Journal of Social Sciences and Humanities Research Fundamentals
26
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TYPE
Original Research
PAGE NO.
26-29
DOI
OPEN ACCESS
SUBMITED
14 February 2025
ACCEPTED
13 March 2025
PUBLISHED
11 April 2025
VOLUME
Vol.05 Issue04 2025
COPYRIGHT
© 2025 Original content from this work may be used under the terms
of the creative commons attributes 4.0 License.
Activation of Students'
Independent Cognitive
Activity by Homologous
Substitution
Tashimov Nurlan Erpolotovich
Associate Professor of Tashkent State Pedagogical University named after
Nizami, Uzbekistan
Abstract:
This article highlights the importance of
activating the independent cognitive activity of
university students using the homologous substitution
method. Solutions to positional and metric problems
were considered using the homologous substitution
method, which replaces cases of three-dimensional
spatial forms with those that are homologously
compatible with them.
Keywords:
Applied geometry, descriptive geometry,
homologous substitution, students, independent
cognitive activity, activation, plot transformation, one-
to-one correspondence, surfaces, surfaces rotation,
positional and metric tasks.
Introduction:
Today in our republic, the rapid
development of science and the rapid updating of
knowledge, the introduction of high-quality and
efficient equipment, technologies, the introduction of a
modern information and communication system in all
areas, including the educational process, require
improved teaching of applied geometry, especially for
higher education institutions.
In the science of applied geometry, when constructing
drawings of various geometric shapes, along with the
methods of plot transformation, the homologous
substitution method can be used.
In this article, we will provide the following information
about the reduction of the homologous substitution
method and the theoretical foundations of its creation.
Let us consider the construction of the central
projection of triangle ABC on the plane Q through the
spatial center S (Fig.1). In this case, the rays SA, SB and
SC passing through the point S intersect with the plane
P, forming the points SA∩P=A', SB∩P=B' and SC∩P=C'.
The connection of these points gives the triangle A'B'C'.
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Journal of Social Sciences and Humanities Research Fundamentals
Let the line of intersection of the planes of triangles
ABC and A'B'C' be a straight li
ne a=P∩Q. It has an
intersection, while the traces of the sides of triangle
A'B'C' on the straight line are the points B'C'∩a=1',
A'B'∩a=2', and C'A'∩a=3' the sides of triangle ABC are
the intersection points of the straight line BC∩a=1,
AB∩a=2, and CA∩a=3. In this case, the points 1≡11',
2≡21' and 3≡31' will be superimposed on the line A.
If we project the ends of triangle ABC through an
arbitrary center S1 in space onto the second side P1 of
the plane P, we get triangle A1'B1'C1'.
In this case, the projection of the center S onto the plane
P1 will be S1'. As can be seen from the drawing, even in
this case, the traces of the 1', 2' and 3' sides of triangle
A'B'C' on the cut line, a will be at the same point as the
traces of the sides of triangle ABC. The lines connecting
the corresponding vertices of triangles A'B'C' and
A1'B1'C1' pass through the point S'. This will establish an
unambiguous correspondence in the space between the
points of triangles ABC and A'B'C' and the points of
triangles A'B'C' and A1'B1'C1'.
Fig.1
Based on the above constructions and conclusions, the
homologous replacement method will be issued as
follows.
On the plane, the method of homologous substitution
is defined by the homology center, a pair of
corresponding points, and the homology axis. Any flat
shape defined on a plane based on the above can be
replaced by a flat shape that is homologous to it. Also,
a circle, an ellipse, a parabola, and a hyperbola defined
on a plane can be replaced by their corresponding
homologous second-order curvature.
These conclusions, in turn, are called Desargues'
theorem.
If the rays connecting the corresponding ends of any
triangle on the plane pass through a point, then the
points of intersection of the corresponding sides of
these triangles lie on the same straight line.
In Fig. 2, substituting triangle ABC into triangle A1B1C1,
which is homologous to it, we obtain the inverse
theorem to this theorem as follows.
If the points of intersection of the corresponding sides
of any triangle on the plane lie on the same straight line,
then the lines connecting the corresponding ends of the
triangle pass through one point.
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Journal of Social Sciences and Humanities Research Fundamentals
Fig.2
The following special cases of homologous substitution
can be given, based on cases where the axis and center
of homology are indicated in the drawing.
1. The center of homology S is a point, and although
the homology axis is an irregular straight line a∞, the
lines connecting the corresponding sides of
homological plane shapes are parallel to each other
(Fig. 3a). that is, AB//A1B1, BC//B1C1, and AC//A1C1.
2. The center of homology S∞ is a nonlinear point, and
although the homology axis is a straight line, the rays
of homologous planar shapes connecting the
corresponding points are parallel to each other (Fig.
3b). that is, AA1//BB1//CC1.
3. The center of homology S∞ is located at an
infinitesimal point, and the axis of homology a∞ is a
nonlinear line, then the corresponding points are lines
in contact with homologous planar shapes parallel to
each other, and the corresponding sides of the planar
shapes are also parallel to each other (Fig. 3c). That is,
AB//A1B1, BC//B1C1, AC//A1C1 and AA1//BB1//CC1.
4. If the center of homology S∞ belongs to the
homology axis, then the lines connecting the
corresponding points of homological shapes are parallel
to each other, and the intersection points of the
corresponding sides of homological shapes belong to
the homology axis (Figure 3d). That is, BC=B1C1=3 S,
AC=A1 C1=2 S, AB=A1 B1=1 S
Using the homologous substitution method, it will be
possible to replace the states of three-dimensional
spatial forms with those that are homologous to them.
In this case, you will need to specify the center of
homology, the position of the homological replacement
plane, and the corresponding homology points. By
performing such substitutions, it is possible
to facilitate the solution of general (elliptical in cross-
section) second-order surfaces, taking into account
their geometric position and related positional and
metric problems, by bringing them into rotation.
а)
b)
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Journal of Social Sciences and Humanities Research Fundamentals
c) d)
Fig.3
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