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.
2. Degree of Non-parabolicity of a Convex Rotational Surface. 4 ( point F(0;0;c).
In this case, we assume that The function will satisfy the condition (1), when: ( , ) = ( 2 + 2) ∈ 2( 2 + 2 < 2).
′′( ) > 0, ′( ) > 0 (0 < < ) When ( , ) = 1 ( 2 + 2), > 0, then Ω is a paraboloid of revolution with a focus in the (1) (2) (3) the surface Ω. off the surface Ω. is the area of the region.
We assume that
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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. When ( , ) =
1 ( 2 + 2), then all rays which are parallel to the axis of the applicate, after 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 Ω(2) if their edges and vertices coincide. In this case, we will refer to the focus of the paraboloid as the conditional focus of
Now, let Ω be a certain convex rotational surface with a hypothetical focus at the point F(0;0;c). Let denote the region in the z=c plane where all the parallel axis-applied rays converge after reflecting
Definition. Let a convex rotational surface Ω is described by the equation = ( 2 + 2), ( 2 + 2 ≤ 2). We will refer to the degree on non-parabolicity of the surface Ω as ( )/( 2) where The introduced concept here possesses such an interesting property.
1) the surface described by the equations = ( , ), ( , ) ∈ і ( , ) ∈ 2( ) is the function that satisfies the conditions (2).
2) We assume that ( , ) = | ( )− ( )|.
In this case, M∋ is the set of all surfaces , ℎ = with metrics where ( , ) is a metric space. for ∀ , , ∈ .
The axioms of non-negativity and symmetry for the introduced metrics are obvious, and the triangle inequality is a consequence of this inequality | ( )− ( )| + | ( )− ( )| ≥ | ( )− ( )| 2.1. Non-parabolicity of a Spherical Mirror. √ 2 − 2 − 2, 2 + 2 ≤ 2 ( ≤ ).
To find a hypothetical focus and the degree of non-parabolicity of a spherical segment Ω: = − We must admit, that a spherical segment is a convex surface. To find its hypothetical focus we will write into Ω the paraboloid of revolution: =
2+ 2 +√ 2− 2 . Thus, с = +√ 2− 2.
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The hypothetical focus of a spherical segment depends on its height ℎ = − √ 2 − 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 = /4, but when its height ℎ → 0, then → /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= 0 (incident ray) and the radius of the arc of the circle = − √ 2 − 2 in point K( 0; − √ 2 − 02): = − √ 2 − 02 − the equation = − √ 2 − 02 +
2 02− 2 2 0√ 2− 02 √ 2− 02
0 ( − 0).
The reflected ray is directed perpendicular to the axis of the sector, if the radius of the sector is 0 = /√2. In this case, if the radius of the sector is = /√2 the height is ℎ0 = √2−1 then, among √2 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 = in the distance ( − 0). Reflected in point K the ray will obtain = | 0(−3 √ 2 − 02 + 2 2 + √ 2 − 2√ 2 − 02 | 2( 2 − 2 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.
The degree of non-parabolicity of a spherical segment is equal to = | (3 √ 2− 2−3 2+ 2)| , < /√2.
2( 2−2 2) = ( (3 √ 2− 2−3 2+ 2 2
2( 2−2 2) ) . , ≤ 0), disturbed by the deviation ∆ = sin ( ), 0 ≤ ≤ 0, 2 0 The disturbed surface remains convex till = 1 2 + 4 sin ( 2 0 ).
(4) ′′ = 1 2 - 4 2 02 sin ( 2 0 ) + 2 cos (2 0 0
). < 4 0 , =
max =[0; 0] 0
2 ( ).
Tangent of the angle between the incident ray and the normal and between the reflected ray is Tangent of the angle between incident and reflected rays is 1 = ( 2 + 2 0
4 ( 0
−1 )) . 2 = 2(2 1−(2 + 2 0 + 2 0 ( 4 0 )) ( 4 0 )) −1 −2.
The angular coefficient of the reflected ray is = 1 2 (( 2 + 2 0
4 ( 0 )) − ( 2 + 2 0
4 ( 0
−1 )) ).
Distance of the point of cross-section of the reflected ray and the focus plane z=c from the axis of applicate of perturbance = 0.2
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].
Kyiv : OJSC «Ukrtelekom», 2003, p.496