The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Conversion between 3D Coordinate Systems

Conversion from cartesian to cylinrical coordinates:

Cartesian [x, y, z] Cylindrical [ρ, φ, z']

φ = arctan(y/x) for x > 0
φ = π/2 for x = 0 and y > 0
φ = π + arctan(y/x)  for x < 0
φ = 3π/2 for x = 0 and y < 0
z' = z  

Conversion from cylindrical to cartesian coordinates:

Cylindrical [ρ, φ, z'] Cartesian [x, y, z]

x = ρcos φ
y = ρsin φ
z = z'

Conversion from spherical to cartesian coordinates:

Spherical [r, θ, φ] Cartesian [x, y, z]

x = rsin θcos φ
y = rsin θsin φ
z = rcos θ

Conversion from cartesian to spherical coordinates:

Cartesian [x, y, z] Spherical [r, θ, φ]

Conversion from spherical to cylindrical coordinates:

Spherical [r, θ, φ] Cylindrical [ρ, φ', z']

ρ = rsin θ
φ' = φ
z' = rcos θ

Conversion from cylindrical to spherical coordinates:

Cylindrical [ρ, φ, z] Spherical [r, θ, φ']

θ = arctan(ρ/z)
φ' = φ

Last Update: 2011-01-11