Spherical Coordinates#

Each vector (point) \(P\) in the 3-dimensional Euclidean space \(\mathbb{R}^3\) can be uniquely identified using its magnitude \(r\) (distance from the origin), the angle \(\theta\) it makes with the \(z\)-axis and the angle \(\varphi\) its projection in the \(xy\) plane makes with the \(x\)-axis.

Definition: Radial Distance (Radius)

The number \(r\) is known as radial distance or radius.

Definition: Inclination (Polar Angle)

The number \(\theta\) is known as the inclination or polar angle.

Definition: Azimuth (Azimuthal Angle)

The number \(\varphi\) is known as the azimuth or azimuthal angle.

Theorem: Local Coordinate Basis of Spherical Coordinates

Let \(\tau: (0, +\infty) \times (0, \uppi) \times (0, 2\uppi) \to \mathbb{R}^3\) be the transition map from spherical coordinates to Cartesian coordinates.

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

\[\boldsymbol{\hat{r}} = \begin{bmatrix}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta\end{bmatrix} \qquad \boldsymbol{\hat{\theta}} = \begin{bmatrix}\cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ -\sin \theta\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi \\ 0\end{bmatrix}\]

Proof

We have the following by definition:

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

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

\[J_{\tau}\left(\begin{bmatrix}r \\ \theta \\ \varphi \end{bmatrix}\right) = \begin{bmatrix}\sin \theta \cos \varphi & r \cos \theta \cos \varphi & -r \sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r \cos \theta \sin \varphi & r \sin \theta \cos \varphi \\ \cos \theta & -r \sin \theta & 0\end{bmatrix}\]

For the determinant, we have:

\[\det J_{\tau}\left(\begin{bmatrix}r \\ \theta \\ \varphi \end{bmatrix}\right) = r^2 \sin \theta \cos^2 \theta + r^2 \sin^3 \theta = r^2 \sin \theta\]

Since \(r^2 \sin \theta \ne 0\), we know that the columns of \(J_{\tau}\) form a local coordinate basis:

\[\boldsymbol{r} = \begin{bmatrix}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta\end{bmatrix} \qquad \boldsymbol{\theta} = \begin{bmatrix}r \cos \theta \cos \varphi \\ r \cos \theta \sin \varphi \\ -r \sin \theta\end{bmatrix} \qquad \boldsymbol{\varphi} = \begin{bmatrix}-r \sin \theta \sin \varphi \\ r \sin \theta \cos \varphi \\ 0\end{bmatrix}\]

We just need to normalize them:

\[\boldsymbol{\hat{r}} = \begin{bmatrix}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta\end{bmatrix} \qquad \boldsymbol{\hat{\theta}} = \begin{bmatrix}\cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ -\sin \theta\end{bmatrix} \qquad \boldsymbol{\hat{\varphi}} = \begin{bmatrix}-\sin \varphi \\ \cos \varphi \\ 0\end{bmatrix}\]

Conventions#

The radial distance can be denoted either by \(r\) or \(\rho\). Some people also switch \(\theta\) and \(\varphi\) around, using the former for the azimuthal angle and the latter for the polar angle.

Note: Elevation

Instead of inclination, some people prefer to use elevation. This is the angle between the point and the \(xy\)-plane and is equal to \(\frac{\pi}{2}\) minus the inclination.

If the range of values for the angles is not restricted, then every point has infinitely many different spherical coordinates because adding or subtracting an integer multiple of \(2\pi\) to an angle does not change the point it corresponds to. However, in order to have a coordinate system, coordinates must be unique. To guarantee this, the set of possible values for the angles needs to be restricted. Two of the most common conventions are \(\theta \in [0;\pi]\), \(\varphi \in [0; 2\pi)\) and \(\theta \in [0;\pi]\), \(\varphi \in (-\pi; \pi]\).