CONIC SECTIONS, lines that are obtained by cutting a right circular cone with planes that do not pass through its vertex. There are conic sections three types. 1) The cutting plane intersects all the generatrices of the cone at the points of one of its cavity (Fig., a); the line of intersection is a closed oval curve - an ellipse, a circle like special case An ellipse is obtained when the cutting plane is perpendicular to the axis of the cone. 2) The cutting plane is parallel to one of the tangent planes of the cone (Fig., b); in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity. 3) The cutting plane intersects both cavities of the cone (Fig., c); the line of intersection - a hyperbola - consists of two identical unclosed parts (branches of the hyperbola) extending to infinity, each of which lies on its own cavity of the cone.

IN analytical geometry conic sections are real, non-disintegrating lines of the second order. In cases where a conic section has a center of symmetry (center), that is, it is an ellipse or hyperbola, its equation is Cartesian system coordinates can be reduced (by moving the origin of coordinates to the center) to the form a 11 x 2 + 2a 12 xy + a 22 y 2 = a 33, where a 11, a 12, a 22, and 33 are constants. The equations of these curves can be reduced to more simple view

Aх 2 + Βу 2 = С, (*)

if for the directions of the coordinate axes we choose the so-called main directions - the directions of the main axes (axes of symmetry) of the conic section. If constants A and B have identical signs(coinciding with the sign C), then equation (*) defines an ellipse; if A and B are of different signs, then it is a hyperbole.

The equation of a parabola cannot be reduced to the form (*). With a proper choice of coordinate axes (one coordinate axis is the only axis of symmetry of the parabola, the other is a straight line perpendicular to it, passing through the vertex of the parabola), its equation can be reduced to the form y 2 = 2 px.

Conic sections were known to mathematicians Ancient Greece. The fact that the ellipse, hyperbola and parabola are sections of cones was discovered by Menaechmus (about 340 BC). Most complete essay, dedicated to these curves, is “Conic Sections” by Apollonius of Perga (about 200 BC). Further development The theory of conic sections is associated with the creation in the 17th century of projective (J. Desargues, B. Pascal) and coordinate (R. Descartes, P. Fermat) methods. With proper choice of the coordinate system (the abscissa axis is the axis of symmetry of the conic section, the ordinate axis is tangent to the vertex of the conic section), the equation of the conic section is reduced to the form y 2 = 2рх + λх 2, where р and λ are constants, р≠0. For λ = 0 this equation defines a parabola, for λ<0 - эллипс, при λ>0 is a hyperbole. This property of the conic section, contained in the last equation, was known to the ancient Greek geometers and was the reason for Apollonius of Perga to assign certain types conic section names that have survived to this day: the word “parabola” means application (since in Greek geometry, the transformation of a rectangle of a given area y 2 into an equal rectangle with a given base 2p is called the application of a given rectangle to this base); the word “ellipse” is a disadvantage (application with a disadvantage); the word “hyperbole” is an excess (an application with excess).

The stereometric definition of a conic section can be replaced by planimetric definitions of these curves as sets of points on a plane. So, for example, an ellipse is a set of points for which the sum of the distances to two given points (foci) has the same value. It is possible to give another planimetric definition of a conic section, covering all three types of these curves: a conic section is a set of points, for each of which the ratio of the distances to a given point (focus) to the distance to a given straight line (directrix) is equal to a given positive number(eccentricity) e. At e<1 коническое сечение - эллипс; при е >1 - hyperbole; when e = 1 - a parabola.

Interest in the conic section has always been maintained by the fact that these lines are often found in descriptions various phenomena nature and in human activity. Conic sections acquired special significance after I. Kepler (1609) established through observations, and I. Newton (1687) theoretically substantiated the laws of planetary motion (one of which states that planets and comets solar system move along conic sections, at one of the foci of which the Sun is located).

Lit.: Warden B. L. van der. Awakening Science. 2nd ed. M., 2006; Alexandrov P. S. Lectures on analytical geometry. 2nd ed. M., 2008.

Municipal Educational Institution

Average Comprehensive school №4

Conic sections

Completed

Spiridonov Anton

student of class 11A

Checked

Korobeynikova A. T.

Tobolsk - 2006

Introduction

The concept of conic sections

Types of conic sections

Study

Construction of conic sections

Analytical approach

Application

Application

Bibliography

Introduction.

Purpose: to study conic sections.

Objectives: learn to distinguish between types of conic sections, construct kinetic sections and apply an analytical approach.

Conic sections were first proposed to be used by the ancient Greek geometer Menaechmus, who lived in the 4th century BC, when solving the problem of doubling a cube. This task is associated with the following legend.

One day, a plague epidemic broke out on the island of Delos. The inhabitants of the island turned to the oracle, who said that to stop the epidemic it was necessary to double the golden altar, which had the shape of a cube and was located in the temple of Apollo in Athens. The islanders made a new altar, the ribs of which were twice as large as the ribs of the previous one. However, the plague did not stop. The angry residents heard from the oracle that they misunderstood his instructions - it was not the edges of the cube that needed to be doubled, but its volume, that is, the edges of the cube had to be doubled. In terms geometric algebra, which was used by Greek mathematicians, the problem meant: according to this segment and find segments x and y such that a: x = x: y = y: 2a. Then the length of the segment x will be equal to .

The given proportion can be considered as a system of equations:

But x 2 =ay and y 2 =2ax are equations of parabolas. Therefore, to solve the problem, one must find their intersection points. If we take into account that the equation of the hyperbola xy=2a 2 can also be obtained from the system, then the same problem can be solved by finding the points of intersection of the parabola and the hyperbola.

To obtain conic sections, Menaechmus intersected a cone - acute, rectangular or obtuse - with a plane perpendicular to one of the generatrices. For an acute-angled cone, the section by a plane perpendicular to its generatrix has the shape of an ellipse. An obtuse cone gives a hyperbola, and a rectangular cone gives a parabola.

This is where the names of curves come from, which were introduced by Apollonius of Perga, who lived in the 3rd century BC: ellipse (έλλείψίς), which means a flaw, a deficiency (of the angle of a cone to a straight line); hyperbola (ύπέρβωλη) - exaggeration, preponderance (of a cone angle over a straight line); parabola (παραβολη) - approximation, equality (of the cone angle right angle). Later the Greeks noticed that all three curves could be obtained on one cone by changing the inclination of the cutting plane. In this case, you should take a cone consisting of two cavities and think that they extend to infinity (Fig. 1).

If we draw a section of a circular cone perpendicular to its axis, and then rotate the cutting plane, leaving one point of its intersection with the cone stationary, we will see how the circle will first stretch out, turning into an ellipse. Then the second vertex of the ellipse will go to infinity, and instead of an ellipse you will get a parabola, and then the plane will also intersect the second cavity of the cone and you will get a hyperbola.

The concept of conic sections.

Conic sections are plane curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex. From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. With the exception of degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas or parabolas (Fig. 2).

When rotating right triangle near one of the legs, the hypotenuse with its extensions describes a conical surface, called the surface of a right circular cone, which can be considered as a continuous series of lines passing through the vertex and called generators, all generators resting on the same circle, called the generator. Each of the generators represents the hypotenuse of a rotating triangle (in its known position), extended in both directions to infinity. Thus, each generatrix extends on both sides of the vertex, as a result of which the surface has two cavities: they converge at one point at a common vertex. If such a surface is intersected by a plane, then the section will produce a curve, which is called a conic section. It can be of three types:

1) if a plane intersects a conical surface along all generatrices, then only one cavity is dissected and a closed curve called an ellipse is obtained in the section;

2) if the cutting plane intersects both cavities, then a curve is obtained that has two branches and is called a hyperbola;

3) if the cutting plane is parallel to one of the generatrices, then a parabola is obtained.

If the cutting plane is parallel to the generating circle, then a circle is obtained, which can be considered as a special case of an ellipse. A cutting plane can intersect a conical surface only at one vertex, then the section results in a point, as a special case of an ellipse.

If a plane passing through the vertex intersects both cavities, then the section produces a pair of intersecting lines, considered as a special case of a hyperbola.

If the vertex is infinitely distant, then the conical surface turns into a cylindrical one, and its section by a plane, parallel to the generators, gives a pair of parallel lines as a special case of a parabola. Conic sections are expressed by 2nd order equations, the general form of which is

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

and are called 2nd order curves.

Types of conic sections.

Conic sections can be of three types:

1) the cutting plane intersects all generatrices of the cone at points of one of its cavity; the intersection line is a closed oval curve - an ellipse; a circle as a special case of an ellipse is obtained when the cutting plane is perpendicular to the axis of the cone.

2) The cutting plane is parallel to one of the tangent planes of the cone; in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity.

3) The cutting plane intersects both cavities of the cone; the line of intersection - a hyperbola - consists of two identical open parts extending to infinity (branches of the hyperbola) lying on both cavities of the cone.

Study.

In cases where a conic section has a center of symmetry (center), i.e., is an ellipse or hyperbola, its equation can be reduced (by moving the origin of coordinates to the center) to the form:

a 11 x 2 +2a 12 xy + a 22 y 2 = a 33 .

Further studies of such (called central) conic sections show that their equations can be reduced to an even simpler form:

Ax 2 + Wu 2 = C,

if we choose the main directions for the directions of the coordinate axes - the directions of the main axes (axes of symmetry) of conic sections. If A and B have the same signs (coinciding with the sign of C), then the equation defines an ellipse; if A and B are of different signs, then it is a hyperbole.

The equation of a parabola cannot be reduced to the form (Ax 2 + By 2 = C). With a proper choice of coordinate axes (one coordinate axis is the only axis of symmetry of the parabola, the other is a straight line perpendicular to it, passing through the vertex of the parabola), its equation can be reduced to the form:

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 a 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 F 1 and F 2 (Fig. 3), then the curve described by the point of a pencil sliding along a tightly stretched thread has the shape of an ellipse. Points F 1 and F 2 are called the focuses of the ellipse, and the segments V 1 V 2 and v 1 v 2 between the points of intersection of the ellipse with the coordinate axes - the major and minor axes. If points F 1 and F 2 coincide, then the ellipse turns into a circle (Fig. 3).

Hyperbola. When constructing a hyperbola, point P, the tip of a pencil, is fixed on a thread that slides freely along pegs installed at points F 1 and F 2, as shown in Figure 4, a, the distances are selected so that segment PF 2 is longer than segment PF 1 by a fixed value less than the distance F 1 F 2 . In this case, one end of the thread passes under the peg F 1, and both ends of the thread pass over the peg F 2. (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 (PV 1 Q), making sure that the thread remains taut all the time, and, pulling both ends of the thread down past point F 2, and when point P is below the segment F 1 F 2, holding the thread by both ends and carefully releasing it. We draw the second branch of the hyperbola by first changing the pins F 1 and F 2 (Fig. 4).

The branches of the hyperbola approach two straight lines that intersect between the branches. These straight lines, called asymptotes of the hyperbola, are constructed as shown in Figure 4, b. Corner

the coefficients of these lines are equal to where is the bisector segment of the angle between the asymptotes, perpendicular to the segment F 2 F 1 ; the segment v 1 v 2 is called the conjugate axis of the hyperbola, and the segment V 1 V 2 is its transverse axis. Thus, the asymptotes are the diagonals of a rectangle with sides passing through four points v 1, v 2, V 1, V 2 parallel to the axes. To construct this rectangle, you need to specify the location of points v 1 and v 2. They are at the same distance, equal

from the intersection point of the O axes. This formula assumes the construction of a right triangle with legs Ov 1 and V 2 O and hypotenuse F 2 O.

If the asymptotes of a hyperbola are mutually perpendicular, then the hyperbola is called equilateral. Two hyperbolas that have common asymptotes, but with the transverse and conjugate axes rearranged, 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 (second 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 headmistress. The construction of a parabola using a tensioned thread, based on the definition of Pappus, was proposed by Isidore of Miletus (VI century) (Fig. 5).

Let's position the ruler so that its edge coincides with the directrix, and apply the drawing leg AC to this edge triangle ABC. 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 into 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, since total length the thread is equal to AB, a 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 part of the leg AB, that is, 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.

ANALYTICAL APPROACH

Algebraic classification. In algebraic terms, conic sections can be defined as plane curves whose coordinates in the Cartesian coordinate system satisfy an equation of the second degree. In other words, the equation of all conic sections can be written in general form as

where not all coefficients A, B and C are equal to zero. By using parallel transfer and rotation of the axes, equation (1) can be reduced to the form

ax 2 + by 2 + c = 0

The first equation is obtained from equation (1) for B 2 > AC, the second - for B 2 = AC. Conic sections whose equations are reduced to the first form are called central. Conic sections defined by equations of the second type with q > 0 are called non-central. Within these two categories there are nine various types conic sections depending on the signs of the coefficients.

1) If the coefficients a, b and c have the same sign, then there are no real points whose coordinates would satisfy the equation. Such a conic section is called an imaginary ellipse (or an imaginary circle if a = b).

2) If a and b have the same sign, and c has the opposite sign, then the conic section is an ellipse; when a = b – circle.

3) If a and b have different signs, then the conic section is a hyperbola.

4) If a and b have different signs and c = 0, then the conic section consists of two intersecting lines.

5) If a and b have the same sign and c = 0, then there is only one real point on the curve that satisfies the equation, and a conic section is two imaginary intersecting lines. In this case, we also speak of an ellipse subtended to a point or, if a = b, a circle subtended to a point.

6) If either a or b is equal to zero, and the other coefficients have different signs, then the conic section consists of two parallel lines.

7) If either a or b is equal to zero, and the remaining coefficients have the same sign, then there is not a single real point that satisfies the equation. In this case, they say that a conic section consists of two imaginary parallel lines.

8) If c = 0, and either a or b is also zero, then the conic section consists of two real coincident lines. (The equation does not define any conic section for a = b = 0, since in this case original equation(1) not second degree.)

9) Equations of the second type define parabolas if p and q are different from zero. If p > 0 and q = 0, we obtain the curve from step 8. If p = 0, then the equation does not define any conic section, since the original equation (1) is not of the second degree.

Application

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 major axis equal to small. 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 emanates from a light source placed at the focus of a parabolic reflector parallel rays. That's why parabolic mirrors are used in high-power spotlights and car headlights. A hyperbole is a graph of many important physical relations, for example, Boyle's law (relating pressure and volume ideal gas) and Ohm's law, which specifies electricity as a function of resistance at constant voltage

Application

Bibliography.

1. Alekseev. Abel's theorem in problems and solutions. 2001

2. Bazylev V. T., Dunichev K. I., Ivanitskaya V. P.. Tutorial for 1st year students of physics and mathematics faculties pedagogical institutes. Moscow "enlightenment" 1974

3. Vereshchagin N.K., A. Shen. Lectures on mathematical logic and theory of algorithms. 1999

4. Gelfand I.M.. Lectures on linear algebra. 1998.

5. Gladky A.V. Introduction to modern logic. 2001

6. M.E. Kazaryan. Course of differential geometry (2001-2002).

7. Prasolov V.V.. Geometry of Lobachevsky 2004

8. Prasolov V.V.. Problems in planimetry 2001

9. Sheinman O.K.. Fundamentals of representation theory. 2004

STATE BUDGET

PROFESSIONAL EDUCATIONAL INSTITUTION

CITIES OF MOSCOW

"POLICE COLLEGE"

Abstract on the discipline Mathematics

On the topic: “Conic sections and their applications in technology”

Performed

Cadet of the 15th platoon

Alekseeva A.I.

Teacher

Zaitseva O.N.

Moscow

2016

Content:

Introduction

1. The concept of conic sections………………………………………………………5

2. Types of conic sections……………………………………………...7

3. Research……………………………………………………………..8

4. Properties of conic sections…. ……………………………………….9

5. Construction of conic sections…………………………………….10

6. Analytical approach……………………………………………………………14

7. Application……………………………………………………………….16

8. Across the cone………………………………………………………..17

List of used literature

Introduction

Conic sections were first proposed to be used by the ancient Greek geometer Menaechmus, who lived in the 4th century BC, when solving the problem of doubling a cube. This task is associated with the following legend.

One day, a plague epidemic broke out on the island of Delos. The inhabitants of the island turned to the oracle, who said that to stop the epidemic it was necessary to double the golden altar, which had the shape of a cube and was located in the temple of Apollo in Athens. The islanders made a new altar, the ribs of which were twice as large as the ribs of the previous one. However, the plague did not stop. The angry residents heard from the oracle that they misunderstood his instructions - it was not the edges of the cube that needed to be doubled, but its volume, that is, the edges of the cube had to be doubled.

To obtain conic sections, Menaechmus intersected a cone - acute, rectangular or obtuse - with a plane perpendicular to one of the generatrices. For an acute-angled cone, the section by a plane perpendicular to its generatrix has the shape of an ellipse. An obtuse cone gives a hyperbola, and a rectangular cone gives a parabola.

This is where the names of the curves came from, which were introduced by Apollonius of Perga, who lived in the 3rd century BC: ellipse, which means a flaw, a deficiency (the angle of a cone to a straight line); hyperbole - exaggeration, superiority (of a cone angle over a straight line); parabola - approximation, equality (of a cone angle to a right angle). Later the Greeks noticed that all three curves could be obtained on one cone by changing the inclination of the cutting plane. In this case, you should take a cone consisting of two cavities and think that they extend to infinity (Fig. 1)

If we draw a section of a circular cone perpendicular to its axis, and then rotate the cutting plane, leaving one point of its intersection with the cone stationary, we will see how the circle will first stretch out, turning into an ellipse. Then the second vertex of the ellipse will go to infinity, and instead of an ellipse you will get a parabola, and then the plane will also intersect the second cavity of the cone and you will get a hyperbola.

For a long time conic sections did not find application until astronomers and physicists became seriously interested in them. It turned out that these lines are found in nature (an example of this is the trajectories of celestial bodies) and graphically describe many physical processes(here the hyperbole is in the lead: let us at least remember Ohm’s law and the Boyle-Marriott law), not to mention their application in mechanics and optics. In practice, most often in engineering and construction, one has to deal with an ellipse and a parabola.

Fig.1

diagram

The concept of conic sections

Conic sections are plane curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex. From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. With the exception of degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas or parabolas (Fig. 2).

Fig.2

When a right triangle is rotated about one of its legs, the hypotenuse with its extensions describes a conical surface called the surface of a right circular cone, which can be considered as a continuous series of lines passing through the vertex and called generators, all generators resting on the same circle, called producing. Each of the generators represents the hypotenuse of a rotating triangle (in its known position), extended in both directions to infinity. Thus, each generatrix extends on both sides of the vertex, as a result of which the surface has two cavities: they converge at one point at a common vertex. If such a surface is intersected by a plane, then the section will produce a curve, which is called a conic section. It can be of three types:

1) if a plane intersects a conical surface along all generatrices, then only one cavity is dissected and a closed curve called an ellipse is obtained in the section;

2) if the cutting plane intersects both cavities, then a curve is obtained that has two branches and is called a hyperbola;

3) if the cutting plane is parallel to one of the generatrices, then a parabola is obtained.

If the cutting plane is parallel to the generating circle, then a circle is obtained, which can be considered as a special case of an ellipse. A cutting plane can intersect a conical surface only at one vertex, then the section results in a point, as a special case of an ellipse.

If a plane passing through a vertex intersects both planes, then the section produces a pair of intersecting lines, considered as a special case of a hyperbola.

If the vertex is infinitely distant, then the conical surface turns into a cylindrical one, and its section by a plane parallel to the generators gives a pair of parallel lines as a special case of a parabola. Conic sections are expressed by 2nd order equations, the general form of which is

Ax 2 +Whoo+C + Dx + Ey + F= 0 and are called 2nd order curves.
(conic section)

Types of conical sections .

Conic sections can be of three types:

1) the cutting plane intersects all generatrices of the cone at points of one of its cavity; the intersection line is a closed oval curve - an ellipse; a circle as a special case of an ellipse is obtained when the cutting plane is perpendicular to the axis of the cone.

2) The cutting plane is parallel to one of the tangent planes of the cone; in cross-section, the result is an open curve that goes to infinity - a parabola, lying entirely on one cavity.

3) The cutting plane intersects both cavities of the cone; the line of intersection - a hyperbola - consists of two identical open parts extending to infinity (branches of the hyperbola) lying on both cavities of the cone.

(Fig. 1) parabola (Fig. 2) ellipse (Fig. 3) hyperbola

Study

In cases where a conic section has a center of symmetry (center), i.e., is an ellipse or hyperbola, its equation can be reduced (by moving the origin of coordinates to the center) to the form:

a 11 x 2 +2xy+a 22 y 2 =a 33 .

Further studies of such (called central) conic sections show that their equations can be reduced to an even simpler form:

Oh 2 + Wu 2 = C,

if we choose the main directions for the directions of the coordinate axes - the directions of the main axes (axes of symmetry) of conic sections. If A and B have the same signs (coinciding with the sign of C), then the equation defines an ellipse; if A and B are of different signs, then it is a hyperbole.

Reduce the parabola equation to the form (Ah 2 + Wu 2 = C) it is impossible. With a proper choice of coordinate axes (one coordinate axis is the only axis of symmetry of the parabola, the other is a straight line perpendicular to it, passing through the vertex of the parabola), its equation can be reduced to the form:

y 2 = 2px.

PROPERTIES OF CONIC SECTIONS

Definitions of Pappus. Establishing the focus of a parabola gave Pappus the idea of ​​giving an alternative definition of conic sections in general. Let F - set point(focus), and L is a given straight line (directrix) not passing through F, and DF and DL are the distances from the moving point P to the focus F and directrix L, respectively. Then, as Papp showed, conic sections are defined as geometric places points P for which the ratio DF:DL is a non-negative constant. This ratio is called the eccentricity e of the conic section. When e< 1 коническое сечение - эллипс; при e >1 - hyperbole; when e = 1 - parabola. If F lies on L, then the loci have the form of lines (real or imaginary), which are degenerate conic sections. The striking symmetry of the ellipse and hyperbola suggests that each of these curves has two directrixes and two foci, and this circumstance led Kepler in 1604 to the idea that the parabola also has a second focus and a second directrix - infinitely remote point and straight. In the same way, a circle can be considered as an ellipse, the foci of which coincide with the center, and the directrixes are at infinity. The eccentricity e in this case is zero.

Properties. The properties of conic sections are truly inexhaustible, and any of them can be taken as defining. Important place in the Mathematical Collection of Pappus, Descartes' Geometry (1637) and Newton's Principia (1687) the problem of the geometric location of points relative to four straight lines is occupied. If four straight lines L are given on the plane 1 , L 2 , L 3 and L4 (two of which may coincide) and the point P is such that the product of the distances from P to L 1 and L 2 proportional to the product of the distances from P to L 3 and L 4 , then the locus of points P is a conic section.

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 a 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 F 1 and F 2 (Fig. 3), then the curve described by the point of a pencil sliding along a tightly stretched thread has the shape of an ellipse. F points 1 and F2 are called the foci of the ellipse, and the segments V 1 V 2 and v 1 v 2 between the points of intersection of the ellipse with the coordinate axes - the major and minor axes. If points F 1 and F 2 coincide, then the ellipse turns into a circle (Fig. 3).

Fig.3

Hyperbola. When constructing a hyperbola, point P, the tip of a pencil, is fixed on a thread that slides freely along pegs installed at points F 1 and F 2 , as shown in Figure 4, a, the distances are selected so that the segment PF 2 longer than segment PF 1 by a fixed value less than the distance F 1 F 2 . In this case, one end of the thread passes under the pin F 1 , and both ends of the thread pass over the pin F 2 . (The pencil point 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.) One branch of the hyperbola (PV 1 Q) we draw, making sure that the thread remains taut at all times, and by pulling both ends of the thread down past point F 2 , and when point P is below segment F 1 F 2 , holding the thread at both ends and carefully releasing it. We draw the second branch of the hyperbola by first changing the pegs F 1 and F 2 (Fig. 4).

Fig.4

The branches of the hyperbola approach two straight lines that intersect between the branches. These lines are called asymptotes of a hyperbola. Angle coefficients of these lines are equal to where is the segment of the bisector of the angle between the asymptotes, perpendicular to the segment F 2 F 1 ; segment v 1 v 2 is called the conjugate axis of the hyperbola, and the segment V 1 V 2 – its transverse axis. Thus, the asymptotes are the diagonals of a rectangle with sides passing through four points v 1 , v 2 , V 1 , V 2 parallel to the axes. To construct this rectangle, you need to specify the location of the points v 1 and v 2 . They are at the same distance, equal to the point of intersection of the axes O. This formula assumes the construction of a right triangle with legs Ov 1 and V 2 O and hypotenuse F 2 O.

If the asymptotes of a hyperbola are mutually perpendicular, then the hyperbola is called equilateral. Two hyperbolas that have common asymptotes, but with the transverse and conjugate axes rearranged, 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 (second 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 headmistress. The construction of a parabola using a tensioned thread, based on the definition of Pappus, was proposed by Isidore of Miletus (VI century) (Fig. 5).

Fig.5

ANALYTICAL APPROACH

Algebraic classification. In algebraic terms, conic sections can be defined as plane curves whose coordinates in the Cartesian coordinate system satisfy an equation of the second degree. In other words, the equation of all conic sections can be written in general form as where not all coefficients A, B and C are equal to zero. Using parallel translation and rotation of the axes, equation (1) can be reduced to the form

ax 2 + by 2 + c = 0

or

px 2 +q y = 0.

The first equation is obtained from equation (1) for B2 > AC, the second - for B 2 = AC. Conic sections whose equations are reduced to the first form are called central. Conic sections defined by equations of the second type with q > 0 are called non-central. Within these two categories, there are nine different types of conic sections depending on the signs of the coefficients.

1) If the coefficients a, b and c have the same sign, then there are no real points whose coordinates would satisfy the equation. Such a conic section is called an imaginary ellipse (or an imaginary circle if a = b).

2) If a and b have the same sign, and c has the opposite sign, then the conic section is an ellipse; when a = b - circle.

3) If a and b have different signs, then the conic section is a hyperbola.

4) If a and b have different signs and c = 0, then the conic section consists of two intersecting lines.

5) If a and b have the same sign and c = 0, then there is only one real point on the curve that satisfies the equation, and the conic section is two imaginary intersecting lines. In this case, we also speak of an ellipse subtended to a point or, if a = b, a circle subtended to a point.

6) If either a or b is equal to zero, and the other coefficients have different signs, then the conic section consists of two parallel lines.

7) If either a or b is equal to zero, and the remaining coefficients have the same sign, then there is not a single real point that satisfies the equation. In this case, they say that a conic section consists of two imaginary parallel lines.

8) If c = 0, and either a or b is also zero, then the conic section consists of two real coincident lines. (The equation does not define any conic section for a = b = 0, since in this case the original equation (1) is not of the second degree.)

9) Equations of the second type define parabolas if p and q are different from zero. If p > 0 and q = 0, we obtain the curve from step 8. If p = 0, then the equation does not define any conic section, since the original equation (1) is not of the second degree.

Application

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.

All bodies in the Solar System move around the Sun in ellipses. Celestial bodies entering the solar system from other star systems, move around the Sun in a hyperbolic orbit and, if their movement is not significantly influenced by the planets of the Solar system, they leave it in the same orbit. They move in ellipses around the Earth artificial satellites And natural satellite- Moon, huh spaceships, launched towards other planets, move at the end of the engines along parabolas or hyperbolas (depending on the speed) until the attraction of other planets or the Sun becomes comparable to gravity(Fig. 3).

Across the cone

The ellipse and its special case - the circle, parabola and hyperbola are easy to obtain experimentally. For example, an ice cream cone would be quite suitable for the role of a cone. Mentally draw one of its generatrices and cut the horn under different angles To her. The task is to make only four attempts and obtain all possible conic sections on the slices. It’s even easier to carry out the experiment with a flashlight: depending on its position in space, the cone of light will create spots on the wall of the room different shapes. The boundary of each spot is one of the conic sections. By turning the flashlight in a vertical plane, you will see how one curve replaces another: the circle is stretched into an ellipse, then it turns into a parabola, and it, in turn, into a hyperbola.

A mathematician solves the same problem theoretically by comparing two angles: α - between the axis of the cone and the generatrix and β - between the cutting plane and the axis of the cone. And here is the result: for α< β в сечении получится эллипс или окружность, при α = β - парабола, а при α >β is a branch of a hyperbola. If we consider the generators to be straight lines and not segments, that is, consider an unlimited symmetrical figure from two cones with common top, it will become clear that an ellipse is a closed curve, a parabola consists of one infinite branch, and a hyperbola consists of two.

The simplest conic section - a circle - can be drawn using a thread and a nail. It is enough to tie one end of the thread to a nail stuck into the paper, and the other to a pencil and pull it tight. Having done full turn, the pencil will outline a circle. Or you can use a compass: by changing its solution, you can easily draw a whole family of circles.

LIST OF REFERENCES USED

1.Vereshchagin N.K., A.Shen. Lectures on mathematical logic and theory of algorithms. 1999

2. Prasolov V.V.. Geometry of Lobachevsky 2004

4. Prasolov V.V.. Geometry of Lobachevsky 2004

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.
see also CELESTIAL MECHANICS.
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 are indebted to Apollo and modern names curves - ellipse, parabola and hyperbola. In his constructions, Apollonius used a two-cavity 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 a 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 - the major and minor axes. If points F1 and F2 coincide, then the ellipse turns into a circle.

From the intersection point of the O axes. This formula assumes the construction of a 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 the transverse and conjugate axes rearranged, 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 a 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.

Where the coefficients A, B and C are not all zero. Using parallel translation and rotation of the axes, equation (1) can be reduced to the form ax2 + by2 + c = 0
or px2 + qy = 0. The first equation is obtained from equation (1) with B2 No. AC, the second - with B2 = AC. Conic sections whose equations are reduced to the first form are called central. Conic sections defined by equations of the second type with q No. 0 are called non-central. Within these two categories, there are nine different types of conic sections depending on the signs of the coefficients. 1) If the coefficients a, b and c have the same sign, then there are no real points whose coordinates would satisfy the equation. Such a conic section is called an imaginary ellipse (or an imaginary circle if a = b). 2) If a and b have the same sign, and c has the opposite sign, then the conic section is an ellipse (Fig. 1,a); when a = b - circle (Fig. 6, b).

Where a - constant coefficient. The number a expresses the length of the focal parameter of the parabola. If the angle at the vertex of the cone is acute, then the segment RQ is not equal to the segment AE; but the relation y2 = RQ×QS is equivalent to an equation of the form

Where a and b are constants, or, after shifting the axes, the equation

Which is the equation of an ellipse. The x-intercepts of the ellipse (x = a and x = -a) and the y-intercepts of the ellipse (y = b and y = -b) define the major and minor axes, respectively. If the angle at the vertex of the cone is obtuse, then the curve of intersection of the cone and the plane has the form of a hyperbola, and the equation takes the following form:

Or, after transferring the axes,

In this case, the x-intercepts, given by x2 = a2, define the transverse axis, and the y-intercepts, given by y2 = -b2, define the conjugate axis. If the constants a and b in equation (4a) are equal, then the hyperbola is called equilateral. By rotating the axes, its equation is reduced to the form xy = k.
Now from equations (3), (2) and (4) we can understand the meaning of the names given by Apollonius to the three main conic sections. The terms "ellipse", "parabola" and "hyperbola" come from Greek words, meaning “deficient,” “equal,” and “superior.” From equations (3), (2) and (4) it is clear that for the ellipse y2 (2b2/a) x. In each case, the value enclosed in parentheses is equal to the focal parameter of the curve. Apollonius himself considered only three general type conic sections (types 2, 3 and 9 listed above), but his approach allows a generalization to consider all real second-order curves. If the cutting plane is chosen parallel to the circular base of the cone, then the cross-section will result in a circle. If the cutting plane has only one common point with a cone, its vertex, you get a section of type 5; if it contains a vertex and a tangent to the cone, then we obtain a section of type 8 (Fig. 6,b); if the cutting plane contains two generatrices of the cone, then the section produces a curve of type 4 (Fig. 6a); when the vertex is transferred to infinity, the cone turns into a cylinder, and if the plane contains two generatrices, then a section of type 6 is obtained. If the circle is viewed from an oblique angle, then it looks like an ellipse. The relationship between a circle and an ellipse, known to Archimedes, becomes obvious if the circle X2 + Y2 = a2 is transformed into an ellipse using the substitution X = x, Y = (a/b) y, given by the equation(3a). The transformation X = x, Y = (ai/b) y, where i2 = -1, allows us to write the equation of the circle in the form (4a). This shows that a hyperbola can be viewed as an ellipse with an imaginary minor axis, or, conversely, an ellipse can be viewed as a hyperbola with an imaginary conjugate axis. The relationship between the ordinates of a circle x2 + y2 = a2 and the ellipse (x2/a2) + (y2/b2) = 1 leads directly to Archimedes' formula A = pab for the area of ​​an ellipse. Kepler knew the approximate formula p (a + b) for the perimeter of an ellipse close to a circle, but the exact expression was obtained only in the 18th century. after the introduction of elliptic integrals. As Archimedes showed, the area of ​​a parabolic segment is four-thirds the area of ​​an inscribed triangle, but the length of the arc of a parabola could only be calculated after the 17th century. Differential calculus was invented.
PROJECTIVE APPROACH
Projective geometry is closely related to the construction of perspective. If you draw a circle on a transparent sheet of paper and place it under a light source, then this circle will be projected onto the plane below. Moreover, if the light source is located directly above the center of the circle, and the plane and the transparent sheet are parallel, then the projection will also be a circle (Fig. 8). The position of the light source is called the vanishing point. It is designated by the letter V. If V is not located above the center of the circle or if the plane is not parallel to the sheet of paper, then the projection of the circle takes the shape of an ellipse. With an even greater inclination of the plane, the major axis of the ellipse (projection of the circle) lengthens, and the ellipse gradually turns into a parabola; on a plane parallel to straight line VP, the projection has the form of a parabola; with an even greater inclination, the projection takes the form of one of the branches of the hyperbola.

Collier's Encyclopedia. - Open Society. 2000 .

See what "CONIC SECTIONS" is in other dictionaries:

Conic sections: circle, ellipse, parabola (the section plane is parallel to the generatrix of the cone), hyperbola. A conic section or conic is the intersection of a plane with a circular cone. There are three main types of conic sections: ellipse, ... ... Wikipedia

Curves resulting from the intersection of a cone with a plane in different directions; their types: ellipse, hyperbola, parabola. Complete dictionary foreign words, which have come into use in the Russian language. Popov M., 1907. CONIC SECTIONS so called. curves...... Dictionary of foreign words of the Russian language

Lines of intersection of a circular cone (see. Conical surface) with planes not passing through its vertex. Depending on the relative position cone and cutting plane, three types of conic sections are obtained: ellipse, parabola, hyperbola ... Big Encyclopedic Dictionary

CONIC SECTIONS

- plane curves that are obtained by intersecting a right circular cone with a plane that does not pass through its vertex. From the point of view of analytical geometry, a conic section is the locus of points satisfying a second-order equation. Except in degenerate cases, 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.

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, 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 is formed when a cutting plane intersects all generatrices of the cone at points in one of its cavity; parabola - when the cutting plane is parallel to one of the tangent planes of the cone; hyperbola - when the cutting plane intersects both cavities of the cone.

Studying conic sections as intersections of planes and cones, ancient Greek mathematicians also considered them as trajectories of points on a plane. It was found that an ellipse can be defined as the locus of points, the sum of the distances from which to two given points is constant; parabola - as a locus of points equidistant from a given point and a given 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.

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).

Establishing the focus of a parabola gave Pappus the idea of ​​giving an alternative definition of conic sections in general. Let F be a given point (focus), and L be a given straight line (directrix) not passing through F, and DF and DL the distances from the moving point P to the focus F and directrix L, respectively. Then, as Papp showed, conic sections are defined as the locus of points P for which the ratio DF/DL is a non-negative constant. This ratio is called the eccentricity e of the conic section. When e< 1 коническое сечение - эллипс; при e >1 - hyperbole; when e = 1 - parabola. If F lies on L, then the loci have the form of lines (real or imaginary), which are degenerate conic sections. The striking symmetry of the ellipse and hyperbola suggests that each of these curves has two directrixes and two foci, and this circumstance led Kepler in 1604 to the idea that a parabola also has a second focus and a second directrix - a point at infinity and straight. In the same way, a circle can be considered as an ellipse, the foci of which coincide with the center, and the directrixes are at infinity. The eccentricity e in this case is zero.

LITERATURE
Van der Waerden B.L. Awakening Science. M., 1959 Alexandrov P.S. Lectures on analytical geometry. M., 1968