What is the canonical form of an equation? Second order lines Ellipse and its canonical equation
This is the generally accepted standard form of an 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 tasks. 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 easily visible.
It is obvious that any 1st order line is a straight line. On the second floor, it is no longer the watchman who is waiting for us, but a much more diverse company of nine statues:
Classification of second order lines
Using a special set of actions, any equation of a second-order line is reduced to one of the following forms:
( and – positive real numbers)
1) 
– canonical equation of the ellipse;
2) – canonical equation of a hyperbola;
3) 
– canonical equation of a parabola;
4) – imaginary ellipse;
5) – a pair of intersecting lines;
6) – pair imaginary intersecting lines (with a single valid point of intersection at the origin);
7) – a pair of parallel lines;
8) – pair imaginary parallel lines;
9) – a pair of coincident lines.
Some readers may have the impression that the list is incomplete. For example, in point No. 7, the equation specifies the pair direct, parallel to the axis, and the question arises: where is the equation defining the straight lines, parallel axes ordinate? Answer: it not considered canonical. Straight lines represent the same standard case, rotated by 90 degrees, and the additional entry in the classification is redundant, since it does not bring anything fundamentally new.
Thus there are nine and only nine various types lines of the 2nd order, but in practice they are most often found ellipse, hyperbola and parabola.
Let's look at the ellipse first. As usual, I focus on those points that have great importance to solve 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 “the eccentricity of an ellipse”.
The canonical equation of an 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 to build an ellipse?
Yes, just take it and just draw it. The task occurs frequently, and a significant part of students do not cope with the drawing correctly:
Example 1
Construct an ellipse, given by the equation
Solution: First, let’s bring the equation to canonical form: 
Why bring? One of the advantages of the canonical equation is that it allows you to instantly determine vertices of the ellipse, which are located at points. It is easy to see that the coordinates of each of these points satisfy the equation.
IN in this case :
Line segment called major axis ellipse;
line segment – minor axis;
number 
called semi-major shaft ellipse;
number 
– minor axis.
in our example: .
To quickly imagine what a particular ellipse looks like, just look at the values of “a” and “be” of its canonical equation.
Everything is fine, smooth and beautiful, but there is one caveat: I completed the drawing using the program. And you can make the drawing using any application. However, in harsh reality, there is a checkered piece of paper on the table, and mice dance in circles on our hands. People with artistic talent, of course, can argue, but you also have mice (though smaller ones). It’s not in vain that humanity invented the ruler, compass, 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. It’s 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 general, it is highly desirable to find additional points.
There are two approaches to constructing an ellipse - geometric and algebraic. I don’t like construction using a compass and ruler because the algorithm is not the shortest and the drawing is significantly cluttered. 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 in the draft we quickly express: 
The equation then breaks down into two functions: 
– defines the upper arc of the ellipse; 
– defines the bottom arc of the ellipse.
Any ellipse is symmetrical with respect to the coordinate axes, as well as with respect to the origin. And this is 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 
. It begs to be found for additional points with abscissas 
. Let's tap three SMS messages on the calculator: 
Of course, it’s also nice that if a serious mistake is made in the calculations, it will immediately become clear during construction.
Let us mark points in the drawing (red), symmetrical points on the remaining arcs ( Blue colour) and carefully connect the whole company with a line: 
It is better to draw the initial sketch very thinly, and only then apply pressure with a pencil. The result should be a quite 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. Write an equation for a line that is symmetrical to a line
relative to the plane 
.
8.3.17.
Write down equations for projections onto a plane 
the following lines:
A) 
;
b) 
V) 
.
8.3.18. Find the angle between the plane and the line:
A) 
;
b) 
.
8.3.19.
Find a point symmetrical to the point 
relative to the plane passing through the 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
Let us 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 second-order curve 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 lines).
Equations 1) – 9) are called canonical equations of second order curves.
Solving the problem of reducing the equation of a second-order curve to canonical form involves finding the canonical equation of the curve and the canonical coordinate system. Reduction to canonical form allows one 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 point O by a certain angle j and subsequent parallel translation of the coordinate system.
Second order curve invariants(8.4.1) are 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 second-order curve (8.4.1), the sum of the coefficients for the squared coordinates
,
determinant composed of coefficients of 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
|
Elliptic curve |
sD<0. Эллипс |
|
|
sD>0. Imaginary ellipse |
||
|
A pair of imaginary lines intersecting at a real point |
||
|
Hyperbolic curve |
Hyperbola |
|
|
Pair of intersecting lines |
||
|
Parabolic curve |
Parabola |
|
|
A pair of parallel lines (different, imaginary or coincident) |
Let's take a closer look at the ellipse, hyperbola and parabola.
Ellipse(Fig. 8.1) is the geometric locus of points in the plane for which the sum of the distances to two fixed points 
this plane, called ellipse foci, is a constant value (greater than the distance between the foci). In this case, the coincidence of the ellipse's foci is not excluded. If the foci coincide, then the ellipse is a circle.
The half-sum of the distances from a 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 a 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 canonical ellipse equation, Where 
.
 |
Rice. 8.1
With the specified choice of a rectangular coordinate system, the ellipse is symmetrical with respect to the coordinate axes and the origin. The axes of symmetry of an ellipse are called 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 big And minor axis respectively.
The points of intersection of an ellipse with its axes are called vertices of the ellipse. The vertices of the ellipse have coordinates (a,0), (–a,0), (0,b), (0,–b).
Ellipse eccentricity called number
Since 0£c This shows that eccentricity characterizes the shape of the ellipse: the closer e is to zero, the more the ellipse resembles a circle; as e increases, the ellipse becomes more elongated. 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 ones; 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 statement A. In Chapter XV, § 3, it has already been proven that all ellipses are affinely equivalent to one of them, namely a circle, and all hyperbolas are a hyperbola. This means that all ellipses, respectively all hyperbolas, are affinely equivalent to each other. All imaginary ellipses, being affinely equivalent to a circle - - 1 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 is enough to prove that a parabola given in a certain coordinate system by its canonical equation similar to a parabola To do this, we subject the plane to a similarity transformation with a coefficient -: Then, with our transformation, the curve turns 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 proven that a curve that splits into a pair of intersecting 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 the equation in some affine coordinate system As for curves that split into a pair of parallel lines, each of them can be (even in some rectangular coordinate system) given by the equation for real ones, respectively for imaginary, direct. The transformation of coordinates allows us to put in these equations (or for coinciding straight lines. This implies the affine equivalence of all decaying curves of the second order that have the same name. Let's move on to the proof of statement B. Let us note first of all: with an affine transformation of the plane, the order of the algebraic curve remains unchanged. Further: every decaying curve of the second order is a pair of straight lines, and with an affine transformation, a straight line goes into a straight line, a pair of intersecting lines goes into a pair of intersecting ones, and a pair of parallel lines goes into a pair of parallel ones; in addition, real lines turn into real lines, and imaginary lines into imaginary lines. This follows from the fact that all coefficients in formulas (3) (Chapter XI, § 3), which determine the affine transformation, are real numbers. From what has been said it follows that a line affinely equivalent to a given decaying curve of the second order is a decaying curve of the same name. Let's move on to non-decaying curves. Again, with an affine transformation, a real curve cannot transform into an imaginary one, and vice versa. Therefore, the class of imaginary ellipses is affinely invariant. Let us 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 a certain 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 turn into a parallelogram containing the transformed curve, which, thus, cannot go to infinity and, therefore, is an ellipse. So, a curve affinely equivalent to an ellipse is certainly an ellipse. From what has been proved it follows that a curve affinely equivalent to a hyperbola or parabola cannot be an ellipse (and also, as we know, cannot be a decaying curve. Therefore, it remains only to prove that with an affine transformation of the plane, a hyperbola cannot transform into a parabola, and on the contrary, this, perhaps, most simply follows from the fact that a parabola does not have a center of symmetry, but a hyperbola has one. But since the absence of a center of symmetry for a parabola will be proven 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 two half-planes defined in the plane of a given line d, then it has at least one common point with the line . In fact, we have seen that there is a coordinate system in which a given parabola has the equation Let, relative to this coordinate system, straight line d have the equation By assumption, there are two points on the parabola, one of which, let’s say, lies in the positive half-plane 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 degree 2 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 to preserve generality in such cases it is said that it defines an imaginary linear image. etc. Depending on the values of the coefficients of the general equation (*), it can be transformed by parallel transfer of the origin and rotation of the coordinate system by a certain angle to one of the 9 canonical types given below, each of which corresponds to a certain class of lines. Exactly, unbreakable lines: y 2 = 2px - parabolas, decaying lines: x 2 - a 2 = 0 - pairs of parallel lines, x 2 + a 2 = 0 - pairs of imaginary parallel lines, x 2 = 0 - pairs of coinciding parallel lines. Study of the type of L. v. can be carried out without reducing the general equation to canonical form. This is achieved by joint consideration of the meanings of the so-called. basic invariants of linear v. n. - expressions composed of coefficients of the equation (*), the values of which do not change during parallel translation and rotation of the coordinate system:(S = a 11 + a 22,).
a ij = a ji So, for example, ellipses, like non-disintegrating 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-disintegrating lines (for hyperbolas δ Three main invariants Δ, δ, and S determine the linear movement. p. (except for the case of parallel lines) up to the 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 movements of the plane (metrically equivalent).
are considered equivalent. Connections between various affine classes of linear v. p. 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 drugs. p.: ellipses, hyperbolas and parabolas form one projective class - the class of real oval lines (ovals). A real oval line is an ellipse, a hyperbola or a parabola, depending on how it is located relative to a line at infinity: an ellipse intersects an improper line at two imaginary points, a hyperbola at two different real points, a parabola touches an improper line; there are projective transformations that transform these lines one into another. There are only 5 projective equivalence classes of linear vectors. p. Exactly, 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, degenerating 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
.
Plane lines whose rectangular coordinates of points 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 whose rectangular coordinates of points satisfy an algebraic equation of the 2nd degree. Among the lines of the second order are ellipses (in particular, circles), hyperbolas, and parabolas. * * * LINES OF THE SECOND ORDER LINES OF THE SECOND ORDER,... ... encyclopedic Dictionary Flat lines, rectangular. the coordinates of the points satisfy the algebras. 2nd degree level. Among L. v. etc. ellipses (in particular, circles), hyperbolas, parabolas... Natural science. encyclopedic Dictionary A flat line, Cartesian rectangular coordinates satisfy the algebraic. equation of the 2nd degree Equation (*) may not determine the actual geometric. image, but to preserve generality in such cases they say that it determines... ... Mathematical Encyclopedia A set of points of 3-dimensional real (or complex) space whose coordinates in the Cartesian system satisfy the algebraic. equation of the 2nd degree (*) Equation (*) may not determine the actual geometric. images, in such... ... Mathematical Encyclopedia This word, very often used in the geometry of curved lines, has an unclear meaning. When this word is applied to unclosed and unbranched curved lines, then by a branch of the curve is meant each continuous separate... ... 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. S. d. play an important role in the general theory of second-order lines. When simultaneously projecting an ellipse into its circumference, S. d.... ... Lines that are obtained by cutting a right circular Cone with planes that do not pass through its vertex. K. s. can be of three types: 1) a cutting plane intersects all generatrices of the cone at points of one of its cavity; line... ... Great Soviet Encyclopedia Lines obtained by cutting a right circular cone with planes that do not pass through its vertex. K. s. can be of three types: 1) a cutting plane intersects all generatrices of the cone at points of one of its cavity (Fig., a): line of intersection... ... Mathematical Encyclopedia Geometry section. The basic concepts of geometric geometry are the simplest geometric images (points, straight lines, planes, curves, and second-order surfaces). The main means of research in AG are the coordinate method (see below) and methods... ... Great Soviet Encyclopedia To clarify 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, we have 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) through a given real point, along with a given imaginary line, the complex conjugate line must also pass through it; therefore, this point must coincide with the vertex of the beam of rays that we use to depict the imaginary line. Similarly, in case 3), the real line must be identical with the support of that rectilinear involution of points that 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 it follows that each pair of points in the involution of points representing the point P must be on some pair of lines in the involution of lines , depicting the straight line g, i.e., that both of these involutions should be located perspectively one relative to the other; in addition, it turns out that the arrows of both involutions are also located prospectively. In general, in the analytical geometry of the plane, which also pays attention to the complex region, we will obtain a complete real picture of this plane if, to the set of all its real points and straight lines, we add as new elements the set of involutionary figures discussed above, together with the arrows of their directions. It will be enough here if I outline in general terms 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 a precise formulation of the presence of the just mentioned elements in a region expanded in comparison with the usual geometry. 2) Then the axioms of connection, which state that also in the extended region defined in paragraph 1)! through (every) two points there passes one and only one line and that (every) two lines have one and only one point in common. In this case, similar to what we had above, we have to distinguish each time four cases depending on whether the given elements are real, and it seems very interesting to think through 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 relationships, 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 learned from studying the theory of functions the habit of representing the set of values of a complex variable by all points of the plane. 4) Finally, regarding the axioms of continuity, I will only indicate here how complex points lying as close as desired to some real point are depicted. 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 (i.e., lying in a “crossed manner”) (Fig. 97), so that two points taken from different pairs lie close to one another and to point P; if we now bring the points closer 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 depicted 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 ideas of continuity, it is necessary to work with them in detail. Although this entire construction is quite cumbersome and tedious in comparison with ordinary real geometry, it can yield incomparably more. In particular, it is capable of raising algebraic images, understood as sets of their real and complex elements, to the level of complete geometric clarity, and with its help one can clearly understand in 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 main 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 still lead, despite 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 collection of three complex coordinates, and with it can be operated in exactly the same way as with real points. In fact, such an introduction of imaginary elements, abstaining from any principled reasoning, has always proved fruitful in those cases where we had to deal with imaginary cyclic points or with the circle of spheres. As already mentioned, Poncelet was the first to use imaginary elements in this sense; his followers in this regard were other French geometers, mainly Chals and Darboux; in Germany, a number of geometers, especially Lie, also used this understanding of imaginary elements with great success. With this retreat into the realm of the imaginary, I end the entire second section of my course and turn to a new chapter,
.
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