Imaginary lines. What is the canonical form of an equation? Ellipse and its canonical equation

We will now show that the affine classification of second-order curves is given by the names of the curves themselves, i.e., that the affine classes of second-order curves are the classes:

real ellipses;

imaginary ellipses;

hyperbole;

pairs of real intersecting lines;

pairs of imaginary (conjugate) intersecting;

pairs of parallel real lines;

pairs of parallel imaginary conjugate lines;

pairs of coinciding real lines.

We need to prove two statements:

A. All curves of the same name (that is, all ellipses, all hyperbolas, etc.) are affinely equivalent to each other.

B. Two curves of different names are never affine equivalent.

We prove assertion A. In Chapter XV, § 3, it was already proved that all ellipses are affinely equivalent to one of them, namely, circles and all hyperbolas are hyperbolas. Hence, all ellipses, respectively, all hyperbolas, are affinely equivalent to each other. All imaginary ellipses, being affinely equivalent to a circle - - 1 of radius, are also affinely equivalent to each other.

Let us prove the affine equivalence of all parabolas. We will prove even more, namely that all parabolas are similar to each other. It suffices to prove that the parabola given in some coordinate system by its canonical equation

like a parabola

To do this, we subject the plane to a similarity transformation with a coefficient - :

Then so that under our transformation the curve

goes into a curve

i.e. into a parabola

Q.E.D.

Let's move on to decaying curves. In § formulas (9) and (11), pp. 401 and 402), it was proved that a curve splitting into a pair of intersecting lines in some (even rectangular) coordinate system has the equation

Doing an additional coordinate transformation

we see that any curve decomposing into a pair of intersecting real, respectively, imaginary conjugate, straight lines, has in some affine coordinate system the equation

As for curves splitting into a pair of parallel lines, each of them can be (even in some rectangular coordinate system) given by the equation

for real, respectively

for imaginary, direct. The transformation of coordinates allows us to put in these equations (or for coinciding lines). This implies the affine equivalence of all decaying second-order curves that have the same name.

We turn to the proof of assertion B.

First of all, we note that under an affine transformation of a plane, the order of an algebraic curve remains unchanged. Further: any decaying curve of the second order is a pair of straight lines, and under an affine transformation, a straight line becomes a straight line, a pair of intersecting lines becomes a pair of intersecting ones, and a pair of parallel lines becomes a pair of parallel ones; in addition, the real lines become real, and the imaginary lines become imaginary. This follows from the fact that all coefficients in formulas (3) (Chapter XI, § 3) that define an affine transformation are real numbers.

It follows from what has been said that a line that is affinely equivalent to a given decaying second-order curve is a decaying curve of the same name.

We pass to non-decomposing curves. Again, with an affine transformation, a real curve cannot go into an imaginary one, and vice versa. Therefore, the class of imaginary ellipses is affine invariant.

Consider classes of real non-decomposing curves: ellipses, hyperbolas, parabolas.

Among all curves of the second order, every ellipse, and only an ellipse, lies in some rectangle, while parabolas and hyperbolas (as well as all decaying curves) extend to infinity.

Under an affine transformation, the rectangle ABCD containing the given ellipse will go into a parallelogram containing the transformed curve, which, therefore, cannot go to infinity and, therefore, is an ellipse.

Thus, a curve affinely equivalent to an ellipse is necessarily an ellipse. It follows from what has been proved that a curve that is affinely equivalent to a hyperbola or a parabola cannot be an ellipse (and, as we know, it cannot be a decaying curve either. Therefore, it only remains to prove that under an affine transformation of the plane, a hyperbola cannot pass into a parabola, and on the contrary, this probably follows most simply from the fact that a parabola has no center of symmetry, while a hyperbola does. But since the absence of a center of symmetry for a parabola will be proved only in the next chapter, we will now give a second, also very simple proof affine non-equivalence of hyperbola and parabola.

Lemma. If a parabola has common points with each of the two half-planes defined in the plane of a given line d, then it has at least one common point with the line.

Indeed, we have seen that there is a coordinate system in which the given parabola has the equation

Let, relative to this coordinate system, the straight line d have the equation

By assumption, there are two points on the parabola, one of which, we assume, lies in the positive and the other lies in the negative half-plane with respect to equation (1). Therefore, remembering that we can write

Second order lines

plane lines whose Cartesian rectangular coordinates satisfy a 2nd degree algebraic equation

a 11 x 2 + a 12 xy + a 22 y 2 + 2a 13 x + 2a 23 y + a 11 = 0. (*)

The equation (*) may not determine the actual geometric image, but for the sake of generality in such cases it is said that it determines the imaginary linear representation. n. Depending on the values ​​of the coefficients of the general equation (*), it can be transformed by parallel translation of the origin and rotation of the coordinate system by some angle to one of the 9 canonical forms below, each of which corresponds to a certain class of lines. Exactly,

unbreakable lines:

y 2 = 2px - parabolas,

breaking lines:

x 2 - a 2 \u003d 0 - pairs of parallel lines,

x 2 + a 2 \u003d 0 - pairs of imaginary parallel lines,

x 2 = 0 - pairs of coinciding parallel lines.

Research of a look L. in. can be carried out without reducing the general equation to canonical form. This is achieved by joint consideration of the values ​​of the so-called. basic invariants of the L.v. n. - expressions composed of the coefficients of the equation (*), the values ​​of which do not change with parallel translation and rotation of the coordinate system:

S \u003d a 11 + a 22,(a ij = a ji).

So, for example, ellipses, as non-decaying lines, are characterized by the fact that for them Δ ≠ 0; the positive value of the invariant δ distinguishes ellipses from other types of non-decaying lines (for hyperbolas δ

The three main invariants Δ, δ, and S determine the LV. (except for the case of parallel lines) up to motion (see Motion) of the Euclidean plane: if the corresponding invariants Δ, δ, and S of two lines are equal, then such lines can be combined by motion. In other words, these lines are equivalent with respect to the group of motions of the plane (metrically equivalent).

There are L.'s classifications. from the point of view of other groups of transformations. Thus, relatively more general than the group of motions, the group of affine transformations (See Affine transformations), any two lines defined by equations of the same canonical form are equivalent. For example, two similar L. in. n. (see similarity) are considered equivalent. Connections between different affine classes of linear c.v. allows us to establish a classification from the point of view of projective geometry (see projective geometry), in which elements at infinity do not play a special role. Real non-disintegrating L. in. etc.: ellipses, hyperbolas and parabolas form one projective class - the class of real oval lines (ovals). The real oval line is an ellipse, hyperbola or parabola, depending on how it is located relative to the line at infinity: the ellipse intersects the improper line at two imaginary points, the hyperbola at two different real points, the parabola touches the improper line; there are projective transformations that take these lines one into another. There are only 5 projective equivalence classes of L.v. n. Precisely,

non-degenerate lines

(x 1 , x 2 , x 3- homogeneous coordinates):

x 1 2 + x 2 2 - x 3 2= 0 - real oval,

x 1 2 + x 2 2 + x 3 2= 0 - imaginary oval,

degenerate lines:

x 1 2 - x 2 2= 0 - pair of real lines,

x 1 2 + x 2 2= 0 - a pair of imaginary lines,

x 1 2= 0 - a pair of coinciding real lines.

A. B. Ivanov.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what "Lines of the second order" is in other dictionaries:

Plane lines whose rectangular point coordinates satisfy a 2nd degree algebraic equation. Among the lines of the second order are ellipses (in particular, circles), hyperbolas, parabolas ... Big Encyclopedic Dictionary

Plane lines whose rectangular point coordinates satisfy a 2nd degree algebraic equation. Among the lines of the second order are ellipses (in particular, circles), hyperbolas, parabolas. * * * SECOND-ORDER LINES SECOND-ORDER LINES,… … encyclopedic Dictionary

Flat lines, rectangular the coordinates of the points k px satisfy algebras. urnium of the 2nd degree. Among L. in. n. ellipses (particularly circles), hyperbolas, parabolas… Natural science. encyclopedic Dictionary

Flat line, Cartesian rectangular coordinates to swarm satisfy algebraic. equation of the 2nd degree Equation (*) may not determine the actual geometric. image, but to preserve the generality in such cases, they say that it determines ... ... Mathematical Encyclopedia

The set of points of a 3-dimensional real (or complex) space, the coordinates of which in the Cartesian system satisfy the algebraic. equation of the 2nd degree (*) The equation (*) may not determine the actual geometric. images, in such ... ... Mathematical Encyclopedia

This word, very often used in the geometry of curved lines, has a not quite definite meaning. When this word is applied to non-closed and non-branching curved lines, then the branch of the curve means each continuous individual ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

Lines of the second order, two diameters, each of which bisects the chords of this curve, parallel to the other. SDs play an important role in the general theory of second-order lines. With the parallel projection of an ellipse into the circle of its S. d. ... ...

Lines that are obtained by sectioning a right circular Cone with planes that do not pass through its vertex. K. s. can be of three types: 1) the cutting plane intersects all generators of the cone at the points of one of its cavity; line… … Great Soviet Encyclopedia

Lines that are obtained by sectioning a right circular cone with planes that do not pass through its vertex. K. s. can be of three types: 1) the cutting plane intersects all the generators of the cone at the points of one of its cavity (Fig., a): line of intersection ... ... Mathematical Encyclopedia

Geometry section. The basic concepts of algebraic geometry are the simplest geometric images (points, lines, planes, curves, and second-order surfaces). The main means of research in A. g. are the method of coordinates (see below) and methods ... ... Great Soviet Encyclopedia

Books

• A short course in analytic geometry, Efimov Nikolai Vladimirovich. The subject of study of analytic geometry are figures, which in Cartesian coordinates are given by equations of the first degree or the second. On a plane, these are straight lines and lines of the second order. ...

This is the generally accepted standard form of the equation, when in a matter of seconds it becomes clear what geometric object it defines. In addition, the canonical form is very convenient for solving many practical problems. So, for example, according to the canonical equation "flat" straight, firstly, it is immediately clear that this is a straight line, and secondly, the point belonging to it and the direction vector are simply visible.

Obviously, any 1st order line represents a straight line. On the second floor, there is no longer a janitor waiting for us, but a much more diverse company of nine statues:

Classification of second order lines

With the help of a special set of actions, any second-order line equation is reduced to one of the following types:

( and are positive real numbers)

1) is the canonical equation of the ellipse;

2) is the canonical equation of the hyperbola;

3) is the canonical equation of the parabola;

4) – imaginary ellipse;

5) - a pair of intersecting lines;

6) - couple imaginary intersecting lines (with the only real point of intersection at the origin);

7) - a pair of parallel lines;

8) - couple imaginary parallel lines;

9) is a pair of coinciding lines.

Some readers may get the impression that the list is incomplete. For example, in paragraph number 7, the equation sets the pair direct, parallel to the axis, and the question arises: where is the equation that determines the lines parallel to the y-axis? Answer: it not considered canon. The straight lines represent the same standard case rotated by 90 degrees, and an additional entry in the classification is redundant, since it does not carry anything fundamentally new.

Thus, there are nine and only nine different types of 2nd order lines, but in practice the most common are ellipse, hyperbola and parabola.

Let's look at the ellipse first. As usual, I focus on those points that are of great importance for solving problems, and if you need a detailed derivation of formulas, proofs of theorems, please refer, for example, to the textbook by Bazylev / Atanasyan or Aleksandrov ..

Ellipse and its canonical equation

Spelling ... please do not repeat the mistakes of some Yandex users who are interested in "how to build an ellipse", "the difference between an ellipse and an oval" and "elebs eccentricity".

The canonical equation of an ellipse has the form , where are positive real numbers, and . I will formulate the definition of an ellipse later, but for now it's time to take a break from talking and solve a common problem:

How to build an ellipse?

Yes, take it and just draw it. The assignment is common, and a significant part of the students do not quite competently cope with the drawing:

Example 1

Construct an ellipse given by the equation

Decision: first we bring the equation to the canonical form:

Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices, which are at the points . It is easy to see that the coordinates of each of these points satisfy the equation .

In this case :


Line segment called major axis ellipse;
line segment – minor axis;
number called semi-major axis ellipse;
number – semi-minor axis.
in our example: .

To quickly imagine what this or that ellipse looks like, just look at the values ​​\u200b\u200bof "a" and "be" of its canonical equation.

Everything is fine, neat and beautiful, but there is one caveat: I made the drawing using the program. And you can draw with any application. However, in harsh reality, a checkered piece of paper lies on the table, and mice dance around our hands. People with artistic talent, of course, can argue, but you also have mice (albeit smaller ones). It is not in vain that mankind invented a ruler, a compass, a protractor and other simple devices for drawing.

For this reason, we are unlikely to be able to accurately draw an ellipse, knowing only the vertices. Still all right, if the ellipse is small, for example, with semiaxes. Alternatively, you can reduce the scale and, accordingly, the dimensions of the drawing. But in the general case it is highly desirable to find additional points.

There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like building with a compass and ruler because of the short algorithm and the significant clutter of the drawing. In case of emergency, please refer to the textbook, but in reality it is much more rational to use the tools of algebra. From the ellipse equation on the draft, we quickly express:

The equation is then split into two functions:
– defines the upper arc of the ellipse;
– defines the lower arc of the ellipse.

Any ellipse is symmetrical about the coordinate axes, as well as about the origin. And that's great - symmetry is almost always a harbinger of a freebie. Obviously, it is enough to deal with the 1st coordinate quarter, so we need a function . It suggests finding additional points with abscissas . We hit three SMS on the calculator:

Of course, it is also pleasant that if a serious error is made in the calculations, then this will immediately become clear during the construction.

Mark points on the drawing (red color), symmetrical points on the other arcs (blue color) and carefully connect the whole company with a line:


It is better to draw the initial sketch thinly and thinly, and only then apply pressure to the pencil. The result should be quite a decent ellipse. By the way, would you like to know what this curve is?

To illustrate this with a concrete example, I will show you what corresponds in this interpretation to the following statement: the (real or imaginary) point P lies on the (real or imaginary) line g. In this case, of course, it is necessary to distinguish between the following cases:

1) real point and real line,

2) real point and imaginary line,

Case 1) does not require any special explanation from us; here we have one of the basic relations of ordinary geometry.

In case 2), along with the given imaginary line, the line complex conjugate to it must necessarily pass through the given real point; consequently, this point must coincide with the vertex of the bundle of rays that we use to represent the imaginary line.

Similarly, in case 3) the real line must be identical with the support of that rectilinear involution of points which serves as a representative of the given imaginary point.

The most interesting case is 4) (Fig. 96): here, obviously, the complex conjugate point must also lie on the complex conjugate line, and hence it follows that each pair of points of the involution of points representing the point P must lie on some pair of lines of the involution of lines representing the straight line g, i.e. that both of these involutions must be located perspectively one relative to the other; moreover, it turns out that the arrows of both involutions are also placed in perspective.

In general, in the analytic geometry of the plane, which also pays attention to the complex domain, we obtain a complete real picture of this plane if we add as new elements to the set of all its real points and lines the set of the involutional figures considered above, together with the arrows of their directions. It will suffice here if I outline in general outline what form the construction of such a real picture of complex geometry would take. In doing so, I will follow the order in which the first propositions of elementary geometry are now usually presented.

1) They start with the axioms of existence, the purpose of which is to give an exact formulation of the presence of the elements just mentioned in an area expanded in comparison with ordinary geometry.

2) Then the connection axioms, which state that also in the extended area defined in item 1)! one and only one line passes through (every) two points, and that (any) two lines have one and only one point in common.

At the same time, just as we had above, we have to distinguish four cases each time depending on whether the given elements are real, and it seems very interesting to think over exactly which real constructions with involutions of points and lines serve as an image of these complex relations.

3) As for the axioms of arrangement (order), here, in comparison with the actual relations, completely new circumstances come into play; in particular, all real and complex points lying on one fixed line, as well as all rays passing through one fixed point, form a two-dimensional continuum. After all, each of us learned from the study of the theory of functions the habit of representing the totality of values ​​of a complex variable by all points of the plane.

4) Finally, with regard to the axioms of continuity, I will only indicate here how complex points are depicted, lying as close as you like to some real point. To do this, through the taken real point P (or through some other real point close to it), you need to draw some straight line and consider on it such two pairs of points separating one another (i.e., lying in a "crossed way") pairs of points (Fig. . 97) so that two points taken from different pairs lie close to one another and to the point P; if we now bring the points together indefinitely, then the involution defined by the named pairs of points degenerates, i.e., both of its hitherto complex double points coincide with the point. Each of the two imaginary points represented by this involution (together with one or the other arrow) passes, hence continuous to some point close to P, or even directly to P. Of course, in order to be able to use these notions of continuity to good use, one must work with them in detail.

Although all this construction is rather cumbersome and tedious in comparison with ordinary real geometry, it can give incomparably more. In particular, it is capable of raising to the level of complete geometric clarity algebraic images, understood as sets of their real and complex elements, and with its help one can clearly understand for oneself on the figures themselves such theorems as the fundamental theorem of algebra or Bezout's theorem that two curves orders have, generally speaking, exactly common points. For this purpose, it would be necessary, of course, to comprehend the basic provisions in a much more precise and illustrative form than has been done so far; however, the literature already contains all the material essential for such investigations.

But in most cases, the application of this geometrical interpretation, despite all its theoretical advantages, would lead to such complications that one has to be content with its fundamental possibility and actually return to a more naive point of view, which is as follows: a complex point is a collection of three complex coordinates, and with it can be operated on in exactly the same way as with real points. Indeed, such an introduction of imaginary elements, refraining from any fundamental reasoning, has always proved fruitful in those cases when we have to deal with imaginary cyclic points or with a circle of spheres. As already mentioned, Poncelet began to use imaginary elements in this sense for the first time; his followers in this respect were other French geometers, chiefly Chall and Darboux; in Germany, a number of geometers, especially Lie, also applied this understanding of imaginary elements with great success.

With this digression into the realm of the imaginary, I conclude the entire second section of my course and turn to a new chapter,

8.3.15. Point A lies on a line. Distance from point A to the plane

8.3.16. Write an equation for a straight line symmetrical to a straight line

relative to the plane .

8.3.17. Compose the equations of projections on a plane the following lines:

a) ;

b)

in) .

8.3.18. Find the angle between the plane and the line:

a) ;

b) .

8.3.19. Find a point symmetrical to a point with respect to the plane passing through the lines:

and

8.3.20. Point A lies on a line

Distance from point A to a straight line equals . Find the coordinates of point A.

§ 8.4. SECOND-ORDER CURVES

Let us establish a rectangular coordinate system on the plane and consider the general equation of the second degree

wherein .

The set of all points in the plane whose coordinates satisfy equation (8.4.1) is called crooked (line) second order.

For any curve of the second order, there is a rectangular coordinate system, called canonical, in which the equation of this curve has one of the following forms:

1) (ellipse);

2) (imaginary ellipse);

3) (a pair of imaginary intersecting lines);

4) (hyperbola);

5) (a pair of intersecting lines);

6) (parabola);

7) (pair of parallel lines);

8) (a pair of imaginary parallel lines);

9) (a pair of coinciding lines).

Equations 1) - 9) are called canonical equations of curves of the second order.

The solution of the problem of reducing the equation of a curve of the second order to the canonical form includes finding the canonical equation of the curve and the canonical coordinate system. Reduction to the canonical form allows you to calculate the parameters of the curve and determine its location relative to the original coordinate system. Transition from the original rectangular coordinate system to canonical is carried out by rotating the axes of the original coordinate system around the point O by some angle j and subsequent parallel transfer of the coordinate system.

Curve invariants of the second order(8.4.1) are called such functions of the coefficients of its equation, the values ​​of which do not change when moving from one rectangular coordinate system to another of the same system.

For a curve of the second order (8.4.1), the sum of the coefficients at squared coordinates

,

determinant composed of the coefficients of the leading terms

and third order determinant

are invariants.

The value of the invariants s, d, D can be used to determine the type and compose the canonical equation of the second-order curve.

Table 8.1.

Classification of second-order curves based on invariants

Let's take a closer look at the ellipse, hyperbola, and parabola.

Ellipse(Fig. 8.1) is called the locus of points of the plane for which the sum of the distances to two fixed points this plane, called ellipse tricks, is a constant value (greater than the distance between the foci). This does not exclude the coincidence of the foci of the ellipse. If the foci are the same, then the ellipse is a circle.

The half-sum of the distances from the point of the ellipse to its foci is denoted by a, half the distances between the foci - c. If a rectangular coordinate system on the plane is chosen so that the foci of the ellipse are located on the Ox axis symmetrically with respect to the origin, then in this coordinate system the ellipse is given by the equation

, (8.4.2)

called the canonical equation of the ellipse, where .

Rice. 8.1

With the specified choice of a rectangular coordinate system, the ellipse is symmetrical about the coordinate axes and the origin. The axes of symmetry of an ellipse call it axes, and the center of symmetry is the center of the ellipse. At the same time, the numbers 2a and 2b are often called the axes of the ellipse, and the numbers a and b are called large and semi-minor axis respectively.

The points of intersection of an ellipse with its axes are called the vertices of the ellipse. The vertices of the ellipse have coordinates (a,0), (–a,0), (0,b), (0,–b).

Ellipse eccentricity called a number

Since 0£c

.

This shows that the eccentricity characterizes the shape of the ellipse: the closer e is to zero, the more the ellipse looks like a circle; as e increases, the ellipse becomes more elongated.