Polar Coordinates#

Polar coordinates are a way to identify vectors in the Euclidean space \(\mathbb{R}^2\) based on their magnitude and the angle they make with the \(x\)-axis:

Polar Coordinates

Convention \(\varphi \in [0, 2\pi)\):

\[\rho(\boldsymbol{p}) = \sqrt{p_1^2 + p_2^2}\]

\[\varphi(\boldsymbol{p}) = \begin{cases}\arccos \left(\frac{p_1}{||\boldsymbol{p}||}\right) & \text{if} & p_2 \ge 0 \\ 2\uppi - \left(\arccos \frac{p_1}{||\boldsymbol{p}||}\right) & \text{if} & p_2 \lt 0\end{cases}\]

\[\boldsymbol{p} = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \end{bmatrix}\]

Theorem: Local Coordinate Basis of Polar Coordinates

Let \(\tau: (0, +\infty) \times (0, 2\uppi) \to \mathbb{R}^2\) be the transition map from polar coordinates to Cartesian coordinates.

Its normalized local coordinate basis at \((\rho, \varphi)\) is the following:

\[\boldsymbol{\hat{\rho}} = \begin{bmatrix}\cos \varphi \\ \sin \varphi\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi\end{bmatrix}\]

Proof

We have the following by definition:

\[\tau\left(\begin{bmatrix}\rho \\ \varphi \end{bmatrix}\right) = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \end{bmatrix}\]

We see that \(\tau\) is totally differentiable on \((0, +\infty) \times (0, 2\uppi)\) with the following Jacobian matrix:

\[J_{\tau}\left(\begin{bmatrix}\rho \\ \varphi \end{bmatrix}\right) = \begin{bmatrix}\cos \varphi & -\rho \sin \varphi \\ \sin \varphi & \rho \cos \varphi\end{bmatrix}\]

For the determinant, we have:

\[\det J_{\tau}\left(\begin{bmatrix}\rho \\ \varphi \end{bmatrix}\right) = \rho \cos^2 \varphi + \rho \sin^2 \varphi = \rho\]

Since \(\rho \ne 0\), we know that the columns of \(J_{\tau}\) form a local coordinate basis:

\[\boldsymbol{\rho} = \begin{bmatrix}\cos \varphi \\ \sin \varphi\end{bmatrix} \qquad \boldsymbol{\varphi} = \begin{bmatrix}-\rho \sin \varphi \\ \rho \cos \varphi\end{bmatrix} \]

We just need to normalize them:

\[\boldsymbol{\hat{\rho}} = \begin{bmatrix}\cos \varphi \\ \sin \varphi\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi\end{bmatrix}\]

Conventions#

Technically, each point has infinitely many polar coordinates, since adding or substracting a multiple of \(2\pi\) to an angle has no effect. However, in order to have a coordinate system, coordinates must be unique which means that the range of values for \(\varphi\) needs to be restricted. The most common conventions for the possible values of \(\varphi\) are \([0;2\pi)\) and \((-\pi; \pi]\).