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 = ρ cos φ
y = ρ sin φ
z = z'

Conversion from spherical to cartesian coordinates:

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

x = r sin θ cos φ
y = r sin θ sin φ
z = r cos 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 = r sin θ
φ' = φ
z' = r cos 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