Spherical and Cylindrical Coordinates

Spherical Coordinates

Conversion of Cartesian coordinates to polar coordinates can be expressed as

r = (x² + y² + z²)^1/2                          (1)

tan(Φ) = y / x                        (1a)

cos(Φ) = z / r

           = z / (x² + y² + z²)^1/2          (1b)

Conversion of polar coordinates to Cartesian coordinates can be expressed as

x = r cos(Φ) sin(θ)                     (2)

y = r sin(Φ) sin(θ)                      (2a)

z = r cos(θ)                             (2b)

Cylindrical Coordinates

Conversion of Cartesian coordinates to cylindrical coordinates for a point P can be expressed as

ρ = (x² + y²)^1/2                    (3)

tan(Φ) = y / x                     (3a)

z = z                      (3b)

Conversion of cylindrical coordinates to Cartesian coordinates can be expressed as

x = ρ cos(Φ)                        (4)

y = ρ sin(Φ)                        (4b)

z = z                        (4c)