Mutual arrangement of the straight and plane. Sign of parallelism of the direct and plane Mutual location of the straight and plane in Cuba

Direct belongs planeIf there are two common points or one common point and parallel to any direct lying in the plane. Let the plane in the drawing are set by two intersecting straight. In this plane, it is required to construct two straight m and n in accordance with these conditions ( G. (A b)) (Fig. 4.5).

R E W E. 1. I arbitrarily carry out M 2, since direct belongs to the plane, note the projection of the intersection points with direct but and b. And we determine their horizontal projections, after 1 1 and 2 1 we carry out M 1.

2. After the point to the plane, we carry out N 2 ║M 2 and N 1 ║M 1.

Direct parallel planeIf it is parallel to any direct lying in the plane.

Crossing direct and plane. Three cases of direct and plane location are possible relative to the planes of projections. Depending on this, the direct and plane intersection point is defined.

First case - Direct and plane - projection. In this case, the intersection point in the drawing is available (both of its projections), it only needs to be denoted.

PRI MERS The drawing is set plane with traces σ ( h 0. f 0) - horizontally projectionable position - and straight l. - Frontally projecting position. Determine the point of their intersection (Fig. 4.6).

The intersection point in the drawing is already there - K (K 1 to 2).

Second case - or straight, or plane - projection. In this case, on one of the planes of projections, the projection of the intersection point is already available, it must be denoted, and on the second plane of the projections - to find on the accessories.

PRI MERS. In fig. 4.7, and depicted plane with traces of the front-stocking position and direct l. - general situation. The projection of the intersection point to 2 on the drawing is already available, and the projection to 1 must be found on the point of point to direct l.. On the
Fig. 4.7, b the plane of the overall position, and the straight M - front-scale projecting, then to 2 already eating (coincides with M 2), and to 1 you need to find from the condition of the point of point to the plane. To do this through to spend
straight ( h. - horizontal) lying in the plane.

Third case - And straight, and plane - general position. In this case, to determine the intersection point of the direct and plane, it is necessary to use the so-called intermediary - the plane of projection. For this, the auxiliary secular plane is carried out. This plane crosses the specified line plane. If this line crosses the specified direct, that is, the intersection point of the straight and plane.

PRI MERS. In fig. 4.8 shows the plane of the ABS triangle - general position - and straight l. - general situation. To determine the intersection point K, it is necessary through l. To carry out the frontally projecting plane σ, to construct a line in the triangle of the intersection of δ and σ (in the drawing it is a segment 1.2), to determine to 1 and by accessories - to 2. Then the visibility of direct l. In relation to the triangle on competing points. On P 1 competing points taken points 3 and 4. Visible on P 1 projection of point 4, since it coordinate z is larger than at point 3, therefore, the projection l 1. From this point to 1 will be invisible.

On P 2 competing points taken point 1, belonging to AB, and point 5 owned by l.. Visible will be point 1, since it has a y coordinate more than a point 5, and therefore the projection of the direct l 2.up to 2 invisible.

Location

Sign:if straight, not lying in this plane, parallel to some direct lying in this plane, then it is parallel to this plane.

1. If the plane passes through this direct, parallel other plane, and crosses this plane, then the line intersection line is parallel to this direct.

2. If one of the 2-straight parallel to this, then the other direct or also parallel to this plane or lies in this plane.

Mutual location of the planes. Parallelism of planes

Location

1. The planes have at least 1 common point, i.e. intersect in direct

2. Planes do not intersect, i.e. There are no 1 common point, in this case they are called parallel.

sign

if 2 intersecting straight 1 planes are respectively parallel to 2 direct other planes, then these planes are parallel.

SV-V.

1. If 2 parallel planes are crossed 3, then the lines of their intersection are parallel

2. Segments of parallel straight lines, prisoners between parallel planes are equal.

Perpendicularity of the straight and plane. Sign of perpendicularity of the direct and plane.

Direct name perpendiolandif they intersect under<90.

Lemma:if 1 of 2 parallel direct perpendicular to the third straight line, then the other direct is perpendicular to this straight line.

Straight ordinary perpendicular to the plane,if it is perpendicular to any direct in this plane.

Theorem: If 1 of their 2 parallel direct is perpendicular to the plane, the other direct is perpendicular to this plane.

Theorem:if 2 direct is perpendicular to the plane, then they are parallel.

Sign

If direct is perpendicular to the 2m intersecting direct lying in the plane, it is perpendicular to this plane.

Perpendicular and oblique

We construct the plane and so on, not belonging the plane. Sometimes they will spend a straight, perpendica of the plane. The point of intersection of a straight line with the plane is N. Section AN - perpendicular, carried out by the plane. The so-called is the base of the perpendicular. We are in the TM plane, which does not match N. Section AM - inclined, carried out from TA to the plane. M is the base of oblique. Cut MN - projection oblique on the plane. Perpendicular An is the distance from T.A to the plane. Any distance is part of the perpendicular.

Three perpendicular theorem:

Direct, conducted in the plane through the base of the inclined perpendicular to its projection on this plane, perpendicular to the most oblique.

The angle between the straight and plane

The angle between the straight andthe plane called the angle between this straight and its projection on the plane.

DIHEDRAL ANGLE. The angle between the planes

Dihed corner The figure formed by the straight and 2 half-plates with a total boundary A, not belonging to one plane.

Border A - the edge of the dummy corner.Half-plane - the face of dugran corner.In order to measure the dihedral angle. You need to build a linear angle inside it. We note on the edge of the COURGRAN angle some point and in each face from this point we carry a ray, perpendicular to the edge. The corner of the corner formed by these rays linear bump diugran corner.Their inside the dwarbon angle may be infinitely a lot. All of them have the same value.

Perpendicularity of two planes

Two intersecting planes perpendicular,if the angle between them is 90.

Sign:

If 1 of the 2-planes passes through a straight, perpendicular to another plane, then such planes are perpendicular.

Polyhedra

Polyhedron- The surface composed of polygons and limiting some geometric body. Face - Polygons, from which polyhedra are composed. Ribs - face faces. Vershins - Ends of ribs. Diagonal polyhedron The segment connecting 2 vertices that do not belong to 1 facet. Plane, on both sides of which there are polyhedron points, called . Sust plane.The total part of the polyhedron and the securing area of \u200b\u200bthe Naz cross section of a polyhedron.The polyhedra are convex and concave. Polyhedron called convexIf it is located one way from the plane of each of its facets (tetrahedron, parallepiped, octahedron). In the convex polyhedron, the sum of all flat corners at each top is less than 360.

PRISM

The polyhedron compiled from 2 equal polygons located in parallel planes and P - parallelograms prism.

Polygons A1A2..a (P) and V1V2..V (P) - the foundations of the prism. A1A2V2B1 ... - parallelograms, A (P) A1V1V (P) - side face. Segments A1B1, A2B2..a (P) in (P) - side edges. Depending on the polygon, underlying the prism, prism called P-coal.Perpendicular conducted from any point of one base to the plane of another base called height.If the side edges of the prism are perpendicular to the base, then prism - straight, and if not perpendicular to - then inclined.The height of the direct prism is equal to the length of its side edge. Direct prismanaz properIf its base is the right polygons, all side faces are equal rectangles.

Parallepiped

AVSD // A1B1S1D1, AA1 // BB1 // SS1 // DD1, AA1 \u003d BB1 \u003d ss1 \u003d dd1 (according to bond parallel planes)

Parallepiped consists of 6 parallelograms. Parallelograms called faces.ABSD and A1B1S1D1 - bases, other faces side. Points a in C d A1 B1 C1 D1 - vertices. Segments connecting vertices - ribs. AA1, BB1, SS1, DD1 - side edges.

Diagonal parallepipeda -the segment connecting 2 vertices that do not belong to 1 facet.

Sv-v.

1. Opposite faces of the parallepiped parallel and equal. 2. The diagonal of the parallepiped is intersecting at one point and are divided by this point in half.

PYRAMID

Consider the polygon A1A2..a (P), the point P, not lying in the plane of this polygon. Connect the point p with the tops of the polygon and we obtain the triangles: RA1A2, RA2A3 ..... (P) A1.

The polyhedron compiled from P-Cornel and P-Triangles called a pyramid.Polygon - Base.Triangles - side face.R - top pyramid.Segments A1R, A2R..a (P) R - side edges.Depending on the polygon lying at the base, the pyramid p-coal. Pyramid heightned perpendicular, conducted from the top to the base plane. Pyramid Nazi RightIf its foundation lies the right polygon and height falls into the center of the base. Apothem- Height of the side face of the right pyramid.

TRUNCATED PYRAMID

Consider the pyramid of RA1A2A3A (P). We carry out the securing plane parallel to the base. This plane divides our pyramid into 2 parts: the upper - the pyramid, similar to this, the lower - truncated pyramid. The side surface consists of a trapezium. Side edges join the tops of the base.

Theorem:the area of \u200b\u200bthe side surface of the correct truncated pyramid is equal to the work of the perimeters of the base per apothem.

Right polyhedra

Convex polyhedron Names correctIf all its faces are equal to the right polygons and each of its top converges the same number of ribs. An example of the correct polyhedron of the Oll cube. All its boundary squares, and in each vertex it converges 3 ribs.

Right tetrahedronthere are 4 equilateral triangles. Each vertex - the top of 3 triangles. The sum of flat corners at each vertex 180.

Correct octahedron Cost of 8 Estantiarceptor of triangles. Each vertex is a vertex of 4 triangles. The sum of flat corners at each vertex \u003d 240

Right Ikosahedron Cost of 20 equilateral triangles. Each vertex is a vertex 5 triangle. The sum of flat corners at each vertex 300.

Cubiccost of 6 squares. Each vertex is a vertex 3 squares. The sum of flat corners at each vertex \u003d 270.

Right dodecahedroncOST from 12 regular pentagons. Each vertex - the vertex 3 of the right pentagons. The sum of flat corners at each vertex \u003d 324.

There are no other types of correct polyhedra.

CYLINDER

The body bounded by the cylindrical surface and two circles with the boundaries of L and L1 cylinder.Circles L and L1 bases of the cylinder. Cut MM1, AA1 - forming. Forming the Cylindrical or side surface of the cylinder. Direct, comprehensive ground centers O and O1 cylinder axis.The length of the forming - cylinder height.Radius of the base (R) -Rodius cylinder.

Cylinder cross sections

Axialpasses through the axis and the diameter of the base

Perpendicular to the axis

The cylinder is the body of rotation. It turns out to rotate the rectangle around 1 out of the parties.

CONE

Consider the circle (o; r) and direct or perpendicular to the plane of this circle. Through each point of the circumference L, and the TR will carry out the segments, they are infinitely a lot. They form a conical surface and called form.

R- vertex, OR - the axis of the conical surface.

The body bounded by the conical surface and the circle with the border l called cone. A circle -cone base. Top of a conical surface - Top cone.Forming conical surface - moderating cone. Conical surface - side surface of the cone.Ro - cone axis. Distance from r to o - height cone.Cone is the body of rotation. It turns out to rotate the right triangle around the category.

Cross section of cone

Axial section

Cross section perpendicular axis

Sphere and Shar

Spherenosed Surface consisting of all points of space located at a given distance from this point. This point is center of the sphere.This distance is Radius of the sphere.

Cut connecting 2 points of the sphere and passing through its center dimmed with a diameter of the sphere.

The body limited to the sphere ball.Center, radius and diameter of the sphere center, radius and diameter of the ball.

Sphere and ball - This is the bodies of rotation. Sphere It turns out to rotate the semicircle around the diameter, and ball It turns out the rotation of the semicircle around the diameter.

in the rectangular coordinate system, the equation of the sphere of radius R with the center C (x (0), y (0), Z (0) has the form (x (0)) (2) + (U-y (0)) (2 ) + (zz (0)) (2) \u003d R (2)

Remote element.

remote element.

• a) do not have common points;

Theorem.

Designation of cuts

GOST 2.305-2008 provides the following requirements for the designation of the cut:

1. The position of the securing plane indicates the drawing of the cross section line.

2. For the cross section line, an open line should be applied (thickness from s to 1.5s Length of line 8-20 mm).

3. With a complex section, the strokes are also carried out in the intersection of sequential planes among themselves.

4. On the initial and finite strokes, the arrows indicate the direction of the view, the arrows should be applied at a distance of 2-3 mm from the external end of the stroke.

5. The sizes of the arrows must correspond to the in Figure 14.

6. The initial and end strokes should not cross the circuit of the corresponding image.

7. At the beginning and end of the section of the section, and if necessary, the places of intersection of the split planes put the same capital letter of the Russian alphabet. The letters are applied near the arrows indicating the direction of the view, and in the crossing places from the outer angle (Figure 24).

Figure 24 - Examples of cutting

8. The incision should be marked by the inscription on the type "A - A" (always two letters through a dash).

9. When the secant plane coincides with the object symmetry plane as a whole, and the corresponding images are located on the same sheet in the direct projection link and are not separated by any other images, for horizontal, frontal and profile cuts, no position of the securing plane, and The incision is not accompanied by the inscription.

10. Frontal and profile cuts, as a rule, give a position corresponding to the item adopted for this item on the main image of the drawing.

11. Horizontal, frontal and profile cuts can be located on the site of the corresponding major species.

12. It is allowed to have a cut at anywhere in the drawing field, as well as with a turn with the addition of a conditional graphic notation - the "rotated" icon (Figure 25).

Figure 25 - Conditional graphic designation - "Rotated" icon

Designation of sections like The designation of the cuts and consists of traces of the securing plane and the arrow indicating the direction of the view, as well as the letters affixed from the outside of the arrow (Figure1B, Figure 3). The submitted section is not inscribed and the secular plane is not shown if the cross section line coincides with the axis of the cross section, and the cross section itself is located on the continuation of the sequential plane or in the gap between the parts of the view. For a symmetric superimposed section, the securing plane is also not shown. If the section is asymmetric and located in the break or is superimposed (Figure 2 b), the cross section line is carried out with arrows, but the letters are not denoted.

The cross section is allowed to be placed with a turn by supplying the inscription above the word "rotated". For several identical sections related to one subject, the cross-section lines are denoted by the same letter and draw one section. In cases where the cross section is obtained from individual parts, cuts should be applied.

Direct common position

Direct General Position (Fig. 2.2) is called direct, not parallel to any of these projection planes. Any segment of such direct is projected in this system of planes of projections is distorted. The angles of inclination of this direct projection planes are distorted.

Fig. 2.2.

Direct private position
Direct private position includes straight lines parallel to one or two projectors.
Any line (direct or curve), a parallel plane of projections, is called a level line. In engineering graph, there are three main lines of the level: horizontal, front and profile lines.

Fig. 2.3-A.

The horizontal is called any line, parallel to the horizontal plane of projections (Fig. 2.z - a). The frontal projection of the horizontal is always perpendicular to the communication lines. Any segment of the horizontal on the horizontal plane of projections is projected into a true magnitude. In the true value, it is projected onto this plane and an angle of inclination of the horizontal (straight) to the frontal plane of projections. As an example in Fig. 2.C-A given a visual image and an integrated horizontal drawing. h.inclined to plane P 2 at an angle b. .
Fig. 2.3-B.

Frontallion is called a line parallel to the frontal plane of projections (Fig.2.3-b). The horizontal projection of the frontal is always perpendicular to the communication lines. Any segment of the front on the frontal plane of projections is projected into a true magnitude. In a true magnitude, it is projected onto this plane and the angle of inclination of the front (straight) to the horizontal plane of the projections (angle a.).
Fig. 2.3-B.

The profile line is called a line, a parallel profile plane of projections (Fig. 2.z-c). The horizontal and frontal projection of the profile line are parallel to the communication lines of these projections. Any segment of the profile line (straight) is projected into a profile plane into a true value. The same plane is projected into the true magnitude and corners of the profile direct to the planes of projections P 1 I. P 2. When you specify the profile direct on the complex drawing, you must specify two points of this straight line.

Direct levels parallel to two planes of projections will be perpendicular to the third plane of projections. Such directly called projection. There are three main projecting straight lines: horizontally, frontally and reacting direct.
Fig. 2.3-G. Fig. 2.3-D. Fig. 2.3-E.

Horizontally projecting straight (Fig. 2.z-d) is called direct, perpendicular plane P one . Any segment of this direct is projected onto the plane P P 1 - to the point.

Frontally projecting straight line (Fig.2.z - d) refer to direct, perpendicular plane P 2. Any segment of this direct is projected onto the plane P 1 without distortion, and on the plane P 2 - to the point.

Receiving straight line (Fig. 2.zen) is called direct, perpendicular plane P 3, i.e. straight, parallel projection planes P 1 I. P 2. Any segment of this direct is projected on the plane P 1 I. P 2 without distortion, and on the plane P 3 - to the point.

Main lines in the plane

Among the direct lines belonging to the plane, the special place is occupied by direct, occupying a private position in space:

1. Horizontal H is straight, lying in this plane and parallel horizontal plane of projections (H // P1) (Fig.6.4).

Figure 6.4 horizontal

2. Frontal F - straight lines located in the plane and parallel frontal plane of projections (F // P2) (Fig.6.5).

Figure 6.5 Frontal

3. Profile straight r - direct, which are in this plane and are parallel to the profile plane of projections (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 a plane horizontal, front-front and profile - profile line plane.

Figure 6.6 Profile Straight

4. The line of the largest skate and its horizontal projection form the linear angle j, which is measured by the dwarbon angle, compiled by this plane and the horizontal plane of projections (Fig. 6.7). Obviously, if direct does not have two common points with a plane, then it or parallel to the plane, or crosses it.

Figure 6.7 Line of the largest skate

The kinematic method of formation of surfaces. Surface task in the drawing.

In engineering graph, the surface is considered as a variety of consecutive positions of the line moving in space on a specific law. In the process of surface formation, line 1 may remain unchanged or change its form.
For visibility, the surface image on a comprehensive drawing the law of movement is advisable to set graphically in the form of a family of lines (A, B, C). The law of moving line 1 can be set two (A and B) or one (a) line and additional conditions specifying the law of movement 1.
The moving line 1 is called forming, fixed lines A, B, C - guides.
The surface formation process Consider on the example shown in Fig.3.1.
Here, direct 1. The law of movement of the forming specifies is set to the guide and direct b. It is understood that the forming 1 slides on the guide A, all the time remaining parallel to the direct line b.
This method of formation of surfaces is called kinematic. With it, you can form different surfaces on the drawing. In particular, Fig. 3.1 shows the most common case of a cylindrical surface.

Fig. 3.1.

Another way to form the surface and its images on the drawing is to task the surface by a set of points belonging to it or lines. At the same time, the points and lines are chosen so that they give the opportunity with a sufficient degree of accuracy to determine the surface shape and solve various tasks on it.
Many points or lines defining the surface are called it frame.
Depending on what the surface frame, dots or lines is specified, frameworks are divided into point and linear.
Figure 3.2 shows the surface frame consisting of two orthogonally located family of lines A1, A2, A3, ..., AN and B1, B2, B3, ..., BN.

Fig. 3.2.

Conical sections.

Conical sectionsflat curves that are obtained by intersection of a direct circular cone with a plane that does not pass through its vertex (Fig. 1). From the point of view of the analytical geometry, the conical cross-section is a geometric location of points satisfying the second order equation. With the exception of degenerate cases considered in the last section, conical sections are ellipses, hyperboles or parabolas.

Conical sections are often found in nature and technology. For example, the orbits of the planets appealing around the Sun have the form of ellipses. The circle is a special case of an ellipse, which has a large axis equal to small. The parabolic mirror has the property that all falling rays, parallel to its axes, converge at one point (focus). This is used in most reflector telescopes, where parabolic mirrors are used, as well as in the antennas of radar and special microphones with parabolic reflectors. From the light source placed in the focus of a parabolic reflector, a parallel ray beam occurs. Therefore, parabolic mirrors are used in powerful spotlights and car headlights. The hyperbole is a graph of many important physical relationships, for example, the law of the boiler (binding pressure and the volume of ideal gas) and the OMA law, which defines the electrical current as a function of resistance at constant voltage.

Early History

The opener of the conical sections is presumably considered Meehm (4 in. BC), Pupin Plato and Teacher Alexander Macedonsky. Mehm used a parabola and an equal hyperbole to solve the task of doubling the cube.

Treatise on conical sections written by Aristle and Euclide at the end of 4 c. BC, were lost, but materials from them were included in the famous conical sections of Apollonia Perga (approx. 260-170 BC), which are preserved to our time. Apollonium abandoned the requirement of perpendicularity of the sequential plane of the cone forming and, varying the angle of its inclination, received all the conical sections from one circular cone, direct or inclined. Apolloony We are obliged and modern names of curves - ellipse, parabola and hyperbole.

In its constructions, apollonium used a two-band circular cone (as in Fig. 1), so it became clear for the first time that the hyperbole is a curve with two branches. Since Apollonia, conical sections are divided into three types depending on the inclination of the securing plane to the cone forming. Ellipse (Fig. 1, a) is formed when the securing plane crosses all the forming cones at the points of one cavity; Parabola (Fig. 1, b) - when the securing plane is parallel to one of the tangent planes of the cone; hyperbole (Fig. 1, B) - when the secant plane crosses both cavities of the cone.

Building conical sections

Studying conical sections as an intersection of planes and cones, the ancient Greek mathematics considered them and as the trajectories of points on the plane. It was found that the ellipse can be defined as a geometric location of points, the amount of distances from which up to two specified points is constant; Parabola - as a geometric area of \u200b\u200bpoints equidistant from a given point and a given straight line; Hyperball - as a geometric point of points, the difference between the distances from which up to two specified points is constant.

These definitions of conical sections as flat curves are prompted and a method for constructing them with a strained thread.

Ellipse.

If the ends of the threads of a given length are fixed at points F1 and F2 (Fig. 2), then the curve described by the edge of the pencil, sliding along the tight thread, has the form of an ellipse. The points F1 and F2 are called the focus of the ellipse, and the segments V1V2 and V1V2 between the intersection points of the ellipse with the coordinate axes are greater and small axes. If the points F1 and F2 coincide, then the ellipse turns into a circle.

Fig. 2 Ellipsis

Hyperbola.

When constructing hyperboles, the point P, the point of the pencil, is fixed on the thread, which freely slides along the pectoons installed at points F1 and F2, as shown in Fig. 3, a. The distances are selected so that the PF2 segment exceeds the length of the PF1 segment per fixed value, smaller than F1F2 distances. At the same time, one end of the thread passes under the smell F1 and both the end of the thread pass on top of the switch F2. (The point of the pencil should not slide on the thread, so it needs to be consolidated by making a small loop on the thread and traveled into it.) One branch of hyperboles (PV1Q) We draw, watching the thread remained strained all the time, and sipping both ends Threads down per point F2, and when the point P will be below the segment of F1F2, holding the thread for both ends and carefully worrying (ie, release) it. We draw the second branch of the hyperbole (Pўv2Qў), we draw, previously changed the roles of the sleeper F1 and F2.

fig. 3 hyperbole

The branches of hyperboles are approaching two direct, which intersect between branches. These direct, called hyperbole asymptotes, are built as shown in Fig. 3, b. The angular coefficients of these lines are equal to ± (v1v2) / (v1v2), where V1v2 is a segment of the angle bisector between asymptotes, perpendicular to the F1F2 segment; The V1V2 segment is called the suspension of hyperboles, and the V1V2 segment is its transverse axis. Thus, the asymptotes are diagonals of the rectangle with the parties passing through four points V1, V2, V1, V2 parallel to the axes. To build this rectangle, you must specify the location of the points V1 and V2. They are at the same distance equal

from the intersection point of axes O. This formula involves the construction of a rectangular triangle with OV1 and V2O Cate and F2O hypotenuse.

If the asymptotes of hyperboles are mutually perpendicular, the hyperbole is called an equally. Two hyperboles having common asymptotes, but with a cross-transverse and conjugated axes are called mutually conjugate.

Parabola.

The focuses of the ellipse and hyperboles were still known to Apollonia, but the focus of parabola, apparently, first set the papp (2nd floor 3 century), which determined this curve as a geometric area of \u200b\u200bpoints equidistant from the specified point (focus) and a given straight line, which is called the director. Building a parabola with a strained thread, based on the definition of papp, was proposed by Isidore Miretsky (6th century). We have a ruler so that its edge coincides with the LLў directress (Fig. 4), and applies to this edge with AB drawing triangle ABC. Fill one end of the thread of the AB length in the top of the triangle, and the other is in the focus of Parabola F. stretching the tip of the pencil, press the edge in the variable point P to the free cathette AB of the drawing triangle. As the triangle is moved along the line, the point P will describe a parabola-focus on the focus f and directress LLў, since the total length of the thread is equal to AB, the thread segment adjacent to the free cathelet of the triangle, and therefore the remaining PF thread segment must be equal to the remaining Parts of the AB category, i.e. PA. The intersection point of V parabola with the axis is called the pearabol vertex, direct passing through the F and V, is the parabola axis. If through the focus to spend a straight, perpendicular axis, then the segment of this straight line, cut off by parabola, is called a focal parameter. For ellipses and hyperboles, the focal parameter is defined in the same way.

Answers to Tickets: No. 1 (not fully), 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 performing the drawings in some cases, it is necessary to build an additional selected image of any part of the subject that requires explanation with respect to the form, sizes or other data. This image is called remote element.It is usually enlarged. The remote element can be posted as a view or as a cut.

When constructing a remote element, the corresponding site of the main image is noted by a closed solid thin line, usually by oval or circle, and denote the capital letter of the Russian alphabet on the shelf of the lifting line. The remote element is recorded by type A (5: 1). In fig. 191 shows an example of a remote element. It is possible is possible closer to the appropriate place on the image of the subject.

1. The method of rectangular (orthogonal) projection. The main invariant properties of rectangular projection. Epur Monge.

Orthogonal (rectangular) projection is a special process of projecting parallel when all projecting rays are perpendicular to the plane of projections. Orthogonal projections are inherent in all properties of parallel projections, but with rectangular projection, the projection of the segment, if it is not parallel to the plane of projections, is always less than the segment itself (Fig. 58). This is explained by the fact that the segment itself in space is a hypothenucleosis of a rectangular triangle, and its projection - cathetoma: a "in" \u003d abcos a.

With rectangular projection, the straight angle is projected into a natural value, when both sides are parallel to the plane of projections, and then, only one of its sides is parallel to the plane of projections, and the second side is not perpendicular to this plane of projections.

Mutual arrangement of the straight and plane.

Straight and plane in space can:

• a) do not have common points;
• b) to have exactly one common point;
• c) have at least two common points.

In fig. 30 depicts all these features.

In case a) straight b parallel to the plane: B || .

In the case of b) straight l crosses the plane at one point O; L \u003d O.

In the case of c) straight and belongs to the plane: a or a.

Theorem. If direct b is parallel to at least one direct A, which belongs to the plane, then the direct parallel plane.

Suppose that straight m crosses the plane at the point q. If M is perpendicular to each direct plane passing through the point q, then the straight m is called perpendicular to the plane.

Tram rails illustrate the direct plane of the earth. The power lines are parallel to the land plane, and tree trunks can serve as examples of direct, crossing the surface of the Earth, some perpendicular planes of the Earth, others - non-perpendicular (inclined).

Ticket 16.

The properties of the pyramid, in which dugrani corners are equal.

A) If the lateral faces of the pyramid with its base form equal dihedral angles, then all the heights of the side of the pyramid are equal (in the right pyramid, these are apophems), and the peak of the pyramid is designed to the center of the circle, inscribed in the base polygon.

B) The pyramid may have equal dugrani angles at the base when the circle can enter into a polygon base.

Prism. Definition. Elements. Types of prism.

Prism-this is a polyhedron, two faces of which are equal to polygons located in parallel planes, and the remaining faces are parallelograms.

The faces that are in parallel planes are called basins prism, and the rest of the face - side edges Prism.

Depending on the basis of the prism, there are:

1) Triangular

2) Quadriginal

3) hexagonal

Prism with side ribs perpendicular to its grounds, called direct prism.

Direct prism is called correct if its bases are the right polygons.

Ticket 17.

Property of diagonals of rectangular parallelepiped.

All four diagonals intersect at one point and divide in half.

In rectangular parallelepiped, all diagonals are equal.

In the rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of the three dimensions.

After a diagonal of the base of the AU, we obtain the triangles of the AC 1 C and the DC. Both of them are rectangular: the first because the parallelepiped direct and, therefore, the edge of the SS 1 perpendicular to the base; The second is because the parallelepiped is rectangular and, it means, at the base it is a rectangle. Of these triangles we find:

AC 1 2 \u003d AC 2 + SS 1 2 and AC 2 \u003d AB 2 + Sun 2

Consequently, AC 1 2 \u003d AB 2 + Sun 2 + SS 1 2 \u003d AB 2 + AD 2 + AA 1 2.

Cases of the mutual arrangement of two planes.

Property 1.:

Lines intersection of two parallel planes with a third plane parallel.

Property 2:

Segments of parallel straight lines concluded between two parallel planes are equal in length.

Property 3.

Through each point of space that does not lie in this plane, it is possible to carry out a plane parallel to this plane, and moreover, only one.

Ticket 18.

Property of opposing faces of parallelepiped.

The opposite faces of the parallelepiped are parallel and equal.

for example , the plane of the parallelogram AA 1 in 1 V and DD 1 C 1 C are parallel, since the intersecting straight lines AA and AA 1 of the plane AA 1 in 1, respectively, parallel to the two intersecting direct DC and DD 1 of the plane DD 1 C 1. The parallelograms of AA 1 in 1 V and DD 1 C 1 C are equal (i.e., they can be combined with imposition), since the side of the AB and DC, AA 1 and DD 1 are equal, and the angles A 1 AB and D 1 DC are equal.

Square surfaces of the prism, pyramids, the right pyramid.

Proper pyramid: SPOV. \u003d 3SASB + SOSN.

Direct can belong and do not belong to the plane. It belongs to the plane, if at least two points are lying on the plane. Figure 93 shows the SUM plane (AXB).Straight l. Belongs the plane of SUM, since its points 1 and 2 belong to this plane.

If direct does not belong to the plane, it can be parallel to it or cross it.

Direct parallel plane, if it is parallel to another direct lying in this plane. Figure 93 Direct m || SUM.since it is parallel to the direct l.belonging to this plane.

Direct can cross the plane at various angles and, in particular, to be perpendicular to it. The construction of the line intersection lines with a plane is given in §61.

Figure 93 - Direct plane belonging

The point in relation to the plane can be located as follows: belong or not belong to it. The point belongs to the plane, if it is located on a straight line, located in this plane. Figure 94 shows a comprehensive drawing of the plane of the SUM specified by two parallel straight l. and p.Line is located in the plane m.Point A lies in the SUM plane, as it lies on a straight line m.Point INit does not belong to the plane, as its second projection does not lie on the relevant projections direct.

Figure 94 - Comprehensive drawing of the plane given by two parallel straight

Conical and cylindrical surface

The conical is surfaces formed by the movement of a straightforward forming l. According to the curvilinear guide m.A feature of the formation of a conical surface is that at the same time one point forming is always fixed. This point is a vertex of a conical surface (Figure 95, but).The determinant of the conical surface includes a vertex S.and guide m,wherein l."~ S; l."^ m.

Cylindrical belongs to surfaces formed by direct forming / moving on a curvilinear guide t.parallel to the specified direction S.(Figure 95, b).The cylindrical surface can be viewed as a special case of a conical surface with an infinitely remote vertex. S.

The determinant of the cylindrical surface consists of a guide t.and directions s forming l.while l "|| S; L "^ m.

If the forming cylindrical surface is perpendicular to the plane of projections, then such a surface is called projection.Figure 95, ina horizontally projection cylindrical surface is shown.

On cylindrical and conical surfaces, the specified points are built using forming passing through them. Lines on surfaces, such as line butin Figure 95, inor horizontal h.figure 95, a, b,build with individual points belonging to these lines.

Figure 95 - conical and cylindrical surface

Torch surfaces

Torch is called a surface formed by a straightforward forming l. concerning with the movement in all its provisions of a certain spatial curve t,called rebet Return(Figure 96). Ribbon Ribbed fully sets the torso and is the geometric part of the surface determinant. The algorithmic part is to indicate the tange of the rebar of the return.

The conical surface is a private case of a torso, in which the edge of return t.degenerated into the point S.- The vertex of the conical surface. The cylindrical surface is a private case of the torso, in which the rebon return is a point in infinity.

Figure 96 - Torso Surface

Granded surfaces

Faceted surfaces formed by moving straightforward forming l. By broken guide m.At the same time if one point S.forming fixed, a pyramid surface is created (Figure 97), if the forming parallel to the specified direction S,this creates a prismatic surface (Figure 98).

Elements of faceted surfaces are: Top S.(in the prismatic surface it is in infinity), the edge (part of the plane bounded by one section of the guide m.and extremely relative to it l.) and edge (line intersection line).

The determinant of the pyramidal surface includes a vertex S,through which the formulations and guides pass: L " ~ S; L.^ t.

The determinant of the prismatic surface, except for the guide t,contains direction S,which are parallel all forming L. Surfaces: l || s; l ^ t.

Figure 97 - Pyramid Surface

Figure 98 - Prismatic surface

Closed faceted surfaces formed by some number (at least four) faces are called polyhedra. From the number of polyhedra, there is a group of the correct polyhedra, in which all the faces are correct and congruent polygons, and multifaceted corners at the vertices convex and contain the same number of faces. For example: Hexahedr - cube (Figure 99, but),tetrahedron - the right quadrangle (Figure 99, 6) Octahedron - a polyhedron (Figure 99, in).The shape of various polyhedra has crystals.

Figure 99 - polyhedra

Pyramid- a polyhedron, at the base of which there is an arbitrary polygon, and side faces - triangles with a total vertex S.

At the complex drawing, the pyramid is given by the projections of its vertices and ribs, taking into account their visibility. The visibility of the edge is determined using competing points (Figure 100).

Figure 100 - Determination of the apparel of the edge using competing points

Prism- a polyhedron, which has a base - two identical and mutually parallel polygons, and side faces - parallelograms. If the prism ribs are perpendicular to the base plane, such a prism is called straight. If the prism of the ribs perpendicular to any plane of projections, then its side surface is called projection. Figure 101 provides a comprehensive drawing of a direct quadrangular prism with a horizontally projection surface.

Figure 101 - Comprehensive drawing of a direct quadrangular prism with a horizontally projection surface

When working with the complex drawing of the polyhedron, it is necessary to build on its surface of the line, and since the line has a set of points, then you need to be able to build points on the surface.

Any point on the face surface can be built using an forming passing through this point. Figure 100 in the face ACS.built point M.with the help of forming S-5.

Screw surfaces

Screws include surfaces created by screw moving straightforward forming. Right screw surfaces call helicoids.

Direct Helicoid is formed by the movement of a straightforward forming i. By two guides: screw line t.and its axis i.; At the same time forming l. Crosses the screw axis at right angles (Figure 102, a). Direct Helicoid is used when creating screw stairs, auks, as well as power threads, in machines.

The inclined Helicoid is formed by the movement of the screwing guide t.and its axis I. so that forming l. Crossing the axis i. at a constant angle φ, different from the direct, i.e. in any position forming l. parallel to one of the forming guide cones with an angle at a vertex of 2φ (Figure 102, b).Inclinedhecoids limit the surfaces of the thread turns.

Figure 102 - Helicoids

Surface of rotation

The surfaces of rotation include surfaces resulting with the rotation of the line l. Around direct i. representing the axis of rotation. They can be linear, such as a cone or a rotation cylinder, and non-linear or curvilinear, such as sphere. The determinant of the rotational surface includes the forming l. and axis i. . Each point of forming during rotation describes a circle whose plane is perpendicular to the axis of rotation. Such circumference of the surface of the rotation is called parallels. The largest of the parallels are called equator.Equator. Determines the horizontal surface essay if I _ | _ n 1 . In this case, the parallels are horizontally from the surface.

The curves of the rotation surface resulting from the intersection of the surface with planes passing through the axis of rotation are called meridians.All meridians of one surface of the congruent. Frontal meridian is called the main meridian; It defines the frontal essay of the surface of the rotation. Profile meridian determines the profile essay of the surface of the rotation.

Build a point on curvilinear surfaces of rotation is most convenient with parallels of the surface. Figure 103 Point M.built on parallels H 4.

Figure 103 - Building a point on a curvilinear surface

The surface of the rotation was found the wider application in the technique. They limit the surfaces of the majority of machine-building details.

The conical surface of rotation is formed by the rotation of the straight i.around the line intersecting with it - axis i. (Figure 104, but). Point M.on the surface built using forming l. And parallels h.This surface is also called the rotational cone or direct circular cone.

The cylindrical surface of rotation is formed by the rotation of the straight l. around the axis parallel to her i. (Figure 104, b).This surface is also called a cylinder or direct circular cylinder.

The sphere is formed by the rotation of the circle around its diameter (Figure 104, in). Point A on the surface of the sphere belongs to the main meridian f,point IN- Equator h,and point M.built on auxiliary parallel h ".

Figure 104 - formation of rotation surfaces

The torus is formed by the rotation of the circle or its arc around the axis lying in the circumference plane. 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 out of circle, then such a torus is called open (Figure 105, b).Open torus is called another ring.

Figure 105 - Torah Education

The surface of rotation can be formed by other second-order curves. Ellipsoid rotation (Figure 106, but)it is formed by the rotation of the ellipse around one of its axes; Paraboloid rotation (Figure 106, b.) - the rotation of the parabola around its axis; The rotation hyperboloid is single-grain (Figure 106, in) is formed by the rotation of the hyperboles around the imaginary axis, and the bipoon (Figure 106, g.) - Rotate the hyperbola around the actual axis.

Figure 106 - the formation of surfaces of rotation of second-order curves

In the general case, the surface is depicted not limited in the direction of distribution of forming lines (see Figures 97, 98). To solve specific tasks and obtaining geometric figures, limited to the planes of the cut. For example, in order to obtain a circular cylinder, it is necessary to limit the section of the cylindrical surface with the cutting planes (see Figure 104, b).As a result, we obtain its upper and lower bases. If the plane of the cut is perpendicular to the axis of rotation, the cylinder will be direct if there is no - the cylinder will be inclined.

To get a circular cone (see Figure 104, but) It is necessary to complete the crepe on top and outside it. If the plane of the base of the base of the cylinder is perpendicular to the axis of rotation - the cone will be straight, if not - inclined. If both planes are cut off through the vertex - the cone is obtained truncated.

Using the plane, the cut can be obtained a prism and a pyramid. For example, the hexagon pyramid will be straight if all its ribs have the same slope to the plane of the cut. In other cases it will be inclined. If it is executed fromthe collection of the cutting planes and none of them passes through the vertex - the truncated pyramid.

The prism (see Figure 101) can be obtained by limiting the section of the prismatic surface with two planes of the cut. If the plane of the cut is perpendicular to the edges, such as an eight-marched prism, it is straight, if not perpendicular to inclined.

Choosing the appropriate position of the cutting planes, it is possible to obtain various forms of geometric figures, depending on the conditions of the task being solved.