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10PHYSICS AND MATHEMATICS
RATIO OF FOUR POINTS IN PROJECTIVE GEOMETRY
Beknazarov B.
Master student of Korkyt Ata Kyzylorda state University. Kyzylorda. Kazakhstan
Seitmuratov A.
Doktor of Physical and Matematical Sciences, Professor, Korkyt Ata Kyzylorda State University.
Kyzylorda. Kazakhstan
Abstract
Projective geometry does not consider metric concepts such as the length of the section, the size of the angle, the size of the figure and the surface area, depending on which the bisector, height, median, rectangle, square, distance, etc. no concepts. However, projective geometry cannot be considered as an irrelevant science, completely separate from elementary geometry. The projection method is one of the main methods in geometric research. In general, it is not necessary to always take projections in a perpendicular direction, they can be taken parallel to any definite direction. The article considers the ratio of four points in the predictive geometry. If two points of a projective line are given, then any third point of it is expressed by those two points, because the beam lines defining the points lie in the same plane and their guide vectors are coplanar, so each of the three vectors is defined as a linear combination of the other two.
Keywords: projective lines, bundles, harmonic pairs, harmony, in projective geometry, harmonic quadrants
A :
If two points of a projective line are given, then any third point of it is expressed by those two points, because the beam lines defining the points lie in the same plane and their guide vectors are coplanar. Thus, each of the three vectors is a linear combination of the other two. Thus, if A, B, C are always different points of the same perspective line
C = aA + PB There are numbers a and ft. if
if the numbers a, ft
c1 = aa1 + ftb1 c2 = aa2 + pb2 is from the system. The determ inant here is different from zero, because the coordinate columns A and B are not proportional to the different points.
Help. A, B, C, D - four different points on the projective line
The coordinate columns are the numbers a, ft,
a', p
C = aA+ftB,D = a'A + P'B (1)
so that It states that the number - — is a complex
a a'
ratio of four points A, B, C, B. Usually it
(ABCD) = £••£■ (2)
a a'
denotes by the formula. Thus, to find a complex relationship, the second colon C, D must be expressed by the first colon A, B.
To verify the projectivity of the given concept, it is enough to show that it is independent of the derivation of the coordinate system.
In this regard, we come to the following conclusion:
Theorem 1. Let A, B, C, B be four quadrilateral straight lines. With respect to a system of two projective coordinates, let their coordinate columns be A1, B1 C D1 and, A2, B2 C2 D2 respectively. Then if
C1 = a±A± + ß1B1, C2 = a2^2 + ß2B2, (3) Di = a-iAi + ß'1B1, D2 = CC2A2 + ß'2B2,
if so
^2 a
Proof.(2,2) coordinate columns of points given in the equation in different systems
xA2 = PA1,yB2 = PB1,zC2 = PCx,TD2 = PDl related to the relationship. P is the matrix of transition from one system to another, x, y, z, t are the proportional coefficients. Multiplying the first two of the formulas (1.5) by the matrix P from the left to get the relations
zC2 = a1xA2 + b1yB2, TD2 = k1xA2 + b\yB2
a p = Piy a- = alX p' =
equations appear. Therefore
Pi Pi Pi Pi z P2 &
a?
t
a, a
^ a2
z t
The theorem is proved.
We insisted that all four points be different in the definition of a complex relationship. The non-convergence of points A and B provides equality. We will further ease this restriction. To do this, we add a new number with ro expressions to the real numbers. We
describe the number ro with the following properties: m m
— = ro (m t 0); — = 0 (m t ro); m^ ro = ro (m ro0)
Now we only need three points A, B, C to be different, point D can face each of them
(Since C t A a n d C t B a r e — t
a
0 and - t ro );
a
1. D = A. In this case ft- t 0, a-t 0 so ft': a' = 0
aP P' and -:—= ro,
u u
Thus (A,B,C,D) = œ..
2. D = B . Similar to the found (A, B, C, D) = œ.
3. D = C .In this case - = - means (A, B,C,D) = 1.
a a'
It is worth noting that the first three points are different, and the fourth, the complex relationship of the quartet facing one of them does not take values other than these œ, 0,1 values.
4. The existence and loneliness of a point that has a known complex relationship with a given three points. Expression of projective coordinates by complex relations.
Theorem 1. Let A, B, C be points on the projective line and let it be known that (A, B, C, D) = h. With such a condition, point D is defined unambiguously.
Proof. If h = œor h = 0, the position of point D is determined, as described in the previous paragraph, and it coincides with point A or B. Therefore, we assume that h ^ œand h ^ 0.
(3.1), in particular, pay attention to the formulas C = aA+ PB.D = a'A +
Let's translate. Here — is unknown, - is known,
w a
plus - ± 0,
a a
After setting the value of the complex relationship, (A,B,C,D)=?-:--=h or Si = ±.
a a, a, ah
Now find the coordinate value of point D with precision from the second to the multiplier of the formula (1.3), and therefore determine its position on the line.
The theorem is proved.
It follows from this theorem that a complex relationship takes place at the coordinates of a point on a projective line.
However, we do not use this system, but the coordinate system introduced in paragraph 1 of 2, which does not require the extension of the concept of number. The following theorem shows the relationship of projective coordinates to a complex relationship.
Help. The four points A, B, C, D, described by the condition (A,, B, C, D) = -1, are called harmonic.
Such quartets play an important role in projective geometry and have a number of interesting properties. Here are some of them.
Property 1. In a harmonious quartet, pairs are separated. The correctness of this statement comes directly from the theorem of point 4.
Property 2. Harmony retains its meaning in the movement of pairs or in substitutions that do not change the composition of the pairs.
Indeed, according to the theorem of point 5, in the displacement of pairs or in the displacement of elements within a pair, the complex relationship either does not change or is inversely equal. In each of these cases, the complex ratio is -1. From this property, if (A,B,C,D) = -1, (BACD) = (ABDC) = (BADC) = (cDAB) = (DCAB) = (CDBA) = (DCBA) = -1
Example 1. Find the complex ratio A(1; -1), B(-2; 1), C(1; 3),D(1; 0). (ABCDD) on the projective line.
The solution. To find a complex relationship, express columns C and D by formulas A and B according to formulas (1.3), namely
(1) = -(h) + PCD; (0) = +
From the relations, we need to find the numbers a, p, a', P'. These are the equations
(1 = a-2p
{3 = -a + p
(1 = w- 2P'
{0 = -a' + P'
synonymous with the system. Then a = -7, p =
-4, a' = -4,P' = -1.
Substituting these values into (1.2) we obtain
N P & -4-1 4 (ABCD) =-■■- = —-.— = -
a a' -7 -1
1
Example 2. Let's calculate the ratio (DBAC) for the points in the first example.
The solution. Method 1. To calculate the value of (DBAC)
A = SD + PB,C = S'D + p-B
Let's find the coefficients in the formulas. Use them as in Example 1
or, C = —1A - 4B,D = -A-B
Then
A= —D — B,C = —1(—D — B) — 4B = 1D + 3B
and (DBAC) = —:3 = -.
Method 2. According to the theorem on the change of the complex ratio of the displacement of points, as (ABCD)is known
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