# 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.

NameCoordsBasisProsCons
Cartesian $x,y,z$ $\hat\imath,\hat\jmath,\hat{k}$
• No singularities or discontinuities
• Fixed for all time
• Easy differentiation
• Independent of origin choice
• No insight into dynamics
Cylindrical
or polar
$R,\theta,z$
or $r,\theta$
$\hat{e}_R,\hat{e}_\theta,\hat{e}_z$
or $\hat{e}_r,\hat{e}_\theta$
• Ideal for circular and helical movement
• Gives insight into centripetal and coriolis acceleration
• Good if there is a clear motion center with 2D (single-axis) rotation
• Singularity when $r = 0$
• Dependent on origin choice
• Complex differentiation
Spherical $r,\theta,\phi$ $\hat{e}_r,\hat{e}_\theta,\hat{e}_\phi$
• Ideal for rotations in 3D
• Good if there is a clear motion center with 3D (multi-axis) rotation
• Singularity when $r = 0$, $\phi = 0$, or $\phi = \pi$
• Dependent on origin choice
• Complex differentiation
Tangential/
normal
$\hat{e}_t,\hat{e}_n,\hat{e}_b$
• Decomposes acceleration into speed and direction changes
• Allows computation of radius of curvature
• Independent of origin choice and orientation
• Discontinuous when curve direction changes
• Not uniquely defined when $\vec{a}$ is parallel to $\vec{v}$
• Complex differentiation

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.

## 3D Motion

 Movement: saddle Viviani eight clover Lissajous deltoid pentagram Show: Coords: none Cartesian cylindrical spherical Basis: none Cartesian cylindrical spherical tang./norm. Origin: $O_1$ $O_2$ Components:

Comparison of coordinate systems and bases in 3D.