The content of the article
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. A hyperbola is a graph of many important physical relationships, such as Boyle's law (relating pressure and volume of an ideal gas) and Ohm's law, which defines electricity as a function of resistance at constant voltage.
EARLY HISTORY
The discoverer of conic sections is supposedly considered to be Menaechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menaechmus used a parabola and an equilateral hyperbola to solve the problem of doubling a cube.
Treatises on conic sections written by Aristaeus and Euclid at the end of the 4th century. BC, were lost, but materials from them were included in famous Conic sections Apollonius of Perga (c. 260–170 BC), which have survived to this day. Apollonius abandoned the requirement that the secant plane of the cone's generatrix be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We also owe the modern names of curves to Apollonius - ellipse, parabola and hyperbola.
In his constructions, Apollonius used a two-sheet circular cone (as in Fig. 1), so for the first time it became clear that a hyperbola is a curve with two branches. Since the time of Apollonius, conic sections have been divided into three types depending on the inclination of the cutting plane to the generatrix of the cone. Ellipse (Fig. 1, A) 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, V) – 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 F 1 and F 2 (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 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 points F 1 and F 2 coincide, then the ellipse turns into a circle.
Hyperbola.
When constructing a hyperbola, the point P, the point of a pencil, is fixed on a thread that slides freely along pegs installed at 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 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.) One branch of the hyperbola ( PV 1 Q) we draw, making sure that the thread remains taut at all times, and pulling both ends of the thread down past the point F 2 and when point P will be below the segment F 1 F 2, holding the thread at both ends and carefully etching (i.e. releasing) it. The second branch of the hyperbola ( Pў V 2 Qў ) we draw, having previously swapped 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 asymptotes of the hyperbola, are constructed as shown in Fig. 3, b. The angular coefficients of these lines are equal to ± ( v 1 v 2)/(V 1 V 2), where v 1 v 2 – segment of the bisector of the angle between 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. 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
from the point of intersection of the axes O. This formula assumes the construction 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 rearranged transverse and conjugate axes, are called mutually conjugate.
Parabola.
The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola was apparently first established by Pappus (2nd half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called the director. The construction of a parabola using a stretched thread, based on the definition of Pappus, was proposed by Isidore of Miletus (6th century). Place the ruler so that its edge coincides with the directrix LLў (Fig. 4), and apply the leg to this edge A.C. drawing triangle ABC. We fasten one end of the thread with a length AB at the top B triangle, and the other at the focus of the parabola F. Using the tip of a pencil to stretch the thread, press the tip at a variable point P to the free leg AB drawing triangle. As the triangle moves along the ruler, the point P will describe the arc of a parabola with focus F and the headmistress LLў , since the total length of the thread is 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, i.e. PA. Intersection point V parabola with an axis is called the vertex of the parabola, the line passing through F And V, – 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.
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 is a given point (focus), and L– a given straight line (directrix) not passing through F, And D F And D L– distance from the moving point P to focus F and headmistresses L respectively. Then, as Pappus showed, conic sections are defined as the locus of points P, for which the relation D F/D L is a non-negative constant. This ratio is called eccentricity e conical section. At e e > 1 – hyperbola; at e= 1 – parabola. If F lies on L, then the geometric loci have the form of straight 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. Eccentricity e in this case is equal to zero.
Dandelen design.
The foci and directrixes of a conic section can be clearly demonstrated by 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 a conic section be formed by the intersection of a certain plane p with a two-cavity straight circular cone with apex at a point O. Let us inscribe two spheres into 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 is under it. Each generatrix of the cone touches both spheres, and the locus of the points of contact looks like two circles C 1 and C 2 located in parallel planes p 1 and p 2. Let P– an arbitrary point on a conic section. Let's draw straight lines PF 1 , PF 2 and extend the straight line P.O.. These lines are tangent to the spheres at 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. Hence, PF 1 + PF 2 = PR 1 + PR 2 = R 1 R 2. Since the plane p 1 and p 2 parallel, line segment R 1 R 2 has a constant length. Thus, the value PR 1 + PR 2 is the same for all point positions P, and point P belongs to the geometric locus of points for which the sum of the 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 straight lines along which the plane p intersects planes p 1 and p 2 , are the directrixes of the constructed ellipse. If p intersects 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 shape of a hyperbola with foci F 1 and F 2 and straight lines - intersection lines p With p 1 and p 2 – as headmistresses. If the conic section is a parabola, as shown in Fig. 5, V, 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 defining. Important place in Mathematical meeting Pappa (approx. 300), Geometry Descartes (1637) and Beginnings Newton (1687) was occupied with the problem of the geometric location of points relative to four straight lines. If four lines are given on a plane L 1 , L 2 , L 3 and L 4 (two of which may be the same) and a period P is such that the product of the 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 were unable to solve the problem of the locus of points relative to four straight lines, Descartes created analytical geometry to obtain a solution and generalize it.
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
px 2 + qy = 0.
The first equation is obtained from equation (1) with B 2 № A.C., the second – at B 2 = A.C.. 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 various types conic sections depending on the signs of the coefficients.
2831) if the odds 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– 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, V).
4) If a And b have different signs and c= 0, then the conic section consists of two intersecting lines (Fig. 6, A).
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 talk about an ellipse contracted to a point or, if a = b, contracted to a point on the 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 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 equal to zero, then the conic section consists of two real coincident 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 p No. 0, a q= 0, we get 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.
Derivation of equations of conic sections.
Any conic section can also be defined as a curve along which a plane intersects a quadratic surface, i.e. with a surface given by a second degree equation f (x, y, z) = 0. Apparently, conic sections were first recognized in this form, and their names ( see below) are due to the fact that they were obtained by intersecting a plane with a cone z 2 = x 2 + y 2. Let ABCD– the base of a right circular cone (Fig. 7) with a right angle at the apex V. Let the plane FDC intersects the generatrix VB at the point F, base – in a straight line CD and the surface of the cone - along the curve DFPC, Where P– any point on the curve. Let's draw through the middle of the segment CD– point E– straight E.F. and diameter AB. Through the point P draw a plane parallel to the base of the cone, intersecting the cone in a circle R.P.S. and direct E.F. at the point Q. Then QF And QP can be taken, accordingly, as the abscissa x and ordinate y points P. The resulting curve will be a parabola.
The construction shown in Fig. 7, can be used for output general equations conic sections. The square of the length of a perpendicular segment restored from any point of the diameter to the intersection with the circle is always equal to the product of the lengths of the diameter segments. That's why
y 2 = RQ H QS.
For a parabola, a segment RQ has a constant length (since at any position of the point P it is equal to the segment A.E.), and the length of the segment QS proportional x(from the ratio QS/E.B. = QF/F.E.). It follows that
Where a– constant coefficient. 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 not equal to the segment A.E.; but the ratio y 2 = RQ H QS is equivalent to an equation of the form
Where a And b– 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 transferring the axes,
In this case, the points of intersection with the axis x, given by the relation x 2 = a 2, determine the transverse axis, and the points of intersection with the axis y, given by the relation y 2 = –b 2, determine the conjugate axis. If constant 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 y 2 b 2 / a) x, for a parabola y 2 = (a) x and for hyperbole y 2 > (2b 2 /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 types of conic sections (types 2, 3 and 9 listed above), but his approach can be generalized 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 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 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. 6, A); 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 you look at a circle from an oblique angle, it looks like an ellipse. The relationship between a circle and an ellipse, known to Archimedes, becomes obvious if the circle X 2 + Y 2 = a 2 using substitution X = x, Y = (a/b) y transform into the ellipse given by equation (3a). Conversion X = x, Y = (ai/b) y, Where i 2 = –1, allows us to write the equation of a 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.
Relationship between ordinates of a circle x 2 + y 2 = a 2 and ellipse ( x 2 /a 2) + (y 2 /b 2) = 1 directly leads to Archimedes' formula 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 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 indicated 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 a straight line V.P., 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.
Each point on the original circle corresponds to a certain 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 infinitely distant.
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 during projection. This relationship can be found using the following construction. Through any point P plane, draw two tangents to any circle and connect the tangent points with a straight line p. Let another line passing through the point P, intersects the circle at points C 1 and C 2 and straight p- at the point Q(Fig. 9). In planimetry it is proven that PC 1 /PC 2 = –QC 1 /QC 2. (The minus sign arises due to the fact that the direction of the segment QC 1 is opposite to the directions of other segments.) In other words, 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 four segments is equal to - 1. If the circle is projected into a conic section and the same notation is retained for the corresponding points, then the harmonic ratio ( PC 1)(QC 2)/(PC 2)(QC 1) will remain equal to - 1. Point P called a line pole p relative to the conic section, and the straight line p– polar point P relative to the conic section.
When the point P approaches a conic section, the polar tends to take the position of a tangent; if point P lies on a conic section, then its polar coincides with the tangent to the conic section at the point P. If the point P is located inside the conic section, then its polar can be constructed as follows. Let's draw 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 1 . Let's draw through the point P another straight line that intersects the conic section at two other points; let us assume 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 the point P approaching the center O central conic section, then polar p moving away from O. When the point P coincides with O, then its polar becomes infinitely distant, or ideal, straight on the plane.
SPECIAL BUILDINGS
Of particular interest to astronomers is the following simple construction of ellipse points using a compass and ruler. Let an arbitrary straight 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 draw through the point Q horizontal line, and through R– a vertical line, and denote their intersection point P P when rotating a straight line OQR around the point O there will be an ellipse. Corner f between the straight line OQR and the major axis is called the eccentric angle, and the constructed ellipse is conveniently specified by 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. An arbitrary straight 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, draw tangents intersecting OS at the point T And OR- at the point Q. Let a vertical line passing through a point T, and a horizontal line passing through the point Q, intersect at a point P. Then the locus of the points P when rotating a segment OR around O will be a hyperbola given by 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 N. Copernicus (1473–1543) noted, can be constructed using epicyclic motion. If a circle rolls without slipping along inside another circle with twice the diameter, then each point P, which does not lie on the smaller circle, but is motionless relative to it, will describe an ellipse. If the point P is on a smaller circle, then the trajectory of this point is a degenerate case of an ellipse - the diameter of the larger circle. An even simpler construction of the ellipse was proposed by Proclus in the 5th century. If the ends A And B line segment AB of a given length slide along two fixed intersecting straight lines (for example, along coordinate axes), then each internal point P the segment will describe an ellipse; the Dutch mathematician F. van Schooten (1615–1660) showed that any point in the plane of intersecting lines, fixed relative to a sliding segment, will also describe an ellipse.
B. Pascal (1623–1662) at the age of 16 formulated the now famous Pascal theorem, which states: the three intersection points of opposite sides of a hexagon inscribed in any conic section lie on the same straight line. Pascal derived more than 400 corollaries from this theorem.
A conical surface is a surface formed by straight lines - the generators of the cone - passing through a given point - the vertex of the cone - and intersecting a given line - the guide of the cone. Let the cone guide have the equations
and the vertex of the cone has coordinates. The canonical equations of the generators of the cone as straight lines passing through the point ) and through the point of the guide will be;
Eliminating x, y and z from the four equations (3) and (4), we obtain the desired equation of the conical surface. This equation has a very simple property: it is homogeneous (that is, all its terms are of the same dimension) with respect to differences. In fact, let us first assume that the vertex of the cone is at the origin. Let X, Y and Z be the coordinates of any point on the cone; they therefore satisfy the cone equation. After replacing X, Y and Z in the equation of the cone, respectively, through XX, XY, XZ, where X is an arbitrary factor, the equation must be satisfied, since XX, XY and XZ are the coordinates of the point of the line passing through the origin of coordinates to the point, i.e. forming a cone. Consequently, the equation of the cone will not change if we multiply all the current coordinates by the same number X. It follows that this equation must be homogeneous with respect to the current coordinates.
If the vertex of the cone lies at a point, we will transfer the origin of coordinates to the vertex, and according to what has been proven, the transformed equation of the cone will be homogeneous with respect to the new coordinates, i.e., with respect to
Example. Write an equation for a cone with a vertex at the origin and a direction
The canonical equations of the generators passing through the vertex (0, 0, C) of the cone and the point of the guide will be:
Let's eliminate x, y and from the four given equations. Replacing through c, we determine and y from the last two equations.
Second order surfaces- these 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 a certain rectangular coordinate system, is defined by the equation:Equation (1) is called canonical equation of an ellipsoid.
Let us establish the geometric form of the ellipsoid. To do this, consider sections of this ellipsoid by planes parallel to the plane Oxy. Each of these planes is determined 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) define an imaginary ellipse, i.e., the points of intersection of the plane z=h does not exist with this ellipsoid. , That and line (2) degenerates into points (0; 0; + c) and (0; 0; - c) (the planes touch the ellipsoid). , then equations (2) can be represented aswhence it follows that the plane z=h intersects the ellipsoid along an ellipse with semi-axes
And . As the values decrease, and increase and reach their highest values at , i.e. in the section of the ellipsoid by the coordinate plane Oxy the largest ellipse with semi-axes and is obtained.A similar picture is obtained when a given surface is intersected by planes parallel to the coordinate planes Oxz And Oyz.
Thus, the sections considered make it possible to depict the ellipsoid as a closed oval surface (Fig. 156). Quantities a, b, c are called axle shafts ellipsoid. When a=b=c ellipsoid is spheroth.
2. Single-strip hyperboloid.
A single-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 single-strip hyperboloid.
Let's set the type of surface (3). To do this, consider a section of its coordinate planes Oxy (y=0)AndOyx (x=0). We obtain, accordingly, the equations
AndNow consider sections of this hyperboloid by planes z=h parallel to the coordinate plane Oxy. The resulting line 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 infinite increase
the quantities a* and b* increase infinitely.Thus, the considered sections make it possible to depict a single-strip hyperboloid in the form of 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 single-strip hyperboloid.
3. Two-sheet hyperboloid.
A two-sheet 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-sheet hyperboloid.
Let us establish the geometric appearance of the surface (5). To do this, consider its sections by coordinate planes Oxy and Oyz. We obtain, accordingly, the equations
Andfrom which it follows that hyperbolas are obtained in 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 when
>c (c>0) the plane z=h intersects the hyperboloid along an ellipse with semi-axes and . As the values of a* and b* increase, they also increase. equations (6) are satisfied by the coordinates of only two points: (0;0;+с) and (0;0;-с) (the planes touch the given surface). equations (6) define an imaginary ellipse, i.e. There are no points of intersection of the z=h plane with this hyperboloid.The quantities a, b and c are called the semi-axes of a two-sheet 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.
Let us consider sections of this surface by coordinate planes Oxy and Oyz. We obtain, accordingly, the equations
Andfrom which it follows that the sections yield parabolas that are symmetrical about the Oz axis, with vertices at the origin. (8)
from which it follows that at . As h increases, the values of a and b also increase; at h=0 the ellipse degenerates into a point (the plane z=0 touches the given hyperboloid). At 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; numbers p and q are its parameters.
In the case of p=q, equation (8) defines a circle with center on the Oz axis, i.e. an elliptical paraboloid can be considered 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)Students most often encounter surfaces of the 2nd order in the first year. At first, problems on this topic may seem simple, but as you study higher mathematics and delve deeper into the scientific side, you can finally lose track of what is happening. In order for this not to happen, you need to not just memorize, but understand how this or that surface is obtained, how changes in coefficients affect it and its location relative to the original coordinate system, and how to find a new system (one in which its center coincides with the origin coordinates, but parallel to one of the coordinate axes). Let's start from the very beginning.
Definition
A 2nd order surface is called a GMT, the coordinates of which satisfy the general equation of the following form:
It is clear that each point belonging to the surface must have three coordinates in some designated basis. Although in some cases the locus of points can degenerate, for example, into a plane. This only means that one of the coordinates is constant and equal to zero throughout the entire range of permissible values.
The full written form of the above equality looks like this:
A 11 x 2 +A 22 y 2 +A 33 z 2 +2A 12 xy+2A 23 yz+2A 13 xz+2A 14 x+2A 24 y+2A 34 z+A 44 =0.
A nm are some constants, x, y, z are variables corresponding to the affine coordinates of a point. In this case, at least one of the constant factors must not be equal to zero, that is, not any point will correspond to the equation.
In the vast majority of examples, many numerical factors are still identically equal to zero, and the equation is significantly simplified. In practice, determining whether a point belongs to a surface is not difficult (it is enough to substitute its coordinates into the equation and check whether the identity holds). The key point in such work is to bring the latter to canonical form.
The equation written above defines any (all listed below) 2nd order surfaces. Let's look at examples below.
Types of surfaces of 2nd order
The equations of 2nd order surfaces differ only in the values of the coefficients A nm. From the general form, at certain values of the constants, various surfaces can be obtained, classified as follows:
- Cylinders.
- Elliptical type.
- Hyperbolic type.
- Conical type.
- Parabolic type.
- Planes.
Each of the listed types has a natural and imaginary form: in the imaginary form, the locus of real points either degenerates into a simpler figure or is absent altogether.
Cylinders
This is the simplest type, as the relatively complex curve lies only at the base, acting as a guide. Generators are straight lines perpendicular to the plane in which the base lies.
The graph shows a circular cylinder, a special case of an elliptical cylinder. In the XY plane, its projection will be an ellipse (in our case, a circle) - a guide, and in XZ - a rectangle - since the generators are parallel to the Z axis. To obtain it from the general equation, it is necessary to give the following values to the coefficients:
Instead of the usual notations x, y, z, x's with a serial number are used - this does not have any meaning.
In fact, 1/a 2 and the other constants indicated here are the same coefficients indicated in the general equation, but it is customary to write them exactly in this form - this is the canonical representation. In what follows, this entry will be used exclusively.
This defines a hyperbolic cylinder. The scheme is the same - the hyperbole will be the guide.
A parabolic cylinder is defined slightly differently: its canonical form includes a coefficient p, called a parameter. In fact, the coefficient is q=2p, but it is customary to divide it into the two factors presented.
There is another type of cylinder: imaginary. No real point belongs to such a cylinder. It is described by the equation of an elliptic cylinder, but instead of one there is -1.
Elliptical type
The ellipsoid can be stretched along one of the axes (along which it depends on the values of the constants a, b, c indicated above; obviously, the larger axis will correspond to a larger coefficient).
There is also an imaginary ellipsoid - provided that the sum of the coordinates multiplied by the coefficients is equal to -1:
Hyperboloids
When a minus appears in one of the constants, the equation of the ellipsoid turns into the equation of a one-sheet hyperboloid. You must understand that this minus does not have to be located in front of the x3 coordinate! It only determines which of the axes will be the axis of rotation of the hyperboloid (or parallel to it, since when additional terms appear in the square (for example, (x-2) 2), the center of the figure shifts, as a result, the surface moves parallel to the coordinate axes). This applies to all 2nd order surfaces.
In addition, you need to understand that the equations are presented in canonical form and they can be changed by varying the constants (while maintaining the sign!); at the same time, their appearance (hyperboloid, cone, and so on) will remain the same.
Such an equation is given by a two-sheet hyperboloid.
Conical surface
In the cone equation, there is no unity - it is equal to zero.
Only a limited conical surface is called a cone. The picture below shows that, in fact, there will be two so-called cones on the chart.
Important note: in all considered canonical equations, constants are assumed to be positive by default. Otherwise, the sign may affect the final graph.
The coordinate planes become planes of symmetry of the cone, the center of symmetry is located at the origin.
In the equation of an imaginary cone there are only pluses; it owns one single real point.
Paraboloids
Surfaces of order 2 in space can take different shapes even with similar equations. For example, paraboloids come in two types.
x 2 /a 2 +y 2 /b 2 =2z
An elliptical paraboloid, when the Z axis is perpendicular to the drawing, will be projected into an ellipse.
x 2 /a 2 -y 2 /b 2 =2z
Hyperbolic paraboloid: in sections with planes parallel to ZY, parabolas will be obtained, and in sections with planes parallel to XY, hyperbolas will be obtained.
Intersecting planes
There are cases when 2nd order surfaces degenerate in the plane. These planes can be arranged in various ways.
First let's look at intersecting planes:
x 2 /a 2 -y 2 /b 2 =0
With this modification of the canonical equation, we simply get two intersecting planes (imaginary!); all real points are on the axis of the coordinate that is absent in the equation (in the canonical one - the Z axis).
Parallel planes
If there is only one coordinate, 2nd order surfaces degenerate into a pair of parallel planes. Don't forget, any other variable can take the place of the player; then planes parallel to other axes will be obtained.
In this case they become imaginary.
Coincident planes
With such a simple equation, a pair of planes degenerates into one - they coincide.
Don't forget that in the case of a three-dimensional basis, the above equation does not specify the straight line y=0! It's missing the other two variables, but that just means their value is constant and equal to zero.
Construction
One of the most difficult tasks for a student is precisely the construction of 2nd order surfaces. It is even more difficult to move from one coordinate system to another, taking into account the angles of inclination of the curve relative to the axes and the offset of the center. Let's repeat how to consistently determine the future appearance of a drawing in an analytical way.
To construct a 2nd order surface, you need to:
- bring the equation to canonical form;
- determine the type of surface under study;
- build based on the values of the coefficients.
Below are all the types considered:
To reinforce this, we will describe in detail one example of this type of task.
Examples
Let's say we have the equation:
3(x 2 -2x+1)+6y 2 +2z 2 +60y+144=0
Let's bring it to canonical form. Let's select complete squares, that is, we will arrange the available terms in such a way that they are a decomposition of the square of the sum or difference. For example: if (a+1) 2 =a 2 +2a+1, then a 2 +2a+1=(a+1) 2. We will perform a second operation. In this case, it is not necessary to open the brackets, since this will only complicate the calculations, but it is necessary to remove the common factor 6 (in brackets with the full square of the game):
3(x-1) 2 +6(y+5) 2 +2z 2 =6
The variable zet appears in this case only once - you can leave it alone for now.
Let's analyze the equation at this stage: all unknowns have a plus sign in front of them; Dividing by six leaves one. Consequently, we have before us an equation defining an ellipsoid.
Notice that 144 was factored into 150-6, and then -6 was moved to the right. Why did it have to be done this way? Obviously, the largest divisor in this example is -6, therefore, in order for a unit to remain on the right after dividing by it, it is necessary to “set aside” exactly 6 from 144 (the fact that the unit should be on the right is indicated by the presence of a free term - a constant not multiplied to the unknown).
Let's divide everything by six and get the canonical equation of the ellipsoid:
(x-1) 2 /2+(y+5) 2 /1+z 2 /3=1
In the previously used classification of 2nd order surfaces, a special case is considered when the center of the figure is at the origin of coordinates. In this example it is offset.
We assume that each bracket with unknowns is a new variable. That is: a=x-1, b=y+5, c=z. In the new coordinates, the center of the ellipsoid coincides with the point (0,0,0), therefore, a=b=c=0, whence: x=1, y=-5, z=0. In the initial coordinates, the center of the figure lies at the point (1,-5,0).
The ellipsoid will be obtained from two ellipses: the first in the XY plane and the second in the XZ plane (or YZ - it doesn’t matter). The coefficients by which the variables are divided are squared in the canonical equation. Therefore, in the above example, it would be more correct to divide by the root of two, one and the root of three.
The minor axis of the first ellipse, parallel to the Y axis, is equal to two. The major axis is parallel to the X axis - two roots of two. The minor axis of the second ellipse, parallel to the Y axis, remains the same - it is equal to two. And the major axis, parallel to the Z axis, is equal to two roots of three.
Using the data obtained from the original equation by converting it to canonical form, we can draw an ellipsoid.
Summing up
The topic covered in this article is quite extensive, but in fact, as you can now see, it is not very complicated. Its development, in fact, ends at the moment when you memorize the names and equations of surfaces (and, of course, what they look like). In the example above, we examined each step in detail, but bringing the equation to canonical form requires minimal knowledge of higher mathematics and should not cause any difficulties for the student.
Analyzing the future schedule based on existing equality is a more difficult task. But to solve it successfully, it is enough to understand how the corresponding second-order curves are constructed - ellipses, parabolas and others.
Cases of degeneration are an even simpler section. Due to the absence of some variables, not only the calculations are simplified, as mentioned earlier, but also the construction itself.
As soon as you can confidently name all types of surfaces, vary constants, turning a graph into one shape or another, the topic will be mastered.
Good luck in your studies!
