How to Change from Cartesian Coordinates to Cylindrical Coordinates?

In Cartesian coordinate system we used to represent the coordinates as the measure of lengths on the respective axis. This makes it necessary for us to differentiate between the Cartesian and the Cylindrical coordinates because the measure of the coordinates in case of cylindrical coordinates is not the same for as units of length. The cylindrical coordinates are written as (R, α, A) whereas the Cartesian coordinates are written as (x, y, z). In Cartesian system 'x' represents the length of the coordinate on x – axis, 'y' represents the length of the coordinate on y – axis and 'z' represents the length of the coordinate on z – axis. In cylindrical coordinate system R and A represent the measures of length and can be written as x, y and z of Cartesian coordinates whereas the third measure 'α' represents the angle whose measure can be evaluated in either degrees or radians. “α” does not represent length. Formulae we use to know how to change from cartesian coordinates to cylindrical coordinates as follows:

R = √ (x² + y²),

α = sin^-1 (y /R) if x ≥ 0,

And α = pi - sin^-1 (y /R) if x < 0,

And A = z,

For converting the cylindrical coordinates back to Cartesian coordinates we use the following formulae:

x = R cos α,

y = R sinα,

And z = A,

Let us consider an example: Suppose we have a Point in Cartesian system as (2, 2, 2), then find its cylindrical form:
Solution:

R = √ (x² + y²) = √ (2² + 2²) = 2 √2

α = sin^-1 (y /R) = sin-1 (2 /2 √2) = sin-1 (1 /√2) = 45⁰ or pi /4,

And A = z = 2,

Therefore, corresponding cylindrical coordinates are: (2√2, 45⁰, 2).