Rectangular, Cylindrical, Sperical coordinates

Rectangular

(x,y,z)

Cylindrical

(r,θ,z)

Spherical

(ϕ,θ,ρ)

Rectangular → Cylindrical

r = sqrt(x² + y²)
θ = arctan(y/x)

z = z

Rectangular → Spherical

ρ = sqrt(x² + y² + z²)

θ = arctan(y/x)

ϕ = arccos(z sqrt(x² + y² + z²))

Cylindrical → Rectangular

x = r cos(θ)

y = r sin(θ)

z = z

Cylindrical → Spherical

ρ = sqrt(r² + z²)

θ = θ

ϕ = arctan(r/z)

Spherical → Rectangular

x = ρ sin(ϕ) cos(θ)

y = ρ sin(ϕ) sin(θ)

z = ρ cos(ϕ)

Spherical → Cylindrical

r = ρ sin(ϕ)

θ = θ

z = ρ cos(ϕ)