8. Construct three projections of points according to their coordinates: A(25,20,15), B(20,25,0) and C(35,0,10). Indicate the position of the points in relation to the planes and axes of projections. Which point is closer to the P3 plane?
4.1.9. Material points A and B begin to fall simultaneously. What position will point B be in when point A touches the ground? Determine the visibility of points. Plot points in new position.
4.1.10. Construct three projections of point A, if the point lies in the P 3 plane, and the distance from it to the P 1 plane is 20 mm, to the P 2 plane - 30 mm. Write down the coordinates of the point.
Straight
4.2.1. How can a straight line be defined in a drawing?
4.2.2. Which line is called a line in general position?
4.2.3. What position can a straight line occupy relative to the projection planes?
4.2.4. In what case does the projection of a straight line turn to a point?
4.2.5. What is characteristic of a complex straight level drawing?
4.2.6. Determine the relative position of these lines.
a…b a…b a…b
4.2.7. Construct projections of a straight line segment AB with a length of 20 mm, parallel to the planes: a) P 2; b) P 1; c) Ox axis. Indicate the angles of inclination of the segment to the projection planes.
4.2.8. Construct projections of segment AB using the coordinates of its ends: A(30,10,10), B(10,15,30). Construct projections of point C dividing the segment in the ratio AC:CB = 1:2.
4.2.9. Determine and record the number of edges of this polyhedron and their position relative to the projection planes.
4.2.10. Through point A, draw a horizontal and a frontal line intersecting straight line m.
4.2.11. Determine the distance between line b and point A
4.2.12. Construct projections of a segment AB with a length of 20 mm, passing through point A and perpendicular to plane a) P 2; b) P 1; c) P 3.
Stereometry
Mutual arrangement of straight lines and planes
In space
Parallelism of lines and planes
Two lines in space are called parallel , if they lie in the same plane and do not intersect.
A straight line and a plane are called parallel , if they do not intersect.
The two planes are called parallel , if they do not intersect.
Lines that do not intersect and do not lie in the same plane are called interbreeding .
Sign of parallelism between a line and a plane. If a line that does not belong to a plane is parallel to some line in this plane, then it is parallel to the plane itself.
Sign of parallel planes. If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel.
Sign of crossing lines. If one of two lines lies in a plane, and the other intersects this plane at a point not belonging to the first line, then these lines intersect.
Theorems on parallel lines and parallel planes.
1. Two lines parallel to a third line are parallel.
2. If one of two parallel lines intersects a plane, then the other line also intersects this plane.
3. Through a point outside a given line, you can draw a line parallel to the given one, and only one.
4. If a line is parallel to each of two intersecting planes, then it is parallel to their line of intersection.
5. If two parallel planes are intersected by a third plane, then the lines of intersection are parallel.
6. Through a point not lying in a given plane, you can draw a plane parallel to the given one, and only one.
7. Two planes parallel to the third are parallel to each other.
8. Segments of parallel lines contained between parallel planes are equal.
Angles between straight lines and planes
The angle between a straight line and a plane the angle between a straight line and its projection onto a plane is called (the angle in Fig. 1).
Angle between intersecting lines is the angle between intersecting lines parallel to the given intersecting lines.
Dihedral angle is a figure formed by two half-planes with a common line. Half-planes are called edges , straight – edge dihedral angle.
Linear angle dihedral angle is the angle between half-lines belonging to the faces of the dihedral angle, emanating from one point on the edge and perpendicular to the edge (the angle in Fig. 2).
The degree (radian) measure of a dihedral angle is equal to the degree (radian) measure of its linear angle.
Perpendicularity of lines and planes
Two straight lines are called perpendicular if they intersect at right angles.
A straight line intersecting a plane is called perpendicular this plane if it is perpendicular to any line in the plane passing through the point of intersection of this line and the plane.
The two planes are called perpendicular , if intersecting, they form right dihedral angles.
Sign of perpendicularity of a line and a plane. If a line intersecting a plane is perpendicular to two intersecting lines in this plane, then it is perpendicular to the plane.
Sign of perpendicularity of two planes. If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.
Theorems on perpendicular lines and planes.
1. If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
2. If two lines are perpendicular to the same plane, then they are parallel.
3. If a line is perpendicular to one of two parallel planes, then it is also perpendicular to the other.
4. If two planes are perpendicular to the same line, then they are parallel.
Perpendicular and oblique
Theorem. If a perpendicular and inclined lines are drawn from one point outside the plane, then:
1) oblique ones having equal projections are equal;
2) of the two inclined ones, the one whose projection is larger is greater;
3) equal obliques have equal projections;
4) of the two projections, the one that corresponds to the larger oblique one is larger.
Three Perpendicular Theorem. In order for a straight line lying in a plane to be perpendicular to an inclined one, it is necessary and sufficient that this straight line be perpendicular to the projection of the inclined one (Fig. 3).
Theorem on the area of the orthogonal projection of a polygon onto a plane. The area of the orthogonal projection of a polygon onto a plane is equal to the product of the area of the polygon and the cosine of the angle between the plane of the polygon and the projection plane.
Construction.
1. On a plane a we conduct a direct A.
3. In plane b through the point A let's make a direct b, parallel to the line A.
4. A straight line has been built b parallel to the plane a.
Proof. Based on the parallelism of a straight line and a plane, a straight line b parallel to the plane a, since it is parallel to the line A, belonging to the plane a.
Study. The problem has an infinite number of solutions, since the straight line A in the plane a is chosen randomly.
Example 2. Determine at what distance from the plane the point is located A, if straight AB intersects the plane at an angle of 45º, the distance from the point A to the point IN belonging to the plane is equal to cm?
Solution. Let's make a drawing (Fig. 5):
AC– perpendicular to the plane a, AB– inclined, angle ABC– angle between straight line AB and plane a. Triangle ABC– rectangular because AC– perpendicular. The required distance from the point A to the plane - this is the leg AC right triangle. Knowing the angle and hypotenuse cm, we will find the leg AC:
Answer: 3 cm.
Example 3. Determine at what distance from the plane of an isosceles triangle is a point located 13 cm from each of the vertices of the triangle if the base and height of the triangle are equal to 8 cm?
Solution. Let's make a drawing (Fig. 6). Dot S away from the points A, IN And WITH at the same distance. So, inclined S.A., S.B. And S.C. equal, SO– the common perpendicular of these inclined ones. By the theorem of obliques and projections AO = VO = CO.
Dot ABOUT– the center of a circle circumscribed about a triangle ABC. Let's find its radius:
Where Sun– base;
AD– the height of a given isosceles triangle.
Finding the sides of a triangle ABC from a right triangle ABD according to the Pythagorean theorem:
Now we find OB:
Consider a triangle SOB: S.B.= 13 cm, OB= = 5 cm. Find the length of the perpendicular SO according to the Pythagorean theorem:
Answer: 12 cm.
Example 4. Given parallel planes a And b. Through the point M, which does not belong to any of them, straight lines are drawn A And b that cross a at points A 1 and IN 1 and the plane b– at points A 2 and IN 2. Find A 1 IN 1 if it is known that MA 1 = 8 cm, A 1 A 2 = 12 cm, A 2 IN 2 = 25 cm.
Solution. Since the condition does not say how the point is located relative to both planes M, then two options are possible: (Fig. 7, a) and (Fig. 7, b). Let's look at each of them. Two intersecting lines A And b define a plane. This plane intersects two parallel planes a And b along parallel lines A 1 IN 1 and A 2 IN 2 according to Theorem 5 about parallel lines and parallel planes.
Triangles MA 1 IN 1 and MA 2 IN 2 are similar (angles A 2 MV 2 and A 1 MV 1 – vertical, corners MA 1 IN 1 and MA 2 IN 2 – internal crosswise lying with parallel lines A 1 IN 1 and A 2 IN 2 and secant A 1 A 2). From the similarity of triangles follows the proportionality of the sides:
Option a):
Option b):
Answer: 10 cm and 50 cm.
Example 5. Through the point A plane g a direct line was drawn AB, forming an angle with the plane a. Via direct AB a plane is drawn r, forming with a plane g corner b. Find the angle between the projection of a straight line AB to the plane g and plane r.
Solution. Let's make a drawing (Fig. 8). From point IN drop the perpendicular to the plane g. Linear dihedral angle between planes g And r- this is a right angle AD DBC, based on the perpendicularity of a line and a plane, as well as Based on the perpendicularity of planes, a plane r perpendicular to the plane of the triangle DBC, since it passes through the line AD. We construct the desired angle by dropping the perpendicular from the point WITH to the plane r, let's denote it Find the sine of this angle of a right triangle MYSELF. Let us introduce an auxiliary segment a = BC. From a triangle ABC: From a triangle Navy we'll find
Then the required angle
Answer:
Tasks for independent solution
I level
1.1. Through a point, draw a line perpendicular to two given intersecting lines.
1.2. Determine how many different planes can be drawn:
1) through three different points;
2) through four different points, no three of which lie on the same plane?
1.3. Through the vertices of the triangle ABC lying in one of two parallel planes, parallel lines are drawn intersecting the second plane at points A 1 , IN 1 , WITH 1 . Prove the equality of triangles ABC And A 1 IN 1 WITH 1 .
1.4. From the top A rectangle ABCD perpendicular restored AM to its plane.
1) prove that triangles MBC And MDC– rectangular;
2) indicate among the segments M.B., M.C., M.D. And M.A. segment of the greatest and shortest length.
1.5. The faces of one dihedral angle are correspondingly parallel to the faces of the other. Determine the relationship between the values of these dihedral angles.
1.6. Find the value of the dihedral angle if the distance from a point taken on one face to the edge is 2 times greater than the distance from the point to the plane of the second face.
1.7. From a point separated from the plane by a distance, two equal inclined slopes are drawn, forming an angle of 60º. Oblique projections are mutually perpendicular. Find the lengths of the inclined ones.
1.8. From the top IN square ABCD perpendicular restored BE to the plane of the square. Angle of inclination of the triangle plane ACE to the plane of the square is equal j, the side of the square is A ACE.
Level II
2.1. Through a point that does not belong to one of the two intersecting lines, draw a line intersecting both given lines.
2.2. Parallel lines A, b And With do not lie in the same plane. Through the point A on a straight line A perpendiculars to straight lines are drawn b And With, intersecting them at the points respectively IN And WITH. Prove that the line Sun perpendicular to straight lines b And With.
2.3. Through the top A right triangle ABC a plane is drawn parallel to Sun. Legs of a triangle AC= 20 cm, Sun= 15 cm. The projection of one of the legs onto the plane is 12 cm. Find the projection of the hypotenuse.
2.4. In one of the faces of the dihedral angle equal to 30º there is a point M. The distance from it to the edge of the corner is 18 cm. Find the distance from the projection of the point M to the second face to the first face.
2.5. Ends of the segment AB belong to the faces of a dihedral angle equal to 90º. Distance from points A And IN to the edge are equal respectively AA 1 = 3 cm, BB 1 = 6 cm, distance between points on the edge Find the length of the segment AB.
2.6. From a point located at a distance from the plane A, two inclined ones are drawn, forming angles of 45º and 30º with the plane, and an angle of 90º between themselves. Find the distance between the bases of the inclined ones.
2.7. The sides of the triangle are 15 cm, 21 cm and 24 cm. Point M removed from the plane of the triangle by 73 cm and located at the same distance from its vertices. Find this distance.
2.8. From the center ABOUT circle inscribed in a triangle ABC, a perpendicular is restored to the plane of the triangle OM. Find the distance from the point M to the sides of the triangle, if AB = BC = 10 cm, AC= 12 cm, OM= 4 cm.
2.9. Distances from point M to the sides and vertex of the right angle are 4 cm, 7 cm and 8 cm respectively. Find the distance from the point M to the plane of a right angle.
2.10. Through the base AB isosceles triangle ABC the plane is drawn at an angle b to the plane of the triangle. Vertex WITH removed from the plane by a distance A. Find the area of the triangle ABC, if the base AB of an isosceles triangle is equal to its height.
Level III
3.1. Rectangle Layout ABCD with the parties A And b bent diagonally BD so that the planes of the triangles BAD And BCD became mutually perpendicular. Find the length of the segment AC.
3.2. Two rectangular trapezoids with angles of 60º lie in perpendicular planes and have a larger common base. The larger sides are 4 cm and 8 cm. Find the distance between the vertices of the straight lines and the vertices of the obtuse angles of the trapezoids if the vertices of their acute angles coincide.
3.3.Cube given ABCDA 1 B 1 C 1 D 1 . Find the angle between the straight line CD 1 and plane BDC 1 .
3.4. On the edge AB Cuba ABCDA 1 B 1 C 1 D 1 point taken R- the middle of this rib. Construct a section of the cube with a plane passing through the points C 1 P.D. and find the area of this section if the edge of the cube is equal to A.
3.5. Through the side AD rectangle ABCD a plane is drawn a so that the diagonal BD makes an angle of 30º with this plane. Find the angle between the plane of the rectangle and the plane a, If AB = A, AD = b. Determine at what ratio A And b the problem has a solution.
3.6. Find the locus of points equidistant from the lines defined by the sides of the triangle.
Prism. Parallelepiped
Prism is a polyhedron whose two faces are equal n-gons (bases) , lying in parallel planes, and the remaining n faces are parallelograms (side faces) . Lateral rib The side of a prism that does not belong to the base is called the side of the prism.
A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called inclined . Correct A prism is a right prism whose bases are regular polygons.
Height prism is the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. Diagonal section is called a section of a prism by a plane passing through two lateral edges that do not belong to the same face. Perpendicular section is called a section of a prism by a plane perpendicular to the side edge of the prism.
Lateral surface area of a prism is the sum of the areas of all lateral faces. Total surface area is called the sum of the areas of all faces of the prism (i.e. the sum of the areas of the side faces and the areas of the bases).
For an arbitrary prism the following formulas are true::
Where l– length of the side rib;
H- height;
P
Q
S side
S full
S base– area of the bases;
V– volume of the prism.
For a straight prism the following formulas are correct:
Where p– base perimeter;
l– length of the side rib;
H- height.
parallelepiped called a prism whose base is a parallelogram. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called inclined . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped with all edges equal is called cube
The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since a parallelepiped is a prism, its main elements are defined in the same way as they are defined for prisms.
Theorems.
1. The diagonals of a parallelepiped intersect at one point and bisect it.
2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the squares of its three dimensions:
3. All four diagonals of a rectangular parallelepiped are equal to each other.
For an arbitrary parallelepiped the following formulas are valid:
Where l– length of the side rib;
H- height;
P– perpendicular section perimeter;
Q– Perpendicular cross-sectional area;
S side– lateral surface area;
S full– total surface area;
S base– area of the bases;
V– volume of the prism.
For a right parallelepiped the following formulas are correct:
Where p– base perimeter;
l– length of the side rib;
H– height of a right parallelepiped.
For a rectangular parallelepiped the following formulas are correct:
Where p– base perimeter;
H- height;
d– diagonal;
a,b,c– measurements of a parallelepiped.
The following formulas are correct for a cube:
Where a– rib length;
d- diagonal of the cube.
Example 1. The diagonal of a rectangular parallelepiped is 33 dm, and its dimensions are in the ratio 2: 6: 9. Find the dimensions of the parallelepiped.
Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. by the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Let us denote by k proportionality factor. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. Let us write formula (3) for the problem data:
Solving this equation for k, we get:
This means that the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.
Answer: 6 dm, 18 dm, 27 dm.
Example 2. Find the volume of an inclined triangular prism, the base of which is an equilateral triangle with a side of 8 cm, if the side edge is equal to the side of the base and inclined at an angle of 60º to the base.
Solution . Let's make a drawing (Fig. 3).
In order to find the volume of an inclined prism, you need to know the area of its base and height. The area of the base of this prism is the area of an equilateral triangle with a side of 8 cm. Let us calculate it:
The height of a prism is the distance between its bases. From the top A 1 of the upper base, lower the perpendicular to the plane of the lower base A 1 D. Its length will be the height of the prism. Consider D A 1 AD: since this is the angle of inclination of the side edge A 1 A to the base plane, A 1 A= 8 cm. From this triangle we find A 1 D:
Now we calculate the volume using formula (1):
Answer: 192 cm 3.
Example 3. The lateral edge of a regular hexagonal prism is 14 cm. The area of the largest diagonal section is 168 cm 2. Find the total surface area of the prism.
Solution. Let's make a drawing (Fig. 4)
The largest diagonal section is a rectangle A.A. 1 DD 1 since diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of the prism, it is necessary to know the side of the base and the length of the side edge.
Knowing the area of the diagonal section (rectangle), we find the diagonal of the base.
Since then
Since then AB= 6 cm.
Then the perimeter of the base is:
Let us find the area of the lateral surface of the prism:
The area of a regular hexagon with side 6 cm is:
Find the total surface area of the prism:
Answer:
Example 4. The base of a right parallelepiped is a rhombus. The diagonal cross-sectional areas are 300 cm2 and 875 cm2. Find the area of the lateral surface of the parallelepiped.
Solution. Let's make a drawing (Fig. 5).
Let us denote the side of the rhombus by A, diagonals of a rhombus d 1 and d 2, parallelepiped height h. To find the area of the lateral surface of a right parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, because ABCD- rhombus H = AA 1 = h. That. Need to find A And h.
Let's consider diagonal sections. AA 1 SS 1 – a rectangle, one side of which is the diagonal of a rhombus AC = d 1, second – side edge AA 1 = h, Then
Similarly for the section BB 1 DD 1 we get:
Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we obtain the equality We obtain the following:
Let us express from the first two equalities and substitute them into the third. We get: then
1.3. In an inclined triangular prism, a section is drawn perpendicular to the side edge equal to 12 cm. In the resulting triangle, two sides with lengths cm and 8 cm form an angle of 45°. Find the lateral surface area of the prism.
1.4. The base of a right parallelepiped is a rhombus with a side of 4 cm and an acute angle of 60°. Find the diagonals of the parallelepiped if the length of the side edge is 10 cm.
1.5. The base of a right parallelepiped is a square with a diagonal equal to cm. The lateral edge of the parallelepiped is 5 cm. Find the total surface area of the parallelepiped.
1.6. The base of an inclined parallelepiped is a rectangle with sides 3 cm and 4 cm. A side edge equal to cm is inclined to the plane of the base at an angle of 60°. Find the volume of the parallelepiped.
1.7. Calculate the surface area of a rectangular parallelepiped if two edges and a diagonal emanating from one vertex are 11 cm, cm and 13 cm, respectively.
1.8. Determine the weight of a stone column in the shape of a rectangular parallelepiped with dimensions of 0.3 m, 0.3 m and 2.5 m, if the specific gravity of the material is 2.2 g/cm 3.
1.9. Find the diagonal cross-sectional area of a cube if the diagonal of its face is equal to dm.
1.10. Find the volume of a cube if the distance between two of its vertices that do not lie on the same face is equal to cm.
Level II
2.1. The base of the inclined prism is an equilateral triangle with side cm. The side edge is inclined to the plane of the base at an angle of 30°. Find the cross-sectional area of the prism passing through the side edge and the height of the prism if it is known that one of the vertices of the upper base is projected onto the middle of the side of the lower base.
2.2. The base of the inclined prism is an equilateral triangle ABC with a side equal to 3 cm. Vertex A 1 is projected into the center of triangle ABC. Rib AA 1 makes an angle of 45° with the base plane. Find the lateral surface area of the prism.
2.3. Calculate the volume of an inclined triangular prism if the sides of the base are 7 cm, 5 cm and 8 cm, and the height of the prism is equal to the smaller height of the base triangle.
2.4. The diagonal of a regular quadrangular prism is inclined to the side face at an angle of 30°. Find the angle of inclination to the plane of the base.
2.5. The base of a straight prism is an isosceles trapezoid, the bases of which are 4 cm and 14 cm, and the diagonal is 15 cm. The two lateral faces of the prism are squares. Find the total surface area of the prism.
2.6. The diagonals of a regular hexagonal prism are 19 cm and 21 cm. Find its volume.
2.7. Find the measurements of a rectangular parallelepiped whose diagonal is 8 dm and forms angles of 30° and 40° with its side faces.
2.8. The diagonals of the base of a right parallelepiped are 34 cm and 38 cm, and the areas of the side faces are 800 cm 2 and 1200 cm 2. Find the volume of the parallelepiped.
2.9. Determine the volume of a rectangular parallelepiped in which the diagonals of the side faces emerging from one vertex are 4 cm and 5 cm and form an angle of 60°.
2.10. Find the volume of a cube if the distance from its diagonal to an edge that does not intersect with it is mm.
Level III
3.1. In a regular triangular prism, a section is drawn through the side of the base and the middle of the opposite side edge. The base area is 18 cm 2, and the diagonal of the side face is inclined to the base at an angle of 60°. Find the cross-sectional area.
3.2. At the base of the prism lies a square ABCD, all of whose vertices are equidistant from the vertex A 1 of the upper base. The angle between the side edge and the base plane is 60°. The side of the base is 12 cm. Construct a section of the prism with a plane passing through vertex C, perpendicular to edge AA 1 and find its area.
3.3. The base of a straight prism is an isosceles trapezoid. The diagonal cross-sectional area and the area of parallel side faces are respectively 320 cm 2 , 176 cm 2 and 336 cm 2 . Find the lateral surface area of the prism.
3.4. The area of the base of a right triangular prism is 9 cm 2, the area of the side faces is 18 cm 2, 20 cm 2 and 34 cm 2. Find the volume of the prism.
3.5. Find the diagonals of a rectangular parallelepiped, knowing that the diagonals of its faces are 11 cm, 19 cm and 20 cm.
3.6. The angles formed by the diagonal of the base of a rectangular parallelepiped with the side of the base and the diagonal of the parallelepiped are equal to a and b, respectively. Find the lateral surface area of the parallelepiped if its diagonal is d.
3.7. The area of the section of the cube that is a regular hexagon is equal to cm 2. Find the surface area of the cube.
Loading...