=Paper=
{{Paper
|id=Vol-3628/short7
|storemode=property
|title=The Degree of Non-parabolicity of the Surface, Close to a Rotational Paraboloid
|pdfUrl=https://ceur-ws.org/Vol-3628/short7.pdf
|volume=Vol-3628
|authors=Vasyl Kryven,Lubov Tsymbaliuk,Volodymyr Valiashek,Andriy Boyko,Nadija Kryva
|dblpUrl=https://dblp.org/rec/conf/ittap/KryvenTVBK23
}}
==The Degree of Non-parabolicity of the Surface, Close to a Rotational Paraboloid==
https://ceur-ws.org/Vol-3628/short7.pdf
The Degree of Non-parabolicity of the Surface Close to a
Rotational Paraboloid
Vasyl Kryven1, Lubov Tsymbaliuk2, Volodymyr Valiashek3, Andriy Boyko4, Nadija Kryva5
Ternopil Ivan Puluj National Technical University, 56 Ruska St, Ternopil, UA46001, Ukraine
Abstract
A measure of deviation from parabolicity of a convex smooth surface of rotation is introduced. The
focus of the surface introdused is the one of the paraboloid of rotation, its axis and vertex coincide with the
axis of the original surface. The relative area of the region filled with rays falling parallel to the axis of
symmetry and reflecting from the surface is given and adopted as the measure of non-parabolicity. The
measure of non-parabolicity of the spherical segment and the wave-like perturbed paraboloid of rotation was
calculated.
Keywords
Convex rotational surface, reflector-type aerial, degree of parabolicity
1. Introduction
The interest in parabolic surfaces in engineering is primarily driven by their applications in antenna
technology for satellite communication. The parabolic antenna was invented by the German physicist
Heinrich Hertz in 1887. Hertz used cylindrical parabolic reflectors for sparking dipole antennas
excitation during his experiments. Hertz successfully demonstrated the existence of electromagnetic
waves, which had been predicted by Maxwell 22 years earlier.
Italian inventor Guglielmo Marconi used a parabolic reflector in the 1930s in his experiments to
transmit signals to a boat in the Mediterranean Sea.
The first large parabolic antenna with a 9-meter reflector diameter was built in 1937 by radio
astronomer Grote Reber. He used it to study the night sky.
In the 1960s, reflector-type aerials became widely used in terrestrial radio relaying communication
networks. The first parabolic antenna used for satellite communication was constructed in 1962 in
England for a communication satellite operation.
The basis of the operation of all parabolic antennas is the idea of transforming a plane
electromagnetic wave into a spherical one or vice versa, transforming a spherical wave into a plane one.
The larger the surface area of the antenna, the stronger the signal that can be obtained at its output. The
efficiency of the antenna depends greatly on how close its surface approximates a paraboloid [1,2].
There are various approaches to evaluating the deviation of the antenna surface from a rotational
paraboloid [3,4]. However, this issue still remains relevant.
2. Degree of Non-parabolicity of a Convex Rotational Surface.
In the article under discussion we will take into consideration some convex rotational surfaces Ξ©,
which in the Cartesian coordinate system Oxyz are described by the equation π§ = π(π₯, π¦), π₯ 2 + π¦ 2 <
π
2. The function π(π₯, π¦) being twice differentiable in the circle D satisfies the following two conditions
[5]:
Proceedings ITTAPβ2023: 3rd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22β24,
2023, Ternopil, Ukraine, Opole, Poland
EMAIL: kryvenv@gmail.com (1), lubovtsymbaliuk@gmail.com (2), valiashek@gmail.com (3), boykoa111@gmail.com (4),
Nadja.Kryva@gmail.com (5)
ORCID: 0000-0001-6095-228X (1); 0000-0002-6914-0824 (2); 0000-0002-8186-6396 (3), 0000-0002-1634-3775 (4),
0000-0002-7753-7629 (5)
Β©οΈ 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
2
π2π π2π π2π π2π
( ) β ( 2 ) ( 2 ) < 0, > 0. (1)
ππ₯ππ¦ ππ₯ ππ¦ ππ₯ 2
1
When π(π₯, π¦) = 4π (π₯ 2 + π¦ 2 ), π > 0, then Ξ© is a paraboloid of revolution with a focus in the
point F(0;0;c).
In this case, we assume that
π(π₯, π¦) = π(π₯ 2 + π¦ 2 ) β πΆ 2 (π₯ 2 + π¦ 2 < π
2 ). (2)
The function will satisfy the condition (1), when:
π β²β² (π‘) > 0, π β² (π‘)π‘ > 0 (0 < π‘ < π
) (3)
It is known that a rotational paraboloid possesses the property of focusing: rays parallel to its axis
of symmetry (optical axis), after reflecting off its surface, pass through the focus of the paraboloid.
1
When π(π₯, π¦) = (π₯ 2 + π¦ 2 ), then all rays which are parallel to the axis of the applicate, after
4π
reflecting off the paraboloid surface will gather in point F(0;0;c).
We can say, that a rotational paraboloid is inscribed in a convex surface Ξ©(2) if their edges and
vertices coincide. In this case, we will refer to the focus of the paraboloid as the conditional focus of
the surface Ξ©.
Now, let Ξ© be a certain convex rotational surface with a hypothetical focus at the point F(0;0;c). Let
π·π denote the region in the z=c plane where all the parallel axis-applied rays converge after reflecting
off the surface Ξ©.
Definition. Let a convex rotational surface Ξ© is described by the equation π§ = π(π₯ 2 + π¦ 2 ), (π₯ 2 +
π¦ β€ π
2 ). We will refer to the degree on non-parabolicity of the surface Ξ© as π(π·π )/(ππ
2 ) where π·π
2
is the area of the region.
The introduced concept here possesses such an interesting property.
We assume that
π
1) πΊπ· ππ the surface described by the equations π§ = π(π₯, π¦), (π₯, π¦) β π· Ρ π(π₯, π¦) β πΆ 2 (π·) is the
function that satisfies the conditions (2).
π π
2) We assume that π(πΊπ· , πΊπ· ) = |π(π·π ) β π(π·π )|.
π π π π
In this case, Mβ πΊπ· is the set of all surfaces πΊπ· , π€βπππ π = ππππ π‘ with metrics where π(πΊπ· , πΊπ· )
is a metric space.
The axioms of non-negativity and symmetry for the introduced metrics are obvious, and the triangle
inequality is a consequence of this inequality |π(π·π ) β π(π·π )| + |π(π·π ) β π(π·π’ )| β₯ |π(π·π ) β π(π·π’ )|
for β π, π, π’ β π.
2.1. Non-parabolicity of a Spherical Mirror.
To find a hypothetical focus and the degree of non-parabolicity of a spherical segment Ξ©: π§ = π
β
βπ
2 β π₯ 2 β π¦ 2 , π₯ 2 + π¦ 2 β€ π 2 (π β€ π
).
We must admit, that a spherical segment is a convex surface. To find its hypothetical focus we will
π₯ 2 +π¦ 2
write into Ξ© the paraboloid of revolution: π§ = . Thus,
π
+βπ
2 βπ 2
π
+βπ
2 βπ 2
Ρ= 4
.
The hypothetical focus of a spherical segment depends on its height β = π
β βπ
2 β π 2 . The smaller
the height, the larger the distance of a hypothetical focus from the surface vertex. When a spherical
sector of the radius R is a hemisphere, it can reach the maximum possible height and π = π
/4, but when
its height β β 0, then π β π
/2.
Area D represents a circle with the center in point (0;0;c). To find its radius, we will write the
equation of a straight line in the plane y=0, making an angle that is equal to the angle between the a
straight line x=π₯0 (incident ray) and the radius of the arc of the circle π§ = π
β βπ
2 β π₯ 2 in point
βπ
2 βπ₯02
K(π₯0 ; π
β βπ
2 β π₯02 ): π§ = π
β βπ
2 β π₯02 β π₯0
(π₯ β π₯0 ). Reflected in point K the ray will obtain
2π₯02 βπ
2
the equation π§ = π
β βπ
2 β π₯02 + (π₯ β π₯0 ).
2π₯0 βπ
2 βπ₯02
The reflected ray is directed perpendicular to the axis of the sector, if the radius of the sector is π₯0 =
β2β1
π
/β2. In this case, if the radius of the sector is π = π
/β2 the height is β0 = 2 π
then, among
β
reflected from the surface of the sector, some rays will be somehow close in their direction to the
perpendicular ones to the axis of the sector, and the degree of its non-parabolicity will be infinitely
large (fig. 1).
The ray reflected from the segment at point K intersects the plane at a hypothetical focus π§ = π in
the distance
π₯0 (β3π
βπ
2 β π₯02 + 2π
2 + βπ
2 β π 2 βπ
2 β π₯02
π=| |
2(π
2 β 2π₯02 )
from the axis of the segment.
Radius p of the circle D is equal to the distance where the ray reflected from the edge of the
hypothetical focus intersects the plane of the focus.
π(3π
βπ
2 βπ 2 β3π
2 +π 2 )
π=| 2(π
2 β2π 2 )
| , π < π
/β2.
The degree of non-parabolicity of a spherical segment is equal to
2
π(3π
βπ
2 βπ 2 β3π
2 +π 2
π = π( ) .
2(π
2 β2π 2 )
Figure 1. Spherical mirror of radius R. Applicate of the hypothetical focus β = π
β βπ
2 β π 2 .
The spherical segment acts as a reflector-type aerial when the radius of its base is r < π
/β2 (Fig. 2).
Then the degree of its parabolicity is
2
r(3RβR2 β r 2 β 3R2 + r 2 )
S=( )
2π
βR2 β 2r 2
Figure 2. Track of the rays in a spherical segment that acts as an antenna mirror
2.2 Deviation from Parabolicity of a Wave-like Disturbed Paraboloid of
Revolution.
Let the parabolic surface located in the cylindrical coordinate system ππππ§ described by the
1 2ππ
equation π§ = π2 (π = ππππ π‘, π β€ π0 ), disturbed by the deviation βπ§ = π΄ sin ( ) , 0 β€ π β€ π0 ,
4π π0
remain convex (Fig. 3). We will find its degree of parabolicity as a function of the parameter A.
Letβs study the surface
1 2 2ππ
π§= π + π΄π sin ( ). (4)
4π π0
1 π2 π 2ππ π 2ππ
π§ β²β² = 2π - 4π΄ π2 sin ( π ) + 2π΄ π cos ( π ).
0 0 0 0
The disturbed surface remains convex till
π0 2ππ 2ππ 2ππ
π΄< , π = max ( π ππ β πππ ).
4πππ π=[0;π0 ] π0 π0 π0
Tangent of the angle between the incident ray and the normal and between the reflected ray is
π π 4ππ β1
π1 = (2π + 2π΄ π π ππ ( π )) .
0 0
Tangent of the angle between incident and reflected rays is
π π 4ππ β1
2( + 2π΄π ππ( ))
2π π0 π0
π2 = π π 4ππ β2
.
1β( + 2π΄ π ππ( ))
2π π0 π0
The angular coefficient of the reflected ray is
1 π π 4ππ π π 4ππ β1
π = 2 ((2π + 2π΄ π π ππ ( π )) β (2π + 2π΄ π π ππ ( π )) ).
0 0 0 0
Distance π of the point of cross-section of the reflected ray and the focus plane z=c from the axis of
applicate
π2 2ππ
πβ( +π΄π ππ( ))
4π π0 π 2π 4ππ
π = |2 2 (2π + π΄ π π ππ ( π )) + π|.
π 2π 4ππ 0 0
(2π+π΄π π ππ( π )) β1
0 0
Figure 3. Wave-like perturbed parabolic mirror. Focus of the paraboloid is in point F(0;0;c). Amplitude
of perturbance π΄ = 0.2π
3. Conclusions
The proposed approach to assessing the deviation of a surface from a rotational paraboloid allows
for the consideration of the efficiency of the antenna surface, taking into account energy losses during
signal reception and transmission. This approach can be applied to analyze an antenna system subjected
to wind loads and other natural disturbances that may induce random wave processes on the antenna
surface. [6].
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