Coordinate system comparison
The different coordinate systems and bases have different strengths and weaknesses, and no single coordinate system or basis is always best. Depending on the application, several different coordinate systems or bases may be used simultaneously for different purposes.
| Name | Coords | Basis | Pros | Cons |
|---|---|---|---|---|
| Cartesian | $x,y,z$ | $\hat\imath,\hat\jmath,\hat{k}$ |
|
|
| Cylindrical or polar |
$R,\theta,z$ or $r,\theta$ |
$\hat{e}_R,\hat{e}_\theta,\hat{e}_z$ or $\hat{e}_r,\hat{e}_\theta$ |
|
|
| Spherical | $r,\theta,\phi$ | $\hat{e}_r,\hat{e}_\theta,\hat{e}_\phi$ |
|
|
| Tangential/ normal |
— | $\hat{e}_t,\hat{e}_n,\hat{e}_b$ |
|
|
Warning: Coordinate systems and bases are different.
Some coordinate systems (e.g., cylindrical, spherical) also have corresponding bases. The tangential/normal basis does not have any associated coordinate system, however. We can mix and match coordinate systems and basis. For example, we may track a point's location in polar coordinates $(r,\theta)$, but express its velocity and acceleration in a tangential/normal basis $\hat{e}_t,\hat{e}_n$.
2D Motion
| Movement: | circle | var-circle | ellipse | arc |
| trefoil | eight | comet | pendulum | |
| Show: | ||||
| Coords: | none | Cartesian | polar | |
| Basis: | none | Cartesian | polar | tangential/normal |
| Origin: | $O_1$ | $O_2$ | ||
| Components: | ||||
Comparison of coordinate systems and bases in 2D.