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CONIC SECTIONS, plane curves, which are obtained by crossing a right circular cone with a plane that does not pass through its top (Fig. 1). From the point of view of analytical geometry, a conic section is the locus of points that satisfy a second-order equation. With the exception of the degenerate cases discussed in the last section, conic sections are ellipses, hyperbolas, or parabolas.

Conic sections are often found in nature and technology. For example, the orbits of the planets revolving around the Sun are ellipses. A circle is a special case of an ellipse, in which the major axis is equal to the minor one. A parabolic mirror has the property that all incident rays parallel to its axis converge at one point (focus). This is used in most reflecting telescopes using parabolic mirrors, as well as in radar antennas and special microphones with parabolic reflectors. A beam of parallel rays emanate from a light source placed at the focus of a parabolic reflector. Therefore, parabolic mirrors are used in powerful spotlights and car headlights. A hyperbola is a graph of many important physical ratios, for example, Boyle's law (linking pressure and volume ideal gas) and Ohm's law, which specifies electricity as a function of resistance at constant voltage.

EARLY HISTORY

The discoverer of conic sections is supposedly Menechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menechmus used a parabola and an isosceles hyperbola to solve the problem of doubling a cube.

Treatises on conic sections written by Aristaeus and Euclid at the end of the 4th century. BC, were lost, but the materials from them were included in the famous Conic sections Apollonius of Perga (c. 260-170 BC), which have survived to our time. Apollonius abandoned the requirement that the secant plane of the generatrix of the cone be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We are indebted to Apollonius modern names curves - ellipse, parabola and hyperbola.

In his constructions, Apollonius used a two-sheeted circular cone (as in Fig. 1), so for the first time it became clear that a hyperbola is a curve with two branches. Since the time of Apollonius, conic sections have been divided into three types, depending on the inclination of the cutting plane to the generatrix of the cone. Ellipse (Fig. 1, a) is formed when the cutting plane intersects all generatrixes 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; hyperbole (Fig. 1, in) - when the cutting plane intersects both cavities of the cone.

CONSTRUCTION OF CONIC SECTIONS

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

These definitions of conic sections as plane curves also suggest a way to construct them using a stretched thread.

Ellipse.

If the ends of a thread of a given length are fixed at points F 1 and F 2 (Fig. 2), then the curve described by the tip of a pencil sliding along a tightly stretched thread has the shape of an ellipse. points F 1 and F 2 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 the points F 1 and F 2 coincide, then the ellipse turns into a circle.

Hyperbola.

When constructing a hyperbola, a point P, the tip of a pencil, is fixed on a thread that slides freely along the pegs installed at the points F 1 and F 2 as shown in fig. 3, a. The distances are chosen so that the segment PF 2 is longer than the segment PF 1 by a fixed amount 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 tip of the pencil should not slide along the thread, so you need to fix it by making a small loop on the thread and threading the tip into it.) One branch of the hyperbola ( PV 1 Q) we draw, making sure that the thread remains taut all the time, and pulling both ends of the thread down past the point F 2 , and when the point P will be below the line F 1 F 2, holding the thread at both ends and carefully easing (i.e. releasing) it. The second branch of the hyperbola ( Pў V 2 Qў) we draw, having previously changed the roles of the pegs F 1 and F 2 .

The branches of the hyperbola approach two straight lines that intersect between the branches. These lines, called the asymptotes of the hyperbola, are constructed as shown in Fig. 3, b. Slopes these lines are equal to ± ( v 1 v 2)/(V 1 V 2), where v 1 v 2 - a segment of the bisector of the angle between the asymptotes, perpendicular to the segment F 1 F 2; line segment v 1 v 2 is called the conjugate axis of the hyperbola, and the segment V 1 V 2 - its transverse axis. So 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 build this rectangle, you need to specify the location of the points v 1 and v 2. They are at the same distance, equal to

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

If the asymptotes of the hyperbola are mutually perpendicular, then the hyperbola is called isosceles. Two hyperbolas having common asymptotes, but with rearranged transverse and conjugate axes, are called mutually conjugate.

Parabola.

The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola, apparently, was first established by Pappus (2nd half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called the director. The construction of a parabola using a stretched thread, based on the definition of Pappus, was proposed by Isidore of Miletus (6th century). Position the ruler so that its edge coincides with the directrix LLў (Fig. 4), and attach the leg to this edge AC drawing triangle ABC. We fix one end of the thread with a length AB at the top B triangle and the other at the focus of the parabola F. Pulling the thread with the tip of a pencil, press the tip at a variable point P to the free skate AB drawing triangle. As the triangle moves along the ruler, the point P will describe the arc of a parabola with focus F and headmistress LLў, since the total length of the thread is equal to AB, the segment of the thread is adjacent to the free leg of the triangle, and therefore the remaining segment of the thread PF must be equal to the rest of the leg AB, i.e. PA. Intersection point V parabola with an axis is called the vertex of the parabola, a straight line passing through F and V, is the axis of the parabola. If a straight line perpendicular to the axis is drawn through the focus, then the segment of this straight line cut off by the parabola is called the focal parameter. For an ellipse and a hyperbola, the focal parameter is defined similarly.

PROPERTIES OF CONIC SECTIONS

Pappus definitions.

Establishing the focus of the parabola led Pappus to the idea of ​​giving an alternative definition of conic sections in general. Let F – given point(focus) and L is a given straight line (directrix) that does not pass through F, and D F and D L– distance from the moving point P to focus F and directors L respectively. Then, as Papp showed, conic sections are defined as locus of points P, for which the ratio D F/D L is a non-negative constant. This ratio is called eccentricity e conical section. At e e > 1 is a hyperbola; at e= 1 is a parabola. If a F lies on L, then the locus has the form of lines (real or imaginary), which are degenerate conic sections.

The conspicuous symmetry of the ellipse and the 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 - a point at infinity and straight. Similarly, the circle can be considered as an ellipse, whose foci coincide with the center, and the directrixes are at infinity. Eccentricity e in this case is zero.

Dandelin's design.

Focuses and directrixes of a conic section can be clearly demonstrated using spheres inscribed in a cone and called Dandelin spheres (balls) in honor of the Belgian mathematician and engineer J. Dandelin (1794–1847), who proposed the following construction. Let the conic section be formed by the intersection of some plane p with a two-cavity right circular cone with apex at a point O. Let us inscribe two spheres in this cone S 1 and S 2 that touch the plane p at points F 1 and F 2 respectively. If the conic section is an ellipse (Fig. 5, a), then both spheres are inside the same cavity: one sphere is located above the plane p and the other below it. Each generatrix of the cone touches both spheres, and the locus of points of contact has the form of two circles C 1 and C 2 located in parallel planes p 1 and p 2. Let P is an arbitrary point on a conic section. Let's draw straight PF 1 , PF 2 and extend the line PO. These lines are tangent to the spheres at the points F 1 , F 2 and R 1 , R 2. Since all tangents drawn to the sphere from one point are equal, then PF 1 = PR 1 and PF 2 = PR 2. Consequently, PF 1 + PF 2 = PR 1 + PR 2 = R 1 R 2. Since the planes p 1 and p 2 parallel, segment R 1 R 2 is of constant length. Thus, the value PR 1 + PR 2 is the same for all point positions P, and point P belongs to the locus of points for which the sum of distances from P before F 1 and F 2 is constant. Therefore, the points F 1 and F 2 - foci of elliptical section. In addition, it can be shown that the lines along which the plane p crosses the plane p 1 and p 2 , are directrixes of the constructed ellipse. If a p crosses both cavities of the cone (Fig. 5, b), then two Dandelin spheres lie on the same side of the plane p, one sphere in each cavity of the cone. In this case, the difference between PF 1 and PF 2 is constant, and the locus of points P has the form of a hyperbola with foci F 1 and F 2 and straight lines - intersection lines p With p 1 and p 2 - as directors. If the conic section is a parabola, as shown in Fig. 5, in, then only one Dandelin sphere can be inscribed in the cone.

Other properties.

The properties of conic sections are truly inexhaustible, and any of them can be taken as decisive. important place in Mathematical meeting Pappa (c. 300), geometries Descartes (1637) and Beginnings Newton (1687) is concerned with the problem of the locus of points with respect to four lines. If four straight lines are given on the plane L 1 , L 2 , L 3 and L 4 (two of which can match) and a dot P is such that the product of distances from P before L 1 and L 2 is proportional to the product of distances from P before L 3 and L 4 , then the locus of points P is a conic section. Mistakenly believing that Apollonius and Pappus failed to solve the problem of the locus of points with respect to four lines, Descartes, in order to obtain a solution and generalize it, created analytic geometry.

ANALYTICAL APPROACH

Algebraic classification.

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

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

ax 2 + by 2 + c = 0

px 2 + qy = 0.

The first equation is obtained from equation (1) with B 2 № AC, the second - at B 2 = AC. Conic sections whose equations are reduced to the first form are called central. Conic sections given 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.

2831) i 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 one sign, and c- opposite, then the conic section is an ellipse (Fig. 1, a); at a = b- circle (Fig. 6, b).

3) If a and b have different signs, then the conic section is a hyperbola (Fig. 1, in).

4) If a and b have different signs and c= 0, then the conic section consists of two intersecting straight lines (Fig. 6, a).

5) If a and b have one 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, one also speaks of an ellipse contracted to a point or, if a = b, contracted to a point of a circle (Fig. 6, b).

6) If either a, or b is equal to zero, and the remaining 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 no real point that satisfies the equation. In this case, the conic section is said to consist of two imaginary parallel lines.

8) If c= 0, and either a, or b is also equal to zero, then the conic section consists of two real coinciding lines. (The equation does not define any conic section at 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 a p No. 0, and q= 0, we obtain the curve from item 8. If, on the other hand, p= 0, then the equation does not define any conic section, since the original equation (1) is not of the second degree.

Derivation of the equations of conic sections.

Any conic section can also be defined as a curve along which a plane intersects with a quadratic surface, i.e. with the surface given by the equation of the second degree f (x, y, z) = 0. Apparently, conic sections were first recognized in this form, and their names ( see below) are related to the fact that they were obtained by crossing the plane with the cone z 2 = x 2 + y 2. Let ABCD- the base of a right circular cone (Fig. 7) with a right angle at the top V. Let the plane FDC intersects generatrix VB at the point F, the base is in a straight line CD and the surface of the cone - along the curve DFPC, where P is any point on the curve. Draw through the middle of the segment CD- point E- direct EF and diameter AB. Through the dot P draw a plane parallel to the base of the cone, intersecting the cone in a circle RPS and direct EF at the point Q. Then QF and QP can be taken, respectively, for the abscissa x and ordinate y points P. The resulting curve will be a parabola.

The construction shown in fig. 7, can be used to output general equations conic sections. The square of the length of a segment of a perpendicular, restored from any point of the diameter to the intersection with the circle, is always equal to the product of the lengths of the segments of the diameter. That's why

y 2 = RQ H QS.

For a parabola, a segment RQ has a constant length (because for any position of the point P it is equal to the segment AE), and the length of the segment QS proportional x(from the relation QS/EB = QF/F.E.). Hence it follows that

where a – constant factor. Number a expresses the length of the focal parameter of the parabola.

If the angle at the apex of the cone is acute, then the segment RQ not equal to cut AE; but the ratio y 2 = RQ H QS is equivalent to an equation of the form

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

which is the equation of an ellipse. Intersection points of the ellipse with the axis x (x = a and x = –a) and the points of intersection of the ellipse with the axis y (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 moving the axes,

In this case, the points of intersection with the axis x, given by the relation x 2 = a 2 , define the transverse axis, and the points of intersection with the axis y, given by the relation y 2 = –b 2 define the mating axis. If constant a and b in equation (4a) are equal, then the hyperbola is called isosceles. 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 "lack", "equal" and "superior". From equations (3), (2) and (4) it is clear that for an ellipse y 2 b 2 / a) x, for the parabola y 2 = (a) x and for hyperbole y 2 > (2b 2 /a) x. In each case, the value enclosed in brackets 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 for a generalization that allows us to consider all real second-order curves. If the cutting plane is chosen parallel to the circular base of the cone, then a circle will be obtained in the section. If the cutting plane has only one common point with the cone, its vertex, then a section of type 5 will be obtained; if it contains a vertex and a tangent to the cone, then we get a section of type 8 (Fig. 6, b); if the cutting plane contains two generatrices of the cone, then a type 4 curve is obtained in the section (Fig. 6, a); when the vertex is transferred to infinity, the cone turns into a cylinder, and if the plane contains two generators, then a section of type 6 is obtained.

When viewed from an oblique angle, a circle looks like an ellipse. The relationship between the circle and the ellipse, known to Archimedes, becomes obvious if the circle X 2 + Y 2 = a 2 using substitution X = x, Y = (a/b) y convert to ellipse, given by the equation(3a). transformation X = x, Y = (ai/b) y, where i 2 = –1, allows us to write the circle equation 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.

Relationship between the ordinates of a circle x 2 + y 2 = a 2 and ellipse ( x 2 /a 2) + (y 2 /b 2) = 1 leads directly to the formula of Archimedes A = p ab for the area of ​​the 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 of 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. In this case, 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 marked with the letter V. If a V located not 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 form of an ellipse. With an even greater inclination of the plane, the major axis of the ellipse (the projection of the circle) lengthens, and the ellipse gradually turns into a parabola; on a plane parallel to a straight line VP, the projection looks like a parabola; with an even greater inclination, the projection takes the form of one of the branches of the hyperbola.

Each point on the original circle corresponds to some point on the projection. If the projection has the form of a parabola or hyperbola, then they say that the point corresponding to the point P, is at infinity or at infinity.

As we have seen, with a suitable choice of vanishing points, a circle can be projected into ellipses of various sizes and with various eccentricities, and the lengths of the major axes are not directly related to the diameter of the projected circle. Therefore, projective geometry does not deal with distances or lengths per se, its task is to study the ratio of lengths that is preserved under projection. This relation can be found using the following construction. through any point P plane we draw two tangents to any circle and connect the points of contact with a straight line p. Let another line passing through the point P, intersects the circle at points C 1 and C 2 , but the straight line p- at the point Q(Fig. 9). Planimetry proves that PC 1 /PC 2 = –QC 1 /QC 2. (The minus sign occurs because the direction of the segment QC 1 opposite to the directions of other segments.) In other words, the points P and Q divide the segment C 1 C 2 externally and internally in the same respect; they also say that the harmonic ratio of the four segments is - 1. If the circle is projected into a conic section and the same designations are kept for the corresponding points, then the harmonic ratio ( PC 1)(QC 2)/(PC 2)(QC 1) will remain equal - 1. Point P called the pole of the line p with respect to a conic section, and a straight line p- polar point P with respect to the conic section.

When dot P approaches a conic section, the polar tends to take the position of a tangent; if point P lies on the conic section, then its polar coincides with the tangent to the conic section at the point P. If point P located inside the conic section, then its polar can be constructed as follows. Let's pass through the point P any straight line intersecting a conic section at two points; draw tangents to the conic section at the points of intersection; suppose that these tangents intersect at a point P one . Let's pass through the point P another straight line that intersects the conic section at two other points; suppose that the tangents to the conic section at these new points intersect at the point P 2 (Fig. 10). Line passing through points P 1 and P 2 , and there is the desired polar p. If point P approaching the center O central conic section, then the polar p moves away from O. When dot P coincides with O, then its polar becomes at infinity, or ideal, straight on the plane.

SPECIAL BUILDINGS

Of particular interest to astronomers is the following simple construction of the points of an ellipse using a compass and straightedge. Let an arbitrary line passing through a point O(Fig. 11, a), intersects at points Q and R two concentric circles centered at a point O and radii b and a, where b a. Let's pass through the point Q horizontal line, and R- a vertical line, and denote their intersection point P P when rotating straight OQR around the dot O will be an ellipse. Corner f between the line OQR and the major axis is called the eccentric angle, and the constructed ellipse is conveniently specified by the parametric equations x = a cos f, y = b sin f. Excluding the parameter f, we obtain equation (3a).

For a hyperbola, the construction is largely similar. Arbitrary line passing through a point O, intersects one of the two circles at a point R(Fig. 11, b). To the point R one circle and to the end point S horizontal diameter of another circle, we draw tangents intersecting OS at the point T and OR- at the point Q. Let the vertical line passing through the point T, and a horizontal line passing through the point Q, intersect at a point P. Then the locus of points P when rotating the segment OR around O there will be a hyperbola given by the parametric equations x = a sec f, y = b tg f, where f- eccentric angle. These equations were obtained by the French mathematician A. Legendre (1752–1833). By excluding the parameter f, we get equation (4a).

An ellipse, as noted by N. Copernicus (1473-1543), can be built using an epicyclic movement. If a circle rolls without sliding along the inside of another circle of twice the diameter, then each point P, not lying on a smaller circle, but fixed relative to it, will describe an ellipse. If point P is on a smaller circle, then the trajectory of this point is a degenerate case of an ellipse - the diameter of a larger circle. An even simpler construction of an ellipse was proposed by Proclus in the 5th century. If ends A and B straight line segment AB of a given length slide along two fixed intersecting straight lines (for example, along the coordinate axes), then each internal point P segment will describe an ellipse; the Dutch mathematician F. van Schoten (1615–1660) showed that any point in the plane of intersecting lines, fixed relative to the sliding segment, will also describe an ellipse.

B. Pascal (1623-1662) at the age of 16 formulated the now famous Pascal's theorem, which says: three points of intersection of opposite sides of a hexagon inscribed in any conic section lie on one straight line. Pascal derived more than 400 corollaries from this theorem.

Surfaces of the second order are surfaces that in a rectangular coordinate system are determined by algebraic equations of the second degree.

1. Ellipsoid.

An ellipsoid is a surface that, in some rectangular coordinate system, is defined by the equation:

Equation (1) is called canonical equation ellipsoid.

Set the geometric view of the ellipsoid. To do this, consider sections of the given ellipsoid by planes parallel to the plane Oxy. Each of these planes is defined by an equation of the form z=h, where h- any number, and the line that is obtained in the section is determined by two equations

(2)

Let us study equations (2) for various values h .

> c(c>0), then equations (2) also define an imaginary ellipse, i.e., intersection points of the plane z=h with the given ellipsoid does not exist. , then and line (2) degenerates into points (0; 0; + c) and (0; 0; - c) (the planes touch the ellipsoid). , then equations (2) can be represented as

whence it follows that the plane z=h intersects the ellipsoid along an ellipse with semiaxes

and . When decreasing, the values ​​of and increase and reach their highest values at , i.e. in the section of the ellipsoid by the coordinate plane Oxy it turns out the largest ellipse with semiaxes and .

A similar picture is obtained when the given surface is intersected by planes parallel to the coordinate planes Oxz and Oyz.

Thus, the considered sections make it possible to depict the ellipsoid as a closed oval surface (Fig. 156). Quantities a, b, c called axle shafts ellipsoid. When a=b=c ellipsoid is spheroth.

2. One-band hyperboloid.

A one-strip hyperboloid is a surface that, in some rectangular coordinate system, is defined by the equation (3)

Equation (3) is called the canonical equation of a one-band hyperboloid.

Set the surface type (3). To do this, consider the section by its coordinate planes Oxy (y=0)andOx(x=0). We obtain, respectively, the equations

and

Now consider sections of this hyperboloid by planes z=h parallel to the coordinate plane Oxy. The line obtained in the section is determined by the equations

or (4)

from which it follows that the plane z=h intersects the hyperboloid along an ellipse with semi-axes

and ,

reaching their lowest values ​​at h=0, i.e. in the section of this hyperboloid, the coordinate axis Oxy produces the smallest ellipse with semi-axes a*=a and b*=b. With an infinite increase

the quantities a* and b* increase infinitely.

Thus, the considered sections make it possible to depict a one-strip hyperboloid as an infinite tube, infinitely expanding as it moves away (on both sides) from the Oxy plane.

The quantities a, b, c are called the semi-axes of a one-strip hyperboloid.

3. Two-sheeted hyperboloid.

A two-sheeted hyperboloid is a surface that, in some rectangular coordinate system, is defined by the equation

Equation (5) is called the canonical equation of a two-sheeted hyperboloid.

Let us establish the geometrical form of the surface (5). To do this, consider its sections by the coordinate planes Oxy and Oyz. We obtain, respectively, the equations

and

from which it follows that hyperbolas are obtained in the sections.

Now consider sections of this hyperboloid by planes z=h parallel to the coordinate plane Oxy. The line obtained in the section is determined by the equations

or (6)

from which it follows that

>c (c>0) the plane z=h intersects the hyperboloid along an ellipse with semi-axes and . As the value increases, a* and b* also increase. Equations (6) are satisfied by the coordinates of only two points: (0; 0; + c) and (0; 0; - c) (the planes touch the given surface). equations (6) define an imaginary ellipse, i.e. there are no intersection points of the z=h plane with the given hyperboloid.

The quantity a, b and c are called the semi-axes of the two-sheeted hyperboloid.

4. Elliptical paraboloid.

An elliptic paraboloid is a surface that, in some rectangular coordinate system, is defined by the equation

(7)

where p>0 and q>0.

Equation (7) is called the canonical equation of an elliptic paraboloid.

Consider the sections of the given surface by the coordinate planes Oxy and Oyz. We obtain, respectively, the equations

and

from which it follows that in the sections, parabolas are obtained, symmetrical about the Oz axis, with vertices at the origin. (eight)

from which it follows that for . As h increases, a and b also increase; for h=0 the ellipse degenerates into a point (the plane z=0 touches the given hyperboloid). For h<0 уравнения (8) определяют мнимый эллипс, т.е. точек пересечения плоскости z=h с данным гиперболоидом нет.

Thus, the considered sections make it possible to depict an elliptical paraboloid in the form of an infinitely convex bowl.

The point (0;0;0) is called the vertex of the paraboloid; the numbers p and q are its parameters.

In the case of p=q, equation (8) defines a circle centered on the Oz axis, i.e. An elliptical paraboloid can be viewed as a surface formed by the rotation of a parabola around its axis (paraboloid of revolution).

5. Hyperbolic paraboloid.

A hyperbolic paraboloid is a surface that, in some rectangular coordinate system, is defined by the equation

(9)

Definition 1. A conical surface or a cone with a vertex at the point M 0 is a surface formed by all straight lines, each of which passes through the point M 0 and through some point of the line γ. The point M 0 is called the top of the cone, the line γ is called the guide. The lines passing through the vertex of the cone and lying on it are called generators of the cone.

Theorem. 2nd order surface with canonical equation

is a cone with a vertex at the origin, guided by an ellipse

Proof.

Let M 1 (x 1 ; y 1 ; z 1) be some point of the surface α different from the origin; ?=OM 1 is a line, M (x; y; z) belongs to ?. Since | | , then, such that

Since, then its coordinates are x 1; y1; z 1 satisfy equation (1). Taking into account conditions (3), we have, where t≠ 0. Dividing both sides of the equation by t2≠ 0, we obtain that the coordinates of an arbitrary point M (x; y; z) of the straight line m=OM 1 satisfy equation (1). It is also satisfied by the coordinates of the point O(0,0,0).

Thus, any point M (x; y; z) of the line m=OM 1 lies on the surface α with equation (1), that is, the line OM 1 =m is a rectilinear generatrix of the surface α.

Let us now consider a section of the surface α by a plane parallel to the Oxy plane with the equation z=c≠ 0:

This section is an ellipse with semiaxes a and b. Therefore, it intersects this ellipse. According to Definition 1, the surface α is a cone with vertex O(0,0,0) (All lines m pass through the origin); the generators of this cone are straight lines m, the guide is the ellipse indicated above.

The theorem has been proven.

Definition 2. A second-order surface with canonical equation (1) is called a second-order cone.

2nd Order Cone Properties.

1º The cone with equation (1) is symmetrical with respect to all coordinate planes, all coordinate axes and the origin (since all variables are contained in equation (1) to the second degree).

2º All coordinate axes have with the cone (1) the only common point - the origin, which serves as its vertex and center at the same time

3º Section of a cone (1) by planes Oxz and Oyz– pairs of straight lines intersecting at the origin; plane Oxy- dot O(0,0,0).

4º Sections of the cone (1) by planes parallel to the coordinate planes, but not coinciding with them, are either ellipses or hyperbolas.

5º If a a = b, then these ellipses are circles, and the cone itself is a surface of revolution. It is called in this case a circular cone.

Definition 3: a conic section is a line along which a circular cone intersects with an arbitrary plane not passing through its vertex. Thus, the canonical sections are the ellipse, the hyperbola, and the parabola.

With the difference that instead of "flat" graphs, we will consider the most common spatial surfaces, and also learn how to correctly build them by hand. I have been looking for software tools for building 3D drawings for quite some time and found a couple of good applications, but despite all the ease of use, these programs do not solve an important practical issue well. The fact is that in the foreseeable historical future, students will still be armed with a ruler with a pencil, and even having a high-quality "machine" drawing, many will not be able to correctly transfer it to checkered paper. Therefore, in the training manual, special attention is paid to the technique of manual construction, and a significant part of the illustrations on the page is a handmade product.

How is this reference material different from analogues?

Having decent practical experience, I know very well which surfaces are most often dealt with in real problems of higher mathematics, and I hope that this article will help you quickly replenish your luggage with relevant knowledge and applied skills, which are 90-95% cases should suffice.

What do you need to know right now?

The most elementary:

First, you need to be able build right spatial Cartesian coordinate system (see the beginning of the article Graphs and properties of functions ) .

What will you gain after reading this article?

Bottle After mastering the materials of the lesson, you will learn how to quickly determine the type of surface by its function and / or equation, imagine how it is located in space, and, of course, make drawings. It's okay if not everything fits in your head from the 1st reading - you can always return to any paragraph as needed later.

Information is within the power of everyone - for its development you do not need any super-knowledge, special artistic talent and spatial vision.

Begin!

In practice, the spatial surface is usually given function of two variables or an equation of the form (the constant of the right side is most often equal to zero or one). The first designation is more typical for mathematical analysis, the second - for analytical geometry . The equation, in essence, is implicitly given function of 2 variables, which in typical cases can be easily reduced to the form . I remind you of the simplest example c :

– plane equation kind.

is the plane function in explicitly .

Let's start with it:

Common Plane Equations

Typical options for the arrangement of planes in a rectangular coordinate system are discussed in detail at the very beginning of the article. Plane equation . Nevertheless, once again we will dwell on equations that are of great importance for practice.

First of all, you have to fully recognize the equations of planes that are parallel to the coordinate planes. Fragments of planes are standardly depicted as rectangles, which in the last two cases look like parallelograms. By default, you can choose any dimensions (within reasonable limits, of course), while it is desirable that the point at which the coordinate axis “pierces” the plane is the center of symmetry:


Strictly speaking, the coordinate axes in some places should have been depicted with a dotted line, but in order to avoid confusion, we will neglect this nuance.

– (left drawing) the inequality defines the half-space farthest from us, excluding the plane itself;

– (medium drawing) the inequality defines the right half-space, including the plane ;

– (right drawing) a double inequality specifies a "layer" located between the planes , including both planes.

For self workout:

Example 1

Draw a body bounded by planes
Compose a system of inequalities that define the given body.

An old acquaintance should come out from under the lead of your pencil cuboid. Do not forget that invisible edges and faces must be drawn with a dotted line. Finished drawing at the end of the lesson.

Please, DO NOT NEGLECT learning tasks, even if they seem too simple. Otherwise, it may turn out that they missed it once, missed it twice, and then spent an hour grinding out a three-dimensional drawing in some real example. In addition, mechanical work will help to learn the material much more efficiently and develop intelligence! It is no coincidence that in kindergarten and elementary school, children are loaded with drawing, modeling, designers and other tasks for fine motor skills of fingers. Forgive me for the digression, but my two notebooks on developmental psychology should not disappear =)

We will conditionally call the following group of planes “direct proportions” - these are planes passing through the coordinate axes:

2) the equation of the form defines a plane passing through the axis;

3) the equation of the form defines a plane passing through the axis.

Although the formal sign is obvious (which variable is missing in the equation - the plane passes through that axis), it is always useful to understand the essence of the events taking place:

Example 2

Build Plane

What is the best way to build? I propose the following algorithm:

First, we rewrite the equation in the form , from which it is clearly seen that the “y” can take any values. We fix the value , that is, we will consider the coordinate plane . The equations set spatial line lying in the given coordinate plane. Let's draw this line on the drawing. The line passes through the origin, so to construct it, it is enough to find one point. Let . Set aside a point and draw a line.

Now back to the plane equation. Since the "y" takes any values, then the straight line constructed in the plane is continuously “replicated” to the left and to the right. This is how our plane is formed, passing through the axis. To complete the drawing, to the left and to the right of the straight line we set aside two parallel lines and “close” the symbolic parallelogram with transverse horizontal segments:

Since the condition did not impose additional restrictions, the fragment of the plane could be depicted slightly smaller or slightly larger.

Once again, we repeat the meaning of the spatial linear inequality using the example. How to determine the half-space that it defines? Let's take a point not owned plane, for example, a point from the half-space closest to us and substitute its coordinates into the inequality:

Received correct inequality, which means that the inequality defines the lower (with respect to the plane ) half-space, while the plane itself is not included in the solution.

Example 3

Build Planes
a) ;
b) .

These are tasks for self-construction, in case of difficulty, use similar reasoning. Brief instructions and drawings at the end of the lesson.

In practice, planes parallel to the axis are especially common. A special case, when the plane passes through the axis, was just in paragraph "b", and now we will analyze a more general problem:

Example 4

Build Plane

Solution: the variable "z" does not explicitly participate in the equation, which means that the plane is parallel to the applicate axis. Let's use the same technique as in the previous examples.

Let us rewrite the plane equation in the form from which it is clear that "Z" can take any values. Let's fix it and in the "native" plane draw the usual "flat" straight line. To build it, it is convenient to take reference points.

Since "Z" takes all values, then the constructed straight line continuously "multiplies" up and down, thereby forming the desired plane . Carefully draw up a parallelogram of reasonable size:

Ready.

Equation of a plane in segments

The most important applied variety. If a all odds general equation of the plane different from zero, then it can be represented as , which is called plane equation in segments. Obviously, the plane intersects the coordinate axes at points , and the great advantage of such an equation is the ease of drawing:

Example 5

Build Plane

Solution: first, we compose the equation of the plane in segments. Throw the free term to the right and divide both parts by 12:

No, this is not a typo and all things happen in space! We examine the proposed surface by the same method that was recently used for planes. We rewrite the equation in the form , from which it follows that "Z" takes any values. We fix and construct an ellipse in the plane. Since "Z" takes all values, then the constructed ellipse is continuously "replicated" up and down. It is easy to understand that the surface endless:

This surface is called elliptical cylinder. An ellipse (at any height) is called guide cylinder, and parallel lines passing through each point of the ellipse are called generating cylinder (which literally form it). axis is axis of symmetry surface (but not part of it!).

The coordinates of any point belonging to a given surface necessarily satisfy the equation .

Spatial the inequality defines the "inside" of the infinite "pipe", including the cylindrical surface itself, and, accordingly, the opposite inequality defines the set of points outside the cylinder.

In practical problems, the most popular case is when guide cylinder is circle :

Example 8

Construct the surface given by the equation

It is impossible to depict an endless “pipe”, therefore art is limited, as a rule, to “cutting”.

First, it is convenient to build a circle of radius in the plane, and then a couple more circles above and below. The resulting circles ( guides cylinder) neatly connected by four parallel straight lines ( generating cylinder):

Don't forget to use dotted lines for invisible lines.

The coordinates of any point belonging to a given cylinder satisfy the equation . The coordinates of any point lying strictly inside the "pipe" satisfy the inequality , and the inequality defines a set of points of the outer part. For a better understanding, I recommend to consider several specific points in space and see for yourself.

Example 9

Construct a surface and find its projection onto a plane

We rewrite the equation in the form from which it follows that "x" takes any values. Let us fix and draw in the plane circle – centered at the origin, unit radius. Since "x" continuously takes all values, then the constructed circle generates a circular cylinder with an axis of symmetry . Draw another circle guide cylinder) and carefully connect them with straight lines ( generating cylinder). In some places, overlays turned out, but what to do, such a slope:

This time I limited myself to a piece of the cylinder in the gap and this is not accidental. In practice, it is often necessary to depict only a small fragment of the surface.

Here, by the way, it turned out 6 generatrices - two additional straight lines "close" the surface from the upper left and lower right corners.

Now let's deal with the projection of the cylinder onto the plane. Many readers understand what a projection is, but, nevertheless, let's spend another five-minute physical education. Please stand up and tilt your head over the drawing so that the tip of the axis looks perpendicular to your forehead. What the cylinder looks like from this angle is its projection onto the plane. But it seems to be an endless strip, enclosed between straight lines, including the straight lines themselves. This projection is exactly domain functions (upper "gutter" of the cylinder), (lower "gutter").

By the way, let's clarify the situation with projections onto other coordinate planes. Let the rays of the sun shine on the cylinder from the side of the tip and along the axis. The shadow (projection) of a cylinder onto a plane is a similar infinite strip - a part of the plane bounded by straight lines ( - any), including the straight lines themselves.

But the projection on the plane is somewhat different. If you look at the cylinder from the tip of the axis, then it is projected into a circle of unit radius with which we started the construction.

Example 10

Construct a surface and find its projections on coordinate planes

This is a task for independent decision. If the condition is not very clear, square both sides and analyze the result; find out exactly what part of the cylinder the function specifies. Use the construction technique that has been repeatedly used above. Brief solution, drawing and comments at the end of the lesson.

Elliptical and other cylindrical surfaces can be offset relative to the coordinate axes, for example:

(on the familiar grounds of an article about 2nd order lines ) - a cylinder of unit radius with a line of symmetry passing through a point parallel to the axis. However, in practice, such cylinders come across quite rarely, and it is absolutely unbelievable to meet a cylindrical surface “oblique” with respect to the coordinate axes.

Parabolic cylinders

As the name suggests, guide such a cylinder is parabola .

Example 11

Construct a surface and find its projections on the coordinate planes.

Couldn't resist this example =)

Solution: We follow the beaten path. Let's rewrite the equation in the form , from which it follows that "Z" can take any value. Let us fix and construct an ordinary parabola on the plane , having previously marked the trivial reference points . Since "Z" takes all values, then the constructed parabola is continuously "replicated" up and down to infinity. We set aside the same parabola, say, at a height (in the plane) and carefully connect them with parallel lines ( generators of the cylinder):

I remind useful technique: if initially there is no confidence in the quality of the drawing, then it is better to first draw the lines thinly and thinly with a pencil. Then we evaluate the quality of the sketch, find out the areas where the surface is hidden from our eyes, and only then we apply pressure to the stylus.

Projections.

1) The projection of a cylinder onto a plane is a parabola. It should be noted that in this case it is impossible to talk about domains of a function of two variables - for the reason that the equation of the cylinder is not reducible to the functional form .

2) The projection of the cylinder onto the plane is a half-plane, including the axis

3) And, finally, the projection of the cylinder onto the plane is the entire plane.

Example 12

Construct parabolic cylinders:

a) , restrict ourselves to a fragment of the surface in the near half-space;

b) in between

In case of difficulties, we are not in a hurry and argue by analogy with the previous examples, fortunately, the technology has been thoroughly worked out. It is not critical if the surfaces turn out to be a little clumsy - it is important to correctly display the fundamental picture. I myself don’t particularly bother with the beauty of the lines, if I get a tolerable “C grade” drawing, I usually don’t redo it. In the sample solution, by the way, one more technique was used to improve the quality of the drawing ;-)

Hyperbolic cylinders

guides such cylinders are hyperbole. This type of surface, according to my observations, is much rarer than the previous types, so I will limit myself to a single schematic drawing of a hyperbolic cylinder:

The principle of reasoning here is exactly the same - the usual school hyperbole from the plane continuously "multiplies" up and down to infinity.

The considered cylinders belong to the so-called surfaces of the 2nd order, and now we will continue to get acquainted with other representatives of this group:

Ellipsoid. Sphere and ball

The canonical equation of an ellipsoid in a rectangular coordinate system has the form , where are positive numbers ( axle shafts ellipsoid), which in the general case different. An ellipsoid is called surface, and body bounded by this surface. The body, as many have guessed, is given by the inequality and the coordinates of any interior point (as well as any point on the surface) necessarily satisfy this inequality. The design is symmetrical with respect to the coordinate axes and coordinate planes:

The origin of the term "ellipsoid" is also obvious: if the surface is "cut" by coordinate planes, then in the sections there will be three different (in the general case)