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volume 4, issue 3, 2025
392
DERIVATION OF THE DISTANCE BETWEEN PARALLEL PLANES
Rustamov Bilol
Abdurashidov Nuriddin
abdurashidovnuriddin9550@gmail.com
Eshtemirov Eshtemir
eshtemireshtemirov577@gmail.com
Lecturer Department of Higher Mathematics,
Denov Institute of Entrepreneurship and Pedagogy
Baltabayeva Saida
Students of Denov Institute of Entrepreneurship and Pedagogy
saidabaltabayeva499@gmail.com
Abstract:
This paper explores the method for determining the distance between two parallel
planes. The formula for calculating the distance, its derivation, and geometric properties are
discussed. The distance between two parallel planes is the shortest distance measured along a
perpendicular line from one plane to the other. Through the formulas provided, examples are
presented to explain how the distance between two parallel planes is calculated. This concept is
widely applied in mathematics and fields such as physics, aiding in solving both theoretical and
practical problems.
Key words:
Two planes, parallel planes, distance, point
.
Introduction
In geometry, the concept of distance between planes holds significant theoretical and practical
value. In particular, calculating the distance between two parallel planes arises frequently in
various scientific and engineering disciplines, including mathematics, physics, engineering
design, and computer graphics. Parallel planes are defined as planes that do not intersect and
share the same normal vector direction, making the shortest distance between them a constant.
Although the formula for finding this distance appears straightforward, it incorporates essential
geometric concepts such as plane equations, normal vectors, and the perpendicular distance from
a point to a plane. In this article, we will explore the mathematical formulation for determining
the distance between two parallel planes, provide a derivation of the formula, and discuss its
practical applications through illustrative examples.[1-4].
Literature Review:
The problem of finding the distance between parallel planes is one of the fundamental topics in
geometry. Numerous studies and scholarly works have mathematically analyzed this problem,
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volume 4, issue 3, 2025
393
proposing various formulas and approaches.
1.
Mirzoev, I. (2015).
Geometry: Theoretical Foundations and Practical Applications.
Tashkent:
Science
and
Technology.
This source provides detailed insights into planes and their position in space. Methods for
calculating the distance between two parallel planes are presented, including theoretical
approaches related to normal vectors and perpendicular distances.
2.
Sharifov, A., & Rakhimov, F. (2018).
Geometric Problems and Their Applications in
Physics.
Tashkent:
National
University
of
Uzbekistan.
In this study, Sharifov and Rakhimov discuss the calculation of distances between planes and
their applications in physics, particularly in electrostatics. The role of the distance between
parallel planes in electrostatic fields and its calculation methods are explained.
3.
G‘ulomov, B. (2020).
Mathematical Models and Geometry. Tashkent: O‘qituvchi.
This book mathematically proves the formula for calculating the distance between parallel planes
and provides examples of its application. The book includes various methodologies to help
understand the formula and its proof.
The literature review indicates that the problem of finding the distance between parallel planes is
widely used across different fields of mathematics, and it holds practical significance in various
sciences, including physics and engineering. [4-6].
Results and Discussion
Let's say that we take mutually parallel planes
�
1
and
�
2
in space.
�
1
: �
1
� + �
1
� + �
1
� + �
1
= 0
�
2
: �
2
� + �
2
� + �
2
� + �
2
= 0
If the point
�(�
1
, �
1
, �
1
)
is determined from the plane
�
1
, the perpendicular
�
2
plane normal
vector drawn from this point is parallel to
� = (�
2
, �
2
, �
2
)
. We draw a straight line parallel to
the vector
�
from M. Its canonical and parametric equations are as follows:
� − �
1
�
2
=
� − �
1
�
2
=
� − �
1
�
2
= �
� = �
2
� + �
1
� = �
2
� + �
1
� = �
2
� + �
1
A straight line perpendicular to the
�
2
plane intersects the plane at the point
� �, �, �
. [7-8].
We put the expression of x, y and z in the parametric equation through the parameter t into the
plane equation
�
2
:
�
2
�
2
� + �
1
+ �
2
�
2
� + �
1
+ �
2
�
2
� + �
1
+ �
2
= 0
�
2
2
� + �
2
�
1
+ �
2
2
� + �
2
�
1
+ �
2
2
� + �
2
�
1
+ �
2
= 0
� �
2
2
+ �
2
2
+ �
2
2
+ �
2
�
1
+ �
2
�
1
+ �
2
�
1
+ �
2
= 0
� =−
�
2
�
1
+ �
2
�
1
+ �
2
�
1
+ �
2
�
2
2
+ �
2
2
+ �
2
2
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394
� = �� =
� − �
1
2
+ �
2
�
2
+ � − �
1
2
� − �
1
= �
2
�
� − �
1
= �
2
�
� − �
1
= �
2
�
� = �� =
�
2
�)
2
+ �
2
�
2
+ �
2
�
2
� = �� = � (�
2
2
+ �
2
2
+ �
2
2
) =
�
2
�
1
+ �
2
�
1
+ �
2
�
1
+ �
2
�
2
2
+ �
2
2
+ �
2
2
From this came the distance between the point and the plane.[8-12].
Now we make the formula for the distance from this point to the plane and the distance from the
plane to the plane. The coordinates of the normal vector in the equation
�
1
�
1
+ �
1
�
1
+ �
1
�
1
+
�
1
= 0
in space cannot be equal to 0. [12-15].
From this
1)
�
1
≠ 0 �
1
=−
�
1
+�
1
+�
1
�
1
,
�
1
= 1, �
1
= 1
,
2)
�
1
= 0 �
1
≠ 0�
1
= 1, �
1
=−
�
1
+�
1
+�
1
�
1
,
3) �
1
= 1
,
�
1
= 0 , �
1
= 0, �
1
≠ 0
,
�
1
= 1, �
1
= 1, �
1
=−
�
1
+�
1
+�
1
�
1
.
If we put in the formula of the distance from the above point to the plane,
1)
�
1
=−
�
1
+�
1
+�
1
�
1
, �
1
= 1, �
1
= 1 A
1
≠ 0;
� =
�
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
�
1
�
2
2
+ �
2
2
+ �
2
2
2)
�
1
= 1 , �
1
=−
�
1
+�
1
+�
1
�
1
,
�
1
= 1 , �
1
≠ 0
;
� =
�
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
�
1
�
2
2
+ �
2
2
+ �
2
2
3)
�
1
= 1, �
1
= 1, �
1
=−
�
1
+�
1
+�
1
�
1
,
�
1
≠ 0;
� =
�
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
2
�
1
�
2
2
+ �
2
2
+ �
2
2
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volume 4, issue 3, 2025
395
Example: Find the distance between two parallel planes
� + 2� − 2� + 2 = 0,
3� + 6� − 6� − 4 = 0.
If we find the distance between two planes in space using coefficients:
� =
�
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
�
1
�
2
2
+ �
2
2
+ �
2
2
� =
−4 2
3
1 +
6 2
3 1 +
−6 −2
3
1
1 3
2
+ 6
2
+(6)
2
=
−4−6 + 6−6 + −6+6
9
=
10
9
. . [15-17 ].
Conclusions
General
Outcome
Beyond the classical formula
� =
�
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
+ �
2
�
1
�
2
�
1
�
1
�
2
2
+ �
2
2
+ �
2
2
this work presented a parametric derivation based on constructing the perpendicular line from a
point on one plane to the other.
Special
Cases
When certain coefficients of the first plane vanish (A₁=0, B₁=0, or C₁=0), three distinct point-
selection strategies were developed. These ensure the method applies to any pair of parallel
planes.
Practical and Theoretical Significance
Practical
: The approach is well- suited for accurately computing distances in
electrostatics, engineering structures, and computer graphics where complex parallel plane
configurations arise.
Theoretical
: The parametric proof reinforces fundamental geometric concepts—normal
vectors, perpendicular projections, and parametric line equations.
Future Work
Extending to distances between hyperplanes in higher- dimensional spaces.
Enhancing computational accuracy and efficiency of the parametric method using
numerical techniques.
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problems from analytical geometry Tashkent-2005, page 546.
3. B.A. Khudayarov "Linear algebra and analytical geometry" Tashkent-2018, 284 pages.
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