This is a generally accepted standard form of an equation, when it becomes clear in a matter of seconds which geometric object it defines. In addition, the canonical view is very convenient for solving many practical tasks. So, for example, according to the canonical equation "Flat" straight, firstly, it is immediately clear that it is a straight line, and secondly, the point belonging to it and the direction vector can be easily seen.
Obviously, any 1st order line is a straight line. On the second floor, however, not a watchman is 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 equation of the second-order line is reduced to one of the following types:
(and are positive real numbers)
1) 
- the canonical equation of the ellipse;
2) - the canonical hyperbole equation;
3) 
- the canonical equation of the parabola;
4) – imaginary ellipse;
5) - a pair of intersecting straight lines;
6) - pair imaginary intersecting lines (with the only valid intersection point at the origin);
7) - a pair of parallel straight lines;
8) - pair imaginary parallel lines;
9) - a pair of coincident straight lines.
Some readers may get the impression that the list is incomplete. For example, in point 7, the equation sets the pair direct parallel to the axis, and the question arises: where is the equation that determines the straight lines parallel to the ordinate? Answer: it not considered canonical... The straight lines represent the same standard case, rotated 90 degrees, and the 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 an ellipse first. As usual, I focus on those moments that have 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 ellipsis”, “the difference between an ellipse and an oval” and “eccentricity of an elebsis”.
The canonical equation of the ellipse has the form, where are positive real numbers, and. I will formulate the very definition of an ellipse later, but for now it's time to take a break from the talking shop and solve a common problem:
How do I build an ellipse?
Yes, take it and just draw it. The task is often encountered, and a significant part of the students do not quite competently cope with the drawing:
Example 1
Construct the ellipse given by the equation
Solution: first we bring the equation to the canonical form: 
Why lead? One of the advantages of the canonical equation is that it allows you to instantly determine ellipse vertices that are in points. It is easy to see that the coordinates of each of these points satisfy the equation.
In this case : 
Section are called major axis ellipse;
section – minor axis;
number 
are called semi-major axis ellipse;
number 
– semi-minor axis.
in our example:.
To quickly imagine what this or that ellipse looks like, it is enough to look at the values "a" and "bs" of its canonical equation.
Everything is fine, foldable and beautiful, but there is one caveat: I made the drawing using the program. And you can complete the drawing with any application. However, in the harsh reality, there is a checkered piece of paper on the table, and mice are dancing in circles on our hands. People with artistic talent, of course, can argue, but you also have mice (though smaller). It's not for nothing that mankind has invented a ruler, compasses, 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 semi-axes. 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 do not like the construction with the help of a compass and a ruler due to not the shortest 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 equation of the ellipse on the draft, quickly express: 
Further, the equation breaks down 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 freebies. Obviously, it is enough to deal with the 1st coordinate quarter, so we need the function 
... Finding additional points with abscissas suggests itself 
... We hit three sms on the calculator: 
Of course, it is also pleasant that if a serious error is made in the calculations, it will immediately become clear during the construction.
Mark the points in the drawing (red), symmetrical points on the remaining 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 give pressure to the pencil. The result should be a decent ellipse. By the way, would you like to know what this curve is?
8.3.15.
Point A lies on a straight line. Distance from point A to plane 
8.3.16. Equate a straight line, symmetrical straight line
relative to the plane 
.
8.3.17.
Draw up the equations of projections to the plane 
the following lines:
a) 
;
b) 
v) 
.
8.3.18. Find the angle between a plane and a straight line:
a) 
;
b) 
.
8.3.19.
Find a point symmetrical to a point 
relative to the plane passing through the straight lines:
and 
8.3.20. Point A lies on a straight line
Distance from point A to straight line 
equals . Find the coordinates of point A.
§ 8.4. SECOND-ORDER CURVES
We establish a rectangular coordinate system on the plane and consider the general equation of the second degree
in which 
.
The set of all points of 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)
(a pair of parallel lines);
8)
(a pair of imaginary parallel lines);
9) (a pair of coinciding straight lines).
Equations 1) - 9) are called canonical equations of curves of the second order.
The solution to the problem of reducing the equation of a second-order curve to the canonical form includes finding the canonical equation of the curve and the canonical coordinate system. Canonicalization allows you to calculate the parameters of a curve and determine its location relative to the original coordinate system. Transition from the original rectangular coordinate system 
to the canonical 
is carried out by rotating the axes of the original coordinate system around the point O by some angle j and subsequent parallel translation of the coordinate system.
By invariants of a curve of the second order(8.4.1) such functions of the coefficients of its equation are called, the values of which do not change when passing from one rectangular coordinate system to another of the same system.
For the second-order curve (8.4.1), the sum of the coefficients at the squares of the coordinates
,
determinant composed of the coefficients at the highest terms
and the third-order determinant
are invariants.
The value of the invariants s, d, D can be used to determine the type and form the canonical equation of a second-order curve.
Table 8.1.
Classification of curves of the second order based on invariants
|
Elliptical type curve |
sD<0. Эллипс |
|
|
sD> 0. Imaginary ellipse |
||
|
A pair of imaginary lines intersecting at a real point |
||
|
Hyperbolic Curve |
Hyperbola |
|
|
A pair of intersecting straight lines |
||
|
Parabolic curve |
Parabola |
|
|
A pair of parallel lines (distinct, imaginary, or coincident) |
Let's consider in more detail 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 foci of an ellipse, there is a constant value (greater than the distance between the foci). This does not exclude the coincidence of the focuses of the ellipse. If the focuses match, 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 of the distances between the foci - by 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 relative to the origin, then in this coordinate system the ellipse is given by the equation
,
(8.4.2)
called the canonical ellipse equation, 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 symmetry axes of the ellipse call it axles, and the center of symmetry - 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 big and semi-minor axis respectively.
The points of intersection of the 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).
Eccentricity ellipse called the number
Since 0 £ c Hence, it can be seen that the eccentricity characterizes the shape of an ellipse: the closer e is to zero, the more the ellipse looks like a circle; with increasing e, the ellipse becomes more elongated. We will now show that the affine classification of curves of the second order is given by the very names of the curves, i.e., that the affine classes of curves of the second order 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 (i.e., all ellipses, all hyperbolas, etc.) are affinely equivalent to each other. B. Two curves of different names are never affinely equivalent. We prove assertion A. In Chapter XV, § 3, it was already proved that all ellipses are affinely equivalent to one of them, namely, the circle and all hyperbolas are hyperbolas.Thus, all ellipses, respectively, all hyperbolas, are affinely equivalent to each other. All imaginary ellipses, being affinely equivalent to the 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, let us subject the plane to a similarity transformation with the coefficient -: Then so that with our transformation, the curve turns into a curve i.e. in a parabola Q.E.D. Moving on to the decaying curves. In § formulas (9) and (11), pp. 401 and 402) it was proved that a curve splitting into a pair of intersecting straight lines in some (even rectangular) coordinate system has the equation By doing an additional coordinate transformation we see that any curve that splits into a pair of intersecting real, respectively, imaginary conjugate, straight lines, has in some affine coordinate system the equation As for curves that split into a pair of parallel straight lines, each of them can be (even in some rectangular coordinate system) given by the equation for valid, respectively for imaginary, direct. The transformation of coordinates makes it possible to put in these equations (or for coinciding straight lines. This implies the affine equivalence of all decaying second-order curves having the same name. We pass to the proof of statement B. Note first of all: under the affine transformation of the plane, the order of the algebraic curve remains unchanged. Further: any split second-order curve is a pair of straight lines, and under an affine transformation, a straight line goes into a straight line, a pair of intersecting straight lines goes into a pair of intersecting ones, and a pair of parallel ones - into a pair of parallel ones; in addition, real lines pass into real ones, and imaginary ones - into imaginary ones. This follows from the fact that all the coefficients in formulas (3) (Chapter XI, § 3), which determine the affine transformation, are real numbers. It follows from what has been said that a line affinely equivalent to a given decaying second-order curve is a decaying curve of the same name. Moving on to non-decaying 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 affinely invariant. Consider the classes of real non-decaying 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. With an affine transformation, the rectangle ABCD containing the given ellipse will be transformed into a parallelogram containing the transformed curve, which, therefore, cannot go to infinity and, therefore, is an ellipse. So, a curve affinely equivalent to an ellipse is certainly an ellipse. It follows from what has been proved that a curve affinely equivalent to a hyperbola or a parabola cannot be an ellipse (and, as we know, there cannot be a decaying curve either. Therefore, it remains only to prove that under an affine transformation of the plane, a hyperbola cannot go to a parabola, and On the contrary, this, perhaps, most easily follows from the fact that the parabola does not have a center of symmetry, while the hyperbola does. affine nonequivalence 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 straight line d, then it has at least one common point with the straight line. Indeed, we have seen that there is a coordinate system in which the given parabola has the equation Let the line d with respect to this coordinate system have the equation By assumption, there are two points on the parabola, one of which, we put, lies in the positive, and the other, 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 an algebraic equation of the 2nd degree a 11 x 2 + a 12 xy + a 22 y 2 + 2a 13 x + 2a 23 y + a 11 = 0. (*) Equation (*) may not define a real geometric image, but in order to preserve generality in such cases, it is said that it defines an imaginary geometric image. p. Depending on the values of the coefficients of the general equation (*), it can be transformed using a parallel translation of the origin and rotation of the coordinate system by some angle to one of the 9 canonical types given below, each of which corresponds to a certain class of lines. Exactly, non-decaying lines: y 2 = 2px - parabolas, decaying lines: x 2 - and 2 = 0 - pairs of parallel straight lines, x 2 + a 2 = 0 - pairs of imaginary parallel lines, x 2 = 0 - pairs of coinciding parallel lines. Research of a type L. century. n. can be carried out without reducing the general equation to the canonical form. This is achieved by joint consideration of the meanings of the so-called. of the basic invariants of L. 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 = 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; a 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 L. v. (except for the case of parallel straight 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 plane movements (metrically equivalent). There are classifications of L. century. from the point of view of other transformation groups. So, 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. century. n. (see similarity)
are considered equivalent. Relationships between various affine classes of L. in. The item allows one to establish a classification from the point of view of projective geometry (see Projective geometry), in which infinitely distant elements do not play a special role. Valid non-decaying L. century. p .: ellipses, hyperbolas and parabolas form one projective class - the class of real oval lines (ovals). A real oval line is an ellipse, hyperbola or parabola, depending on how it is located relative to the infinitely distant straight line: the ellipse intersects the improper straight line at two imaginary points, the hyperbola - at two different real points, the parabola touches the improper straight line; there are projective transformations that transfer these lines to one another. There are a total of 5 projective equivalence classes for an L.V. n. Namely, non-degenerating 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, degenerating lines: x 1 2 - x 2 2= 0 - a 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
.
Plane lines, the rectangular coordinates of the points of which satisfy an algebraic equation of the 2nd degree. Among the lines of the second order are ellipses (in particular, circles), hyperbolas, parabolas ... Big Encyclopedic Dictionary Plane lines, the rectangular coordinates of the points of which satisfy an algebraic equation of the 2nd degree. 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 to px satisfy the algebras. level 2 url. Among L. century. n. ellipses (in particular, circles), hyperbolas, parabolas ... Natural science. encyclopedic Dictionary Plane line, Cartesian rectangular coordinates to swarm satisfy algebraic. equation of the 2nd degree Equation (*) may not determine the actual geometric. the image, but in order to preserve the generality in such cases they say that it determines ... ... Encyclopedia of mathematics The set of points of a 3-dimensional real (or complex) space whose coordinates in the Cartesian system satisfy algebraic. equation of the 2nd degree (*) Equation (*) may not determine the actual geometric. image, in such ... ... Encyclopedia of mathematics This word, very often used in the geometry of curved lines, does not have a completely definite meaning. When this word is applied to open and non-branching curved lines, then by the branch of the curve is meant each continuous separate ... ... Encyclopedic Dictionary of 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. S. d. 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 the section of a straight circular Cone with planes that do not pass through its vertex. K. s. can be of three types: 1) the cutting plane intersects all generatrices of the cone at the points of one of its cavities; line ... ... Great Soviet Encyclopedia Lines that are obtained by the section of a straight circular cone with planes that do not pass through its apex. K. s. can be of three types: 1) the cutting plane intersects all generatrices of the cone at the points of one of its cavities (Fig., a): the line of intersection ... ... Encyclopedia of mathematics Geometry section. The basic concepts of A. g. Are the simplest geometric images (points, lines, planes, curves, and surfaces of the second order). The main research tools in A. g. Are the method of coordinates (see below) and methods ... ... Great Soviet Encyclopedia To clarify this with a specific example, I will show you what corresponds in this interpretation to the following statement: (real or imaginary) point P lies on the (real or imaginary) line g. In this case, of course, one has to distinguish between the following cases: 1) real point and real line, 2) a real point and an imaginary line, Case 1) does not require special clarifications from us; here we have one of the basic relations of ordinary geometry. In case 2), along with the given imaginary line, the complex conjugate line with it must also pass through the given real point; therefore, this point must coincide with the vertex of the beam of rays that we use to represent the imaginary straight 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 a given imaginary point. The most interesting is case 4) (Fig. 96): here, obviously, the complex conjugate point must also lie on the complex conjugate line, and from this it follows that each pair of points of involution of points representing the point P must be on some pair of involution lines of lines representing the straight line g, that is, that both of these involutions must be located in perspective one relative to the other; moreover, it turns out that the arrows of both involutions are also located in perspective. In general, in the analytic geometry of the plane, which also pays attention to the complex region, we will get a complete real picture of this plane if we add as new elements to the totality of all its real points and lines the totality of the involutionary figures discussed above together with the arrows of their directions. It will be sufficient here if I outline in general outline what form the construction of such a real picture of complex geometry would take in this case. In doing so, I will follow the order in which the first sentences 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 extended in comparison with ordinary geometry. 2) Then the axioms of connection, which assert that also in the extended domain defined in item 1)! one and only one straight line passes through (every) two points, and that (every) two straight lines have one and only one point in common. At the same time, similar to what we had above, each time we have to distinguish four cases 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 a representation of these complex relations. 3) As for the axioms of location (order), here, in comparison with the actual relations, completely new circumstances appear on the scene; 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 has taken away from the study of the theory of functions the habit of depicting the totality of values of a complex variable by all points of the plane. 4) Finally, as regards the axioms of continuity, I will only indicate here how complex points are depicted that lie arbitrarily close 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 two pairs of points separating each other (that is, lying "in a crossed manner") (Fig. . 97) so that two points taken from different pairs lie close to one another and to the point P; if we now unboundedly approach the points, then the involution defined by the named pairs of points degenerates, that is, both of its still complex double points coincide with the point Each of the two imaginary points represented by this involution (together with one or the other arrow) goes over, therefore, continuously to some point close to point P, or even directly to point P. Of course, in order to be able to usefully apply these concepts of continuity, it is necessary to work with them in detail. Although all this construction is rather cumbersome and tedious in comparison with the usual real geometry, but it can give incomparably more. In particular, it is able to raise to the level of complete geometric visualization algebraic images, understood as collections of their real and complex elements, and with its help one can clearly understand on the figures themselves such theorems as the basic theorem of algebra or Bezout's theorem that two curves of 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 visual form than has been done so far; however, the literature already contains all the material essential for such research. But in most cases, the application of this geometric interpretation would nevertheless lead, with all its theoretical advantages, to such complications that one has to be content with its fundamental possibility and actually return to a more naive point of view, which consists in the following: a complex point is a set of three complex coordinates, and with it can be operated in the same way as with real points. Indeed, such an introduction of imaginary elements, refraining from any kind of principled reasoning, has always proved fruitful in those cases when we had to deal with imaginary cyclic points or with a circle of spheres. As already mentioned, Poncelet was the first to use imaginary elements in this sense; his followers in this respect were other French geometers, mainly Chal and Darboux; in Germany a number of geometers, especially Lee, also used this understanding of imaginary elements with great success. With this digression into the realm of the imaginary, I finish the entire second section of my course and turn to a new chapter,
.
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