1. Spherical Coordinates
A point can be written in terms of two angles and the distance
1. ρ the distance from the origin to the point
Point φ
2. φ from the angle from the positive z axis down to the point ρ
3. θ the angle from the positive x axis to the projection onto xy plane. θ

2. Converting from Rectangular to Spherical Coordinates
red line =
green line
ρ = z
Final Point φ ρ z x
tan(φ)= θ y
tan(θ)=

3. Converting from Spherical to Rectangular Coordinates
Notice that: z
Final Point
cos(φ) = φ ρ z
sin(φ )= x θ y

4. Converting from Cylindrical to
Spherical Coordinates
Length of the
red line = r z
Final Point φ ρ
length of the green line
ρ = z r
tan(φ)= r θ
θ = θ
The θ from Cylindrical Coordinates is the
same θ in Spherical Coordinates

5. Converting from Spherical to Cylindrical
red line = r z
Final Point
cos(φ) = φ ρ z
sin(φ )= r r θ
θ = θ
The θ from Spherical Coordinates is the
same θ in Cylindrical Coordinates