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Spherical Coordinates

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In three dimensional spaces Spherical Coordinate is used to find the location of a Point that depends on the distance ρ which is measured from the origin and two angles θ and ϕ. When we are going to define a spherical coordinate system then we have to select two orthogonal directions that is the Zenith angle and the point lies on the space. The zenith angles are those angles which are measured from horizontal to the perpendicular. Now we will see the spherical coordinate system.
A point ‘S’ in the spherical coordinates is defined as follows:
In the plane the radius or radial distance denotes the Euclidean distance that is measured from the origin O to S.
The angle lies between the zenith direction and the line segment OS is the inclination or it is also said to be polar angle.
The signed angle that is measured from the azimuth reference direction to the orthogonal projection of the Line Segment OS on the reference plane is the azimuth or it is also known as azimuthal angle.
If we subtract the 90 degrees to the inclination then we get the elevation angle.
The azimuth angle is arbitrary, if the inclination is zero or 180 degrees. If the value of radius is zero then both azimuth and inclination are arbitrary.
In linear Algebra, the vector from the origin O to the point S is said to be the Position vector of S.
Additionally it is necessary to find the unique Set of spherical for each point and its range is also restricted. Some conditions is also given for these angles which are:
The value of r is greater than zero,
= 00 < Ѳ < 1800; this is the standard convention of spherical coordinates.

Spherical Coordinates Conversions

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There are three types of coordinate systems, first one is Cartesian coordinate system used for two dimensional plane, second one is Cylindrical coordinate system used to find the Position of cylindrical plane and third one is Spherical Coordinate system.
Spherical coordinate system is used to calculate the position of three dimensional objects. Three coordinates are there in Spherical coordinate system–
First one is 'ρ' (Pronounced as rho) is the distance from origin to the Point we are calculating and this must be positive i.e. ρ >= 0.
Second one is 'θ' (Pronounced as theta) is the angle between x- axis and line 'r'. This measures the angle from positive x - z plane to the point. It can vary from 0 to 3600.
Last one is 'φ' (Pronounced as phi) is the angle that is needed to rotate down from positive z- axis to get the point and thus value of 'φ' varies from 0 to 1800 only.
We can perform Spherical Coordinates conversion into Cartesian and Cylindrical form:
If we need conversion of spherical coordinates (ρ, θ, φ) into Cylinder coordinates (r, θ, z) then we need to find 'r' and 'z' as 'θ' will be same in both coordinate systems.
By drawing a simple sketch of both coordinate systems one can get value of 'z' and 'r' in terms of 'ρ' and 'φ' as shown below –
r = ρ cos φ,
'θ' will be same in both coordinate systems.
z = ρ cos φ,
Or
ρ2 = r2 + z2,
For spherical coordinate conversion into Cartesian coordinates (x, y, z) we need to follow given formulas–
Relation between Cartesian and Cylindrical coordinates-
x = r cos θ,
y = r sin θ,
And
z = z,
Relation between Cartesian and Spherical coordinates–
x = ρ sin φ cos θ,
y = ρ sin φ sin θ,
z = ρ cos φ,
And
ρ2 = x2 + y2 + z2,
To translate points from Cartesian or Cylindrical coordinate system these conversion formulas are used.

Spherical Coordinates Volume Element

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Coordinate system can be defined as a system which may use one or more than one number or it may also use one or more than one coordinates. One of the type of it is spherical coordinate system which is defined as that system in which Position of a Point located anywhere in coordinate system can be explained by three quantities.
A) Radial distance – Distance of measuring point from origin.
B) Polar angle is the angle which is measured from direction which is known as zenith direction. Zenith is a fixed direction.
C) Azimuth angle- This is the angle which usually passes through origin of spherical coordinate system.
ρ, θ, and φ are Spherical Coordinates used in spherical coordinate system where 'ρ' is radial coordinate, 'θ' is the polar angle, 'φ' is the azimuthal angle.
When a function is integrated with respect to volume in spherical coordinate system then it is referred to Spherical Coordinates volume element. A volume element can be defined as:
dU = ρ (p1 , p2, p3) dp1 dp2 dp3,
Where p1, p2,p3….. are coordinates. For any Set A volume is calculated as:
Volume (A)=∫ ρ (p1 , p2, p3) dp1 dp2 dp3,
Spherical Coordinates Unit Vectors are unit vectors which are represented by r^, θ^ and φ^ and formulated as follows.
ρ^ = (dr / dr) / (|dr / dr|),

[cosθsinφ
= sinθsinφ
cosφ]

θ^ = (dr / dθ) / (|dr / dθ|)

[-sin θ
= cos θ
0]

φ^ = (dr / dφ) / (|dr / dφ|),


[ cos θ cosφ
= sin θ cosφ
- sin φ ]


Spherical Coordinates Triple Integral is used to convert an integral of Cartesian coordinates into spherical coordinates. Following steps are used for this purpose:
(a)The limits must be represented in suitable form.
(b)The integrands must be represented in the suitable form and,
(c) Finally multiply with correct volume element.

Fourier Transform Spherical Coordinates

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Fourier transform is one of the applications which is used approximately in all engineering and physics fundamentals. Great scientist Joseph Fourier gave the concept of Fourier Transform.
Basically Fourier transform is used to show a relation between two Functions which are time and frequency. Fourier transform uses frequency spectrum.
Theorem called Fourier integral theorem describes the relationship between time and frequency.
In Fourier transform function time is represented in form of frequency spectrum, and it is called as frequency Domain representation and if frequency spectrum is represented in form of function time then it is called time domain representation which comes under inverse Fourier transform.
Both transform operation and complex valued function which are generated by it, are involved in Fourier transform.
Fourier transform can be used to transform Cartesian coordinates, spherical coordinates, cylindrical coordinates, polar coordinates and etc.
Fourier Transform Spherical Coordinates can be explained as follows:
Let’s find out the solution of 1/ (|y|2 + 1) where y ε R3,
We can write integral over - ∞ to ∞:∫∫∫ 1/ ( |y|2 + 1) exp(−2πi (y⋅ξ) dy1 dy2 dy3.
It is clear that this functi0n is radial so that in spherical coordinates,
f(ρ, θ, ϕ) = r2 + 1 = f(ρ).
It will simple take the limits of Integration 0 to ∞, 0 to 2π and 0 to π and also
dy1 dy2 dy3 → ρ2 sinθ dθ dϕ dρ.

Dot product should be of form ||y|| ||ξ|| times some trigonometric function of angle between them, and ||y|| is just 'ρ'.
Complete solution of above problem can be obtained using Fourier Transform Spherical Coordinates.