Rectangular to Spherical Coordinates

Coordinate system is a system which involves one or more than one number called Ordered Pair to define a Point located anywhere on a plane. Three parameters are used in any coordinate system to define a point in Coordinate Plane. The spherical coordinate system uses Spherical Coordinates to explain the Position of a point on coordinate plane. Another type of coordinate system is Rectangular Coordinate System which is also known as Cartesian coordinate system.
Three parameters in spherical coordinate system are ρ, θ, and φ. These are called spherical coordinates and three Cartesian coordinates which are used in rectangular coordinate system are x, y, and z coordinates.
In spherical coordinate system, ρ, θ, and φ coordinates are called radial distance, polar angle and azimuth angle respectively.
Let’s see the relationship between rectangular and spherical coordinates:
'x' component of rectangular coordinates in the form of spherical coordinates is given as:
X = ρ sin (θ) cos (φ),
'y' component may be given as:
Y = ρ sin (θ) sin (φ),
And 'z' component will be derived from
Z = ρ cos (θ),
To convert spherical coordinates into rectangular coordinates, we use Pythagorean Theorem:
ρ = √x² + y² + z²,
θ = cos-1 (z/ ρ) = cos-1 (z/ √x2+ y2 + z2) and
φ = tan-1 (y / x).
Let’s try to understand conversion from Rectangular to Spherical Coordinates using following example.
Let rectangular coordinates (√2, 1, 0) be represent by (x, y, z).
Then we can write (x, y, z) = (1, 1,0),
ρ = √x² + y² + z² = 1 + 1 + 0 = 2,
θ = cos^-1 (z/ ρ) = cos^-1 (0 / 2) = cos^-1(0) = π/2,
and
φ = tan^-1 (y / x) = tan^-1 (1 / 1) = π/4.