Mutual arrangement of planes. The relative position of the two planes. Parallelism of planes Relative position of two planes definition

Question 7.

Two planes in space can be either mutually parallel, and in a particular case, coinciding with each other, or intersecting. Mutually perpendicular planes are a special case of intersecting planes and will be discussed below.

Parallel planes. Planes are parallel if two intersecting straight lines of one plane are respectively parallel to two intersecting straight lines of another plane. When solving various problems, it is often necessary to draw a plane β parallel to a given plane α through a given point A.

In fig. 81 plane α is given by two intersecting straight lines a and b. The sought plane β is defined by straight lines a1 and b1, respectively parallel to a and b and passing through a given point A1.

Intersecting planes. The line of intersection of two planes is a straight line, for the construction of which it is sufficient to define two points common to both planes, or one point and the direction of the line of intersection of the planes.

Before considering the construction of the line of intersection of two planes, we will analyze an important and auxiliary problem: we will find the point K of the intersection of a straight line in general position with a projecting plane.

Let, for example, be given a straight line a and a horizontally projecting plane α (Fig. 82). Then the horizontal projection K1 of the desired point must simultaneously lie on the horizontal projection α1 of the plane α and on the horizontal projection a1 of the straight line a, i.e. at the point of intersection of a1 with α1 (Figure 83). The frontal projection K2 of the point K is located on the line of the projection connection and on the frontal projection a2 of the straight line a.

And now let's analyze one of the special cases of intersecting planes, when one of them is projecting.

In fig. 84 shows a plane in general position, defined by a triangle ABC, and a horizontally projecting plane α. Let's find two common points for these two planes. Obviously, these common points for the planes ∆ABS and α will be the points of intersection of the sides AB and BC of the triangle ABC with the projecting plane α. The construction of such points D and E both on the spatial drawing (Fig. 84) and on the diagram (Fig. 85) does not cause difficulties after the example disassembled above.

Connecting the projections of the same name of points D and E, we obtain the projection of the line of intersection of the plane ∆ ABC and plane α.

Thus, the horizontal projection D1Е1 of the line of intersection of the given planes coincides with the horizontal projection of the plane projecting α - with its horizontal trace α1.

Let us now consider the general case. Let two planes of general position α and β be given in space (Fig. 86). To build the line of their intersection, it is necessary, as noted above, to find two points common to both planes.

To determine these points, the given planes are intersected by two auxiliary planes. It is more expedient to take projection planes and, in particular, level planes as such planes. In fig. 86, the first minor plane of the level γ each of these planes intersects the horizontal lines h and h1, which define the point 1 common to the planes α and β. This point is determined by the intersection of contours h2 and h3, along which the auxiliary δ plane intersects each of these planes.

MUTUAL POSITION OF TWO PLANES.

Two planes in space can be located either parallel to each other, or intersect.

Parallel planes... In projections with numerical marks, a sign of the parallelism of the planes on the plan is the parallelism of their contour lines, the equality of the laying and the coincidence of the directions of falling of the planes: pl. S || pl. L - h S || h L, l S = l L, pad. I. (Figure 3.11).

In geology, a flat homogeneous body composed of a rock is called a layer. The layer is bounded by two surfaces, the upper one of which is called the roof, and the lower one - the sole. If the layer is considered at a relatively short length, then the roof and bottom are equated to the planes, obtaining in space a geometric model of two parallel inclined planes.

Plane S is the roof, and plane L is the bottom of the layer (Figure 3.12, a). In geology, the shortest distance between the top and bottom is called true power (in Figure 3.12, a the true power is indicated by the letter H). In addition to the true thickness, other parameters of the rock layer are used in geology: vertical thickness - H in, horizontal thickness - L, apparent thickness - H type. Vertical power in geology, the distance from the top to the bottom of the layer, measured vertically, is called. Horizontal power layer is the shortest distance between the roof and the bottom, measured in the horizontal direction. Apparent power - the shortest distance between the visible fall of the roof and the bottom (the visible fall is called the straight-line direction on the structural plane, that is, a straight line belonging to the plane). Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the apparent power is always greater than the true power. It should be noted that for horizontally lying layers, the true thickness, vertical and visible, coincide.

Consider the technique of constructing parallel planes S and L, spaced from each other at a given distance (Figure 3.12, b).

On the plan with intersecting straight lines m and n the plane S is given. It is necessary to construct the plane L, parallel to the plane S and spaced from it at a distance of 12 m (that is, the true thickness - H = 12 m). Plane L is located under plane S (plane S is the top of the layer, plane L is the bottom).

1) Plane S is set on the plan by projections of contour lines.

2) On the scale of the laying, the line of incidence of the plane S is plotted - u S. Perpendicular to the line u S set aside a given distance of 12 m (true thickness of the H layer). Below the line of incidence of the plane S and parallel to it, draw a line of incidence of the plane L - u L. Determine the distance between the lines of incidence of both planes in the horizontal direction, i.e. the horizontal thickness of the layer L.

3) Deferring the horizontal power from the horizontal on the plan h S, parallel to it, draw a horizontal line of the plane L with the same numerical mark h L. It should be noted that if the L plane is located under the S plane, then the horizontal thickness should be deposited in the direction of the S plane rise.

4) Based on the condition of parallelism of two planes, the horizontal lines of the plane L are drawn on the plan.

Intersecting planes... The sign of the intersection of two planes is usually the parallelism on the plan of the projections of their contours. The line of intersection of two planes in this case is determined by the points of intersection of two pairs of the same name (having the same numerical marks) contour lines (Figure 3.13):; ... Connecting the obtained points N and M with a straight line m, determine the projection of the desired intersection line. If the plane S (A, B, C) and L (mn) are specified on the plan not by contours, then to build their intersection line t it is extremely important to construct two pairs of contour lines with the same numerical elevations, which at the intersection will determine the projections of the points R and F of the desired straight line t(Figure 3.14). Figure 3.15 shows the case when two intersecting

planes S and L horizontally are parallel. The intersection line of such planes will be a horizontal line h... It is worth saying that to find a point A belonging to this straight line, an arbitrary auxiliary plane T is drawn, which intersects the planes S and L. The plane T intersects the plane S in a straight line a(C 1 D 2), and the plane L is in a straight line b(K 1 L 2).

Intersection point of lines a and b belonging to the planes S and L, respectively, will be common for these planes: = A. The elevation of point A can be determined by interpolating the straight lines a and b... It remains to draw a horizontal line through A h 2.9, which is the line of intersection of the planes S and L.

Consider another example (Figure 3.16) of constructing the line of intersection of the inclined plane S with the vertical plane T. The desired straight line m defined by points A and B at which the contours h 3 and h 4 planes S intersect the vertical plane T. From the drawing it can be seen that the projection of the intersection line coincides with the projection of the vertical plane: mº T. In solving geological exploration problems, a section of one or a group of planes (surfaces) by a vertical plane is usually called a cut. The additional vertical projection of the straight line constructed in the considered example m called the profile of the cut made by the plane T in a given direction.

MUTUAL POSITION OF TWO PLANES. - concept and types. Classification and features of the category "MUTUAL LOCATION OF TWO PLANES." 2017, 2018.

No. _____ Date 02.10.14

Subject Geometry

Class 10

Lesson topic:The relative position of the two planes. Parallelism of planes

Lesson objectives: to acquaint with the concept of parallelism of planes, to study the sign of parallelism of a plane and the properties of parallel planes

Lesson type: learning new material

DURING THE CLASSES

1. Organizational moment.

Greeting students, checking the readiness of the class for the lesson, organizing the attention of students, disclosing the general goals of the lesson and its plan.

2. Formation of new concepts and methods of action.

The two planes are calledparallel, if they have no common points, i.e. if α = α (fig. 20).

Theorem 1. Through a point not lying in a plane, you can draw only one plane parallel to this plane.

Proof. Let the plane be givena and point A, A a ... In plane a take two intersecting linesand and b : a , b , a = B (Fig. 21.) Then, by Theorem 1 (§2, Section 2.1.) through the pointA directa 1 and b 1 such that a 1 || a and b 1 || b Hence, according to the axiomCIIIthere is only one plane passing through intersecting straight linesa 1 and b 1 ... It now remains to show that α, i.e. α = .

Let it not be so, i.e. planes intersect in a straight line with.Then at least one of the straight linesa orb not parallel to a straight linewith. For definiteness, we assume thata with anda with = S.

Hence,a 1 c and, as in the proof of Theorem 2 in §2, we havea 1 c = WITH, those.a 1 a = C.

This contradicts the fact that a, ||a ... Therefore, α = α ... The theorem is proved.

Theorem 2. If you intersect two parallel planes with a third plane, then their straight lines of intersection will be parallel, i.e. α, a = α, b = => a|| b(rice.22 ).

So, two planes in space can be mutually located in two versions:

planes intersect in a straight line;

the planes are parallel.

Parallelism of planes

Theorem 3. If two intersecting straight lines of one plane are respectively parallel to two straight lines of another plane, then these planes are parallel.

Theorem 4. Segments of parallel straight lines bounded by parallel planes are equal,between themselves.

3. Application. Formation of skills and abilities.

Objectives: To ensure that students apply the knowledge and methods of action that they need for SR, to create conditions for students to identify individual ways of using what they have learned. Page 24 No. 87,88,89,90 (1)

4. Homework information stage.

Objectives: To provide students with an understanding of the purpose, content and ways of doing homework. Page 22 p3 # 90 (2)

5. Summing up the results of the lesson.

Objective: To provide a qualitative assessment of the work of the class and individual students.

6. Stage of reflection.

At least 1, so at least 1 element is nonzero. Let 1 and 2 intersect, they have a common line, they have a common system, they are not parallel, and so they are compatible, then. Let 1 and 2 be parallel:,. If the Cartesian coordinate system, then - normal vectors. Cosine of the angle between two vectors:

A necessary and sufficient condition for the perpendicularity of two planes:

Or

20. Various ways to define a straight line in space. Straight and plane. 2 straight lines in space. The angle between two straight lines. Comment. A straight line in space cannot be defined by one equation. This requires a system of two or more equations. The first opportunity to compose the equations of a straight line in space is to represent this straight line as the intersection of two non-parallel planes given by the equations A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2= 0, where the coefficients A 1, B 1, C 1 and A 2, B 2, C 2 not proportional: A 1 x + B 1 y + C 1 z + D 1=0; A 2 x + B 2 y + C 2 z + D 2= 0; however, when solving many problems, it is more convenient to use other equations of the straight line that explicitly contain some of its geometric characteristics. М 0 (x 0, y 0, z 0) parallel to the vector a = (l, m, n) Definition. Any nonzero vector parallel to a given line is called its direction vector.For any point M (x, y, z) lying on a given straight line, the vector M 0 M = {x - x 0, y - y 0, z - z 0) is collinear to the direction vector a . Therefore, the equalities take place:

called canonical equations straight in space. In particular, if you want to get the equations of a straight line passing through two points: M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2), the direction vector of such a straight line can be considered the vector M 1 M 2 = {x 2 - x 1, y 2 - y 1, z 2 - z 1), and equations (8.11) take the form:

- equations of a straight line passing through two given points... If we take each of the equal fractions in the equations as some parameter t, you can get the so-called parametric equations of the straight line:

... In order to pass from equations to canonical or parametric equations of a straight line, it is required to find the direction vector of this straight line and the coordinates of any point belonging to it. The direction vector of the straight line is orthogonal to the normals to both planes, therefore, it is collinear with their vector product. Therefore, as the direction vector, you can choose [ n 1 n 2 ] or any vector with proportional coordinates. To find a point lying on a given straight line, you can set one of its coordinates arbitrarily, and find the other two from the equations, choosing them so that the determinant of their coefficients does not equal zero.

Angle between straight lines. The angle between a straight line and a plane. The angle between straight lines in space is equal to the angle between their direction vectors. Therefore, if two straight lines are given by canonical equations of the form

and
the cosine of the angle between them can be found by the formula:

... The conditions for parallelism and perpendicularity of straight lines are also reduced to the corresponding conditions for their direction vectors:

- parallelism condition,

- line perpendicularity condition... The angle φ between the straight line given by the canonical equations

and the plane defined by the general equation Ax + By + Cz + D= 0, can be considered as complementary to the angle ψ between the directing vector of the straight line and the normal to the plane. Then

The condition of parallelism of a straight line and a plane is in this case the condition of perpendicularity of vectors n and a : Al + Bm + Cn= 0, and the condition of perpendicularity of a straight line and a plane- the condition for the parallelism of these vectors: A / l = B / m = C / n.

21. canonical equation of an ellipse. Properties. a line is called, which in some Cartesian rectangular coordinate system is determined by the canonical equation x 2 / a 2 + y 2 / b 2 = 1, provided a≥b> 0. It follows from the equation that for all points of the ellipse │x│≤ a and │у│≤ b. Hence, the ellipse lies in a rectangle with sides 2a and 2b. The points of intersection of the ellipse with the axes of the canonical coordinate system, which have coordinates (a, 0), (-a, 0), (0, b) and (0, -b), are called the vertices of the ellipse. The numbers a and b are called, respectively, the major and minor semiaxes. C1. The axes of the canonical coordinate system are the axes of symmetry of the ellipse, and the origin of the canonical system is its center of symmetry. The appearance of the ellipse is most easily described by comparing it with a circle of radius a centered at the center of the ellipse: x 2 + y 2 = a 2. For each x such that I x I< а, найдутся две точки эллипса с ординатами ±b√1-x 2 /a 2 и две точки окружности с ординатами ±a√1-x 2 / а 2 Пусть точке эллипса соответствует точка окруж­ности с ординатой того же знака. Тогда отношение ординат соответствующих точек равно b/a. Итак, эллипс получается из окружности таким сжатием ее к оси абсцисс, при котором ординаты всех точек уменьшаются в одном и том же отношении b/a. С эллипсом связаны две замечательные точки, называемые его фокусами. Пусть по определению с 2 =a 2 – b 2 и c≥0.Фокусами называются точки F 1 и F 2 с координатами (с, 0) и (-с, 0) в канонической системе координат. Отношение e=c/a называется эксцентриситетом эллипса. Отметим, что < 1. С2. Расстояние от произвольной точки М (х, у), лежащей на эллипсе, до каждого из фокусов является линейной функцией от ее абсциссы х: R 1 =│F 1 M│=a- x, r 2 =│F 2 M│=a+ x. С3. Для того чтобы точка лежала на эл­липсе, необходимо и достаточно, чтобы сумма ее расстояний до фокусов равнялась большой оси эллипса 2а. С эллипсом связаны две замечательные прямые, называемые его директрисами. Их уравнения в канонической системе коор­динат x=a/ , x=-a/ . С4. Для того чтобы точка лежала на эллипсе, необходимо и достаточно, чтобы отношение ее рас­стояния до фокуса к расстоянию до соответствующей ди­ректрисы равнялось эксцентриситету эллипса . Уравнение касательной, проходящая через точку M 0 (x 0 ;y 0) имеет вид: xx 0 /a 2 + yy 0 /b 2 = 1. С5. Касательная к эллипсу в точке M 0 (x 0 ;y 0) есть биссектриса угла, смежного с углом между от­резками, соединяющими эту точку с фокусами

22. Canonical equation of hyperbola. Properties. We called a hyperbola a line, which in a certain Cartesian rectangular coordinate system is determined by the canonical equation x 2 / a 2 - y 2 / b 2 = 1. From this equation it can be seen that for all points of the hyperbola │x│≥a, i.e. all points of the hyperbola lie outside the vertical strip of width 2a. The abscissa axis of the canonical coordinate system intersects the hyperbola at points with coordinates (a, 0) and (-a, 0), called the vertices of the hyperbola. The ordinate axis does not intersect the hyperbola. Thus, the hyperbola consists of two unconnected parts. They are called her branches. The numbers a and b are called, respectively, the real and imaginary semiaxes of the hyperbola. C1. For a hyperbola, the axes of the canonical coordinate system are the axes of symmetry, and the origin of the canonical system is the center of symmetry. To study the shape of the hyperbola, we find its intersection with an arbitrary straight line passing through the origin. We take the equation of the straight line in the form y = kx, since we already know that the straight line x = 0 does not intersect the hyperbola. The abscissas of the crossing points are found from the equation x 2 / a 2 - k 2 x 2 / b 2 = 1. Therefore, if b 2 - a 2 k 2> 0, then x = ± ab / √b 2 - a 2 k 2. This allows you to specify the coordinates of the intersection points (ab / u, abk / u) and (-ab / u, -abk / u), where u = (b 2 - a 2 to 2) 1/2 is denoted.

The straight lines with the equations y = bx / a and y = -bx / a in the canonical coordinate system are called the asymptotes of the hyperbola. C2. The product of the distances from the point of the hyperbola to the asymptotes is constant and equal to a 2 b 2 / (a ​​2 + b 2). C3. If a point moves along the hyperbola so that its abscissa in absolute value increases indefinitely, then the distance from the point to one of the asymptotes tends to zero. Let us introduce a number c, putting c 2 = a 2 + b 2 and c> 0. The points F 1 u F 2 with coordinates (c, 0) and (-c, 0) in the canonical coordinate system are called foci of a hyperbola. The ratio e = c / a, as for an ellipse, is called eccentricity. Hyperbola has e> 1. C4. Distances from an arbitrary point M (x, y) on the hyperbola to each of the foci as follows depend on its abscissa x: r 1 = │F 1 M│ = │a-ex│, r 2 = │F 2 M│ = │a + ex│. C5. In order for the point M to lie on the hyperbola, it is necessary and sufficient that the difference in its distances to the foci in absolute value be equal to the real axis of the hyperbola 2a. Hyperbola directrixes are straight lines defined in the canonical coordinate system by the equations x = a /, x = -a /. C6. For a point to lie on a hyperbola, it is necessary and sufficient that the ratio of its distance to the focus to the distance to the corresponding directrix be equal to the eccentricity. The equation of the tangent to the hyperbola at the point M 0 (x 0, y 0) lying on it has the form: xx 0 / a 2 - yy 0 / b 2 = 1. C7. The tangent to the hyperbola at the point M 0 (x 0, y 0) is the bisector of the angle between the segments connecting this point with the foci.

23. Canonical equation of a parabola. Properties. we called the line, which in some Cartesian rectangular coordinate system is determined by the canonical equation y 2 = 2px, provided p> ​​0. From the equation it follows that for all points of the parabola x≥0. The parabola passes through the origin of the canonical coordinate system. This point is called the apex of the parabola. The focus of a parabola is a point F with coordinates (p / 2, 0) in the canonical coordinate system. The directrix of a parabola is a straight line with the equation x = -p / 2 in the canonical coordinate system. C1. The distance from the point M (x, y) lying on the parabola to the focus is r = x + p / 2. C2. In order for point M to lie on a parabola, it is necessary and sufficient that it be equally distant from the focus and from the directrix, this parabola. The eccentricity e = 1 is attributed to the parabola. By virtue of this agreement, the formula r / d = e is true for an ellipse, for a hyperbola, and for a parabola. We derive the equation of the tangent to the parabola at the point M 0 (x 0, y 0) lying on it, has the form yy 0 = p (x + x 0). C3 The tangent to the parabola at the point Mo is the bisector of the angle adjacent to the angle between the segment that connects Mo to the focus, and the ray emerging from this point in the direction of the parabola axis.

24. Algebraic lines. To set algebraic lines on a plane, then some algebraic ur-th of the form F (x, y) = 0 and some affine coordinate system of a circle on a plane, then those and only those M (x, y) whose coordinates satisfy the equation are considered to lie on given equation. Equations for a surface in space are similarly set. Set algebraic equation of the form F (x, y, z) = 0 (z) with 3 variables and some coordinate system OXYZ, only those points F (x, y, z ) = 0 (z) are the equation of the plane. In this case, we believe that two ur-ies determine the same line or surface, etc., when one of these ur-ies is obtained from the other by multiplying by some numerical factor lambda 0.

25. The concept of an algebraic surface. The study of arbitrary sets of points is a completely immense problem. Def. An algebraic surface is a set of points that in some Cartesian coordinate system can be given by an equation of the form + ... + = 0, where all exponents are non-negative integers. The largest of the sums (of course, here we mean the largest of the sums actually included in the equation, i.e. it is assumed that after reducing similar terms, there will be at least one term with a nonzero coefficient that has such a sum of exponents.) + +, ...., + + is called the degree of the equation , as well as the order of the algebraic surface. This definition means, in particular, that the sphere, the equation of which in the Cartesian rectangular coordinate system has the form (+ (+ (=, is an algebraic surface of the second order. Theorem. An algebraic surface of order p in any Cartesian coordinate system can be given by an equation of the form +… + = 0 of order p.

26. Cylindrical surfaces of the 2nd order. Let the plane be given some line of the 2nd order and a bunch of parallel lines d such that for any d not parallel to, then the set φ of all points of the space belonging to those lines of the bunch that intersect the line γ are called guides, and the lines intersecting φ are generators. Let us derive the equation of a cylindrical surface with respect to an affine coordinate system. Let some K lie in some plane P, the equation of which F (x, y) = 0, in direction a (a 1 a 2 a 3) d is parallel to a. The point M (x, y, z) lies on some generator, and N (x'y'o) is the point of intersection of this generator with the plane P. The vector MN will be collinear with ta, hence MN = ta, x '= x + a 1 t; y '= y + a 2 t; 0 = z + a 3 t therefore t = -z / a 3, then x ’= x- (a 1 z) / a 3; y '= y- (a 2 z) / a 3 F (x'y') = 0 F (x- (a 1 z) / a 3; y- (a 2 z) / a 3. Now it is clear that the equation F (x, y) = 0 is the equation of a cylinder with generators parallel to the Oy axis, and F (y, z) = 0 with generators parallel to the Ox axis. 0 a 2 = 0 a 3 ≠ 0 F (x, y) = 0, therefore, as many lines of the second order, as many cylinders Surfaces: 1. Elliptical cylinder x 2 / a 2 + y 2 / b 2 = 1 2. Hyperbolic cylinder x 2 / a 2 -y 2 / b 2 = 1 3. Parabolic cylinder y 2 = 2πx 4. A pair of intersecting planes x 2 / a 2 -y 2 / b 2 = 0 5. A pair of parallel planes x 2 / a 2 = 1

27. Canonical surfaces of the second order. A surface on which there is a point Mo, possessing the property that together with each point Mo ≠ M contains a straight line (Mo M), such a surface is called canonical or a cone. M o is the top of the cone, and the straight lines are its generators. A function F (x, y, z) = 0 is called homogeneous if F (tx, ty, tz) = φ (t) F (x, y, z), where φ (t) is a function of t. Theorem. If F (x, y, z) is a homogeneous function, then the surface defined by this equation is a canonical surface with a vertex at the origin. Doc. Let an affine coordinate system be given and a canonical equation with the center F (x, y, z) = 0 is given from it. Consider an equation with a vertex at the point O M (x, y, z) = 0, then any point OM from F will have the form M 1 (tx, ty, tz) on the canonical surface. M o M (x, y, z), since it satisfies the surface, then F (tx, ty, tz) = 0 is a homogeneous function φ (t) F (x, y, z) = 0 hence the surface is canonical. Curves of the 2nd order are sections in the final surface of the planes x 2 + y 2 -z 2 = 0 / When the canonical surfaces are cut by planes, we obtain the following lines in the section: a) a plane passing through a point or a pair of merged straight lines and a pair of intersecting straight lines. B) the plane does not pass through the vertex of the cone; therefore, we obtain in the section either an ellipse, or a hyperbola, or a parabola.

28. Surfaces of revolution. Let a Cartesian frame be given in 3-dimensional space. The plane P passes through Oz, γ is given in the Ozy plane, and the angle xOy = φ γ has the form u = f (z). Take a point M from γ with respect to the frame Oxyz. γ - the circumcircle γМ along all points М from γ is called a mapping. The section of the surface of rotation of the plane passing through the axis of rotation is called the meridian. The section of the surface of rotation of a plane perpendicular to the axis of rotation is called parallel. The equation of the surface of revolution x 2 + y 2 = f 2 (z) is the equation of the surface of revolution. 1) If the angle φ = 0, then γ lies in the xOz plane, x 2 + y 2 = f 2 (z) 2) γ lies in the xOy plane and its equation y = g (x), then y 2 + z 2 = g 2 (x) 3) γ lies in the yOz plane and its equation is z = h (y), then z 2 + x 2 = h 2 (y)

29. Ellipsoids. The surface that is obtained by rotating an ellipse around its axes of symmetry. Directing the vector e 3 first along the minor axis of the ellipse, and then along the major axis, we get the ur-th ellipse in the following forms: ... By virtue of the formula ur-i of the corresponding surfaces of revolution will be = 1 (a> c). Surfaces with such ur-s are called compressed (a) and retracted (b) ellipsoids of revolution.

Each point М (x, y, z) on the compressed ellipsoid of revolution is shifted to the plane y = 0 so that the distance from the point to this plane decreases in a constant ratio for all points λ<1. После сдвига точка попадет в положение M’ (x’, y’, z’) , где x’=x, y’=λy, z’=z. Таким образом, точки эллипсоида вращения переходят в точки поверхности с ур-ем , where b = λa. A surface that has this ur-e in some Cartesian coordinate system is called an ellipsoid (c). If by chance it turns out that b = c, we get again an ellipsoid of revolution, but already elongated. The ellipsoid, just like the ellipsoid of revolution from which it is derived, is a closed bounded surface. It can be seen from the equation that the origin of the canonical coordinate system is the center of symmetry of the ellipsoid, and the coordinate planes are its plane of symmetry. The ellipsoid can be obtained from the sphere x 2 + y 2 + z 2 = a 2 by compressions to the planes y = 0 and z = 0 in the ratios λ = b / a and μ = c / a.

30. Hyperboloids.One-sheet hyperboloid of revolution Is the surface of rotation of the hyperbola around the axis that does not intersect it. According to the formula we get the equation of this surface (fig. 48). As a result of the compression of a one-sheet hyperboloid of revolution to the plane y = 0, we obtain a one-sheet hyperboloid with ur-em ... An interesting sv-in a one-sheet hyperboloid is the presence of rectilinear generators in it. The so-called straight lines, all points lying on the surface. Two rectilinear generators pass through each point of a one-sex hyperboloid, ur-th of which can be obtained as follows. Ur-e (8) can be rewritten as ... Consider a straight line with ur-ys μ = λ, λ = μ (9), where λ and μ are some numbers (λ 2 + μ 2 ≠ 0). The coordinates of each point of the straight line satisfy both ur-holes, and therefore ur-th (8), which is obtained by term-by-term multiplication. Therefore, whatever λ and μ may be, the line with ur-s (9) lies on a one-sheet hyperboloid. Thus, system (9) defines a family of rectilinear generators. If, together with the hyperbola, we rotate its asymptotes, then they describe a straight circular cone, called the asymptotic cone of the hyperboloid of revolution. When a hyperboloid of revolution is compressed, its asymptotic cone contracts into an asymptotic cone of a general one-sheet hyperboloid.

Two-sheet hyperboloid. A two-sheeted hyperboloid of revolution is a surface obtained by rotating a hyperbola around the axis that crosses it. According to the formula we get ur-e of a two-sheet hyperboloid of revolution As a result of the compression of this surface to the plane y = 0, a surface with ur-em is obtained (12). The surface, which in some Cartesian rectangular coordinate system has an ur-e of the form (12), is called a two-sheet hyperboloid (Fig. 49). Two branches of the hyperbola here correspond to two unconnected parts ("cavities") of the surface. The asymptotic cone of a two-sheet hyperboloid is defined in the same way as for a one-sheet hyperboloid.

31. Paraboloids.Elliptical paraboloid. Rotating the parabola x 2 = 2pz around its axis of symmetry, we get a surface with ur-em x 2 + y 2 = 2pz. It is called a paraboloid of revolution. Compression to the plane y = 0 transforms the paraboloid of revolution into a surface, the ur-e of which is reduced to the form 2z (14). A surface that has such ur-e in some rectangular Cartesian coordinate system is called an elliptic paraboloid. Hyperbolic paraboloid. By analogy with ur-em (14) we can write ur-e A surface that has such ur-e in some rectangular Cartesian coordinate system is called a hyperbolic paraboloid. From the canonical equation z = x 2 / a 2 - y 2 / b 2 of the hyperbolic paraboloid it follows that the planes Oxz and Oyz are planes of symmetry. The Oz axis is called the axis of the hyperbolic paraboloid .. The lines z = h of the intersection of the hyperbolic paraboloid with the planes z = h represent, for h> 0, hyperbolas x 2 / a * 2 - y 2 / b * 2 = 1 with semiaxes a * = a√h , b * = b√h, and for h<0 – сопряженные гиперболы для гипербол x 2 /a *2 - y 2 /b *2 =1 с полуосями a * = a√-h, b * =b√-h. Используя эти формулы, легко построить «карту» гипер­болического параболоида. Как и в случае эллиптическо­го параболоида, можно убедиться в том, что гиперболический па­раболоид может быть получен путем параллельного перемещения параболы, представляющей собой сечение плоскостью Охz (Оуz), когда ее вершина движется вдоль параболы, являющейся сече­нием параболоида плоскостью Оуz (Охz).

32. Complex numbers. Algebraic form of a complex number. A complex number is an expression of the form z = x + iy, where x and y are real numbers, i is an imaginary unit. The number x is called the real part of the number z and is denoted by Re (z), and the number y is called the imaginary part of the number z and is denoted by Im (z). The numbers z = x + iy and z = x - iy are called conjugate. Two complex numbers z 1 = x 1 + iу 1 and z 2 = x 2 + iу 2 are called equal if their real and imaginary parts are equal. In particular i 2 = -1. Arithmetic operations on a set of complex numbers are defined as follows. 1. Addition: z 1+ z 2 = x 1 + x 2 + i (y 1 + y 2); 2.Subtraction: z 1 -z 2 = x 1 -x 2 + i (y 1 -y 2); 3. Multiplication: z 1 z 2 = (x 1 x 2 -y 1 y 2) + i (x 1 y 2 + x 2 y 1); Division: z 1 / z 2 = ((x 1 x 2 + y 1 y 2) + i (x 2 y 1 - x 1 y 2)) / x 2 2 + y 2 2. For presentation to.ch. are the points of the coordinate plane Oxy. A plane is called complex if each k.ch. z = x + iy the point of the plane z (x, y) is assigned, and this correspondence is one-to-one. The axes Ox and Oy, on which the real numbers z = x + 0i = x and the purely imaginary numbers z = 0 + iy = iy are located, are called, respectively, the real and imaginary axes

33. Trigonometric form of a complex number. Moivre's formula. If the real x and imaginary y parts of a complex number express through the module r = | z| and argument j (x = r cosj, y = r sinj), then any complex number z, except for zero, can be written in trigonometric form z = r (cosj + isinj). Features of the trigonometric form: 1) the first factor is a non-negative number, r³0; 2) the cosine and sine of the same argument are written; 3) the imaginary unit is multiplied by sinj. May also be helpful indicative notation of complex numbers, closely related to trigonometric through Euler's formula: z = re i j. Where e i j is the exponent expansion for the case of a complex exponent. Formula for raising a complex number in trigonometric form to a power. Moivre's formula has the form: z = n = r n (cosnj + isin nj), where r is the modulus, and j is the argument of the complex number.

34. Operations over polynomials. Euclid's Algorithm. General view of the equation of the n-th degree: a 0 x n + a 1 x n -1 +… + a n -1 x + a n = 0 (1). A set of coefficients is determined. (a 0, a 1, ..., a n -1, a n) are arbitrary complex numbers. Consider the left side of (1): a 0 x n + a 1 x n -1 +… + a n -1 x + a n -polynomials of the n-th degree. Two polynomials f (x) and g (x) will be considered equal or identically equal if the coefficients at the same degrees are equal. Any polynomial is determined by a set of coefficients.

Let us define the operations of addition and multiplication over polynomials: f (x) = a 0 + a 1 x +… + a n x n; g (x) = b 0 + b 1 x +… + b s x s n³s; f (x) + g (x) = c 0 + c 1 x + ... + c n x n -1 + c n; c i = a i + b i if i = 0,1 ... n; i> s b i = 0; f (x) * g (x) = d 0 + d 1 x + ... + d n + s x n + s; ; d 0 = a 0 b 0; d 1 = a 0 b 1 + a 0 b 1; d 2 = a 0 b 2 + a 1 b 1 + a 2 b 0. The degree of the product of polynomials is equal to the sum and the operations have the following properties: 1) a k + b k = b k + a k; 2) (a k + b k) + c k = a k + (b k + c k); 3). A polynomial f (x) is called inverse (x) if f (x) * (x) = 1. The division operation is not possible in the set of polynomials. In Euclidean space for a polynomial, there is a division with remainder. f (x) and g (x) exists r (x) and q (x) are uniquely defined. ; ; f (x) = g (x);; ... The degree of the right side of the £ degree g (x), and the degree of the left side from here from here - we came to a contradiction. We prove the first part of the theorem:. Let's multiply g (x) by a polynomial such that the leading coefficients are multiplied.

After k steps.

; ; has a lesser degree q (x). Polynomial q (x) - quotient of f (x), a r (x) -remainder of the division. If f (x) and g (x) have real coefficients, then q (x) and r (x)- are also valid.

35 Divisor of polynomials. GCD. Let there be given two nonzero polynomials f (x) and j (x) with complex coefficients. If the remainder is zero, then f (x) is said to be divisible by j (x) if j (x) is a divisor of f (x). C-va of the polynomial j (x): 1) The polynomial j (x) is a divisor of f (x) if Y (x) exists and f (x) = j (x) * Y (x) (1). j (x) -divisor, Y (x) -partial. Let Y (x) satisfy (1), then from the previous theorem Y (x) is quotient, and the remainder is 0. If (1) is satisfied, then j (x) is a divisor, hence j (x)<= степени f(x). Basic properties of divisibility of a polynomial: 1) ; 2 f (x) and g (x) are divisible by j (x), then they are divisible by j (x); 3) if; 4) if f 1 (x) .. f k (x): j (x) ®f 1 g 1 +… + f k g k: j (x); 5) any polynomial is divisible by any polynomial of degree zero f (x) = a 0 x n + a 1 x n -1 + a n c; 6) if f (x): j (x), then f (x): cj (x); 7) The polynomial cf (x) and only they will be divisors of the polynomial j (x), having the same degree as f (x); 8) f (x): g (x) and g (x): f (x), then g (x) = cf (x); 9) Any divisor of one of f (x) and cf (x), with¹0 will be a divisor for the other. Definition: Greatest common divisor (GCD). A polynomial j (x) will be called gcd f (x) and g (x) if it divides each of them. Zero degree polynomials are always gcd and are coprime. The gcd of nonzero polynomials f (x) and g (x) is called d (x), which is explicit. common divisor and is divisible by any other divisor and common of these polynomials. Gcd f (x) and g (x) = (f (x): g (x)). Algorithm for finding GCD: Let the degree g (x)<= степениf(x) f(x)=g(x)g 1 (x)+r 1 (x) g(x)=r 1 (x)q 2 (x)+r 2 (x)

r k-2 (x) = r k-1 (x) q k (x) + r k (x)

r k-1 (x) = r 2 (x) + q k (x) r k (x) -NOD. Let's prove it. r k (x) divisor r k -1 (x) ®on divisor r k -2 (x) ... ®on divisor g (x) ®on divisor f (x). g (x) g 1 (x) is divisible by rk (x) ® f (x) - g (x) g 1 (x) is divisible by rk (x) ® r 1 (x) is divisible by rk (x) ® r 2 (x) is divisible by rk (x) ®… qk (x): rk (x) is divisible by rk (x).