Changing Coordinate Systems: Rectangular and Spherical
Rectangular Coordinates
(Cartesian Coordinates)
| Spherical Coordinates
|
 |
 |
Consider the following triangles:
Rectangular Coordinates
(Cartesian Coordinates)
| Spherical Coordinates
|
 |
 |
Comparing these we see that
- ρ = sqrt(x2 + y2 + z2)
- φ = cos-1(z/sqrt(x2 + y2+z2))
- z = ρ cos φ
Note that we cannot use the inverse tangent function to find φ because φ lies in the interval [0,2π] and the range of tan-1 is (-π,π).
Also consider the following triangles that lie on the xy plane:
Rectangular Coordinates
(Cartesian Coordinates)
Spherical Coordinates
| |
 |
 |
Comparing these we see that
- x = ρ sin φ cos θ
- y = ρ sin φ sin θ
- θ = cos-1(x/sqrt(x2+y2)) for y>0 and θ = 2π-cos-1(x/sqrt(x2+y2)) otherwise.
Back to Describing Surfaces Using Different Coordinate Systems.