Volume 04 Issue 02-2024
12
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
It is known that the Poincaré interpretation of Lobachevsky’s geometry is used in solving many technical problems, in
problems related to the theory of complex variable functions.
In this article, we show the Poincaré interpretation of the Lobachevsky plane, which is interpreted in a circle in a plane,
using one circle of a two-section hyperboloid, using the method of spatial representation, and the Lobachevsky axiom
and the results derived from it are also valid.
KEYWORDS
Lobachevsky’s axiom, hyperbolic line, inve
rsion, outer product, intersection, asymptotic cone, oscillating cone.
INTRODUCTION
Lobachevsky's geometry, the first non-Euclidean
geometry, only after its interpretations appeared, did
the scientific mind believe that these concepts are
logically correct. There are many geometric
interpretations of Lobachevsky's geometry. The most
famous of them is the Kelly Klein interpretation, and
one of the interpretations related to many technical
issues and used in the theory of complex variables is
the Poincaré interpretation.
The geometrical interpretations of Lobachevsky's
geometry [3], [4] depend on how the basic concepts of
planimetry, "point" and "straight line", are chosen
from each other. Since the basic concepts are accepted
without any definition, each teacher has his own idea
about these concepts. In this article, we will get
acquainted with the method of visualizing the
elements
of
the
Poincaré
interpretation
of
Lobachevsky geometry using spatial forms.
Poincaré's interpretation of Lobachevsky's geometry,
like his Kelly Klein interpretation, is represented by
points inside a circle on a plane.
Research Article
FULFILLMENT OF LOBACHEVSKY'S AXIOM IN EUCLIDEAN SPACE OF
THE POINCARE INTERPRETATION OF LOBACHEVSKY'S GEOMETRY
Submission Date:
February 05, 2024,
Accepted Date:
February 10, 2024,
Published Date:
February 15, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue02-03
Fayzullaev Sherzod
Teacher At The Department Of Mathematics Teaching Methodology At Jizzakh State Pedagogical University,
Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 02-2024
13
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
In our previous works, we considered the spatial
representation of the Poincaré model of the
Lobachevsky plane. In this case, we adopted the
following initial concepts and adopted the following
definition.
In the three-dimensional Euclidean space, let the
surface π, which is one section of a two
-section
hyperboloid, and its asymptotic cone K be given. In this
case, we choose the axis of symmetry of the cone K so
that it coincides with its own axis, and the tip of the
cone coincides with the origin of the coordinates. In
that case, π surface and cone equations K are
expressed in the following form.
0
:
1
:
2
2
2
2
2
2
=
−
+
=
−
+
z
y
x
V
z
y
x
In such a selection method, z=1 is the plane,
𝜋
is the
projection plane for the surface, and intersects with
the cone
𝐾
on a circle with the same radius
1
2
2
=
+
y
x
(Fig. 3).
Figure 3
In this article, as in [2], we consider the points of the
hyperboloid as the points of the Lobachevsky plane. In
addition, let us denote by the set of conic cones
𝐾(𝑊(𝑂
1
))
whose ends are at the origin of the
coordinates, and when all z=1 planes intersect, forming
a circle, and these circles intersect the hyperboloid and
are mutually orthogonal to the circle whose radius is
equal to one. Let's look at the cone
𝐾
belonging to this
set
𝐾(𝑊(𝑂
1
))
. Naturally, this cone will be obtuse.
Theorem. Any conic generator
𝐾 ∈ 𝐾(𝑊(𝑂
1
))
cuts one
segment of the bisector hyperboloid π along the
hyperbolic line
𝑙
.
Proof. Since the generators of the cone
𝐾
are
orthogonal to the unit circle in the z=1 plane, it is
natural that it intersects one segment
𝑧 > 0
of the
two-section hyperboloid
𝜋
. We called this section
hyperbolic. Because this section asymptotically
approaches the V cone generators passing through the
points C and D, which are the points of the circle
section. This follows from the fact that the cone
𝑉
is
the asymptotic cone of the π hyperboloid.
Description.
K
'
∈
K(W(O_1))
conic generators, the line
𝑙
formed by the intersection of the hyperboloid
𝜋
with a
circle is called a straight line in the interpretation of the
Lobachevsky plane on the hyperboloid (Fig. 4).
Volume 04 Issue 02-2024
14
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Figure 4
Now we show that the Lobachevsky axiom and the results derived from it are appropriate for the spatial
representation of the Poincaré interpretation of the Lobachevsky plane.
Let us be given a regular cone
𝐾′ ∈ 𝐾(𝑊(𝑂
1
))
and a point
𝑀
that does not belong to this cone and lies on the
hyperboloid. Let
𝐾
1
, 𝐾
2
∈ 𝐾(𝑊(𝑂
1
))
be given by such large cones. Let the point
𝑀
belong to the generators of these
obtuse cones. The obtuse cone
𝐾′ ∈ 𝐾(𝑊(𝑂
1
))
lies inside both obtuse cones
𝐾
1
, 𝐾
2
∈ 𝐾(𝑊(𝑂
1
))
and has no point in
common with their generators (Fig. 8).
Figure 8
Volume 04 Issue 02-2024
15
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Here, the intersection of the hyperboloid π with the
cone
generators
𝐾′ ∈ 𝐾(𝑊(𝑂
1
))
forms
the
Lobachevsky straight line. The Lobachevsky straight
lines
𝑙
1
and
𝑙
2
formed by the intersection of the
generators
of
the
obtuse
cones
𝐾
1
, 𝐾
2
∈
𝐾(𝑊(𝑂
1
))
and π with the hyperboloid are formed and
they pass through the point
𝑀
. From this, it is possible
to draw two straight lines
𝑙
1
and
𝑙
2
that do not
intersect with this straight line through a straight line
𝑙
and a point
𝑀
not lying on it.
So, the Lobachevsky axiom is fulfilled for the spatial
representation of the Poincaré interpretation of the
Lobachevsky plane.
Now we will consider the following conclusions from
Lobachevsky's axiom.
1-Conclusion. An infinite number of straight lines that
do not intersect with the straight line
l
can be drawn
through the point
𝑀
that does not lie on the given
straight line
𝑙
in the plane.
We show that this result is valid for the spatial
representation of the Poincaré interpretation of the
Lobachevsky plane.
We are given a regular cone
𝐾′ ∈ 𝐾(𝑊(𝑂
1
))
and a
point M that does not belong to this cone and lies on
the hyperboloid. Let there be hypercones
𝐾
1
, 𝐾
2
∈
𝐾(𝑊(𝑂
1
))
whose generators pass through the point
𝑀
, both of which include the hypercone
𝐾′ ∈
𝐾(𝑊(𝑂
1
))
. Let's consider the obtuse cones
𝐾
3
, 𝐾
4
∈
𝐾(𝑊(𝑂
1
))
such that the obtuse cone
𝐾
3
contains the
obtuse cone
𝐾
1
, and the obtuse cone
𝐾
4
contains the
obtuse cone
𝐾
2
, and at the point
𝑀
have it. Then
𝐾
3
, 𝐾
4
∈ 𝐾(𝑊(𝑂
1
))
and the Lobachevsky straight lines
𝑙
3
and
𝑙
4
formed by intersection of the hyperboloid π
pass through the point
𝑀
and the straight line
𝑙
does
not lie passes through vertical angles. These straight
lines
𝑙
3
and
𝑙
4
also do not intersect with the straight
line
𝑙
. Because in order to intersect with the straight
line
𝑙
, the straight line
𝑙
3
must cross the straight line
𝑙
1
at some point
𝑁
. This leads to the violation of
Euclid's 1st postulate. Likewise, the straight line
𝑙
4
does
not intersect with the straight line
𝑙
(Fig. 9).
Figure 9
Volume 04 Issue 02-2024
16
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
From this it can be concluded that infinitely many
straight lines can be drawn through the point
𝑀
, which
is not on the straight line
l
, and does not intersect with
the straight line
l
. Because you can get as many cones
as you like.
2-Conclusion. Two parallel straight lines
𝑙
can be drawn
to a given straight line through a point that does not lie
on it.
In order to show that this result is also valid, we are
given a regular cone
𝐾′ ∈ 𝐾(𝑊(𝑂
1
))
and a point M that
does not belong to this cone and lies on the
hyperboloid. Let
𝐾
1
, 𝐾
2
∈ 𝐾(𝑊(𝑂
1
))
be given cones,
whose generators pass through the point
𝑀
, both
𝐾′ ∈
𝐾(𝑊(𝑂
1
))
and the generator of the asymptotic cone
𝑉
. The Lobachevsky straight lines
𝑙
1
and
𝑙
2
formed by
the intersection of these obtuse cones and
π
hyperboloid are parallel to the straight line
𝑙
. The
reason is that we showed above what parallel straight
lines look like (Fig. 10).
Figure 10
Therefore, the straight lines
𝑙
1
and
𝑙
2
are straight lines
𝑙
passing through the point M and parallel to the
straight line l.
CONCLUSION
In conclusion, it is possible to say that the Poincaré
interpretation of the Lobachevsky plane in Euclidean
space is the circle formed by the intersection of the
two-section hyperboloid with one section and the
asymptotic cone with the
𝑧 = 1
plane, which forms the
orthogonal circles formed by the intersection of the
generators of this ellipse as long as the Lobachevsky
axiom is fulfilled for straight lines and hyperboloid
points formed by the intersection of convex cones.
REFERENCES
1.
Sh. U. Fayzullaev. Puankare talqinining fazoviy
tasviri. “Zamonaviy matematikaning nazariy
asoslari va amaliy masalalar” Respublika ilmiy
-
amaliy anjumani materiallari to’plami. Andijon. 28
-
mart 2022 yil. I-qism.
2.
I. M. Hatamov, SH. U. Fayzullaev. Lobachevskiy
tekisligining gipervoloid ustidagi talqini. Fizika,
matematika va informatika. Ilmiy
–
uslubiy jurnal.
Toshkent
–
2019 yil. 1-son.
Volume 04 Issue 02-2024
17
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
02
Pages:
12-17
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.063
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
3.
Н.В.Ефимов. Высшая геометрия. Москва
.
Физматлит
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Н.Г.Подаева , Д.А. Жук. Лекции по основам
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