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.
Spherical Coordinates ConversionsBack to Top
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,
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,
z = z,
Relation between Cartesian and Spherical coordinates–
x = ρ sin φ cos θ,
y = ρ sin φ sin θ,
z = ρ cos φ,
ρ2 = x2 + y2 + z2,
To translate points from Cartesian or Cylindrical coordinate system these conversion formulas are used.
Spherical Coordinates Volume ElementBack to Top
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|),
θ^ = (dr / dθ) / (|dr / dθ|)
= cos θ
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 CoordinatesBack to Top
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.