# Spherical coordinates

There are good discussions of spherical coordinates in Wikipedia spherical coordinate system and Mathworld spherical coordinate system.

There is more information in the docstring for the `sphere2cart()` function.

## Terms

Origin

  Origin of the sphere

P

  The point represented by spherical coordinates

OP

  The line connecting the origin and P

radial distance

  or radius. The Euclidean length of OP.

z-axis

  The vertical of the sphere. If we consider the sphere as a globe, then the z-axis runs from south to north. This is the zenith direction of the sphere.

Reference plane

  The plane containing the origin and orthogonal to the z-axis (zenith direction)

y-axis

  The horizontal axis of the sphere, orthogonal to the z-axis, on the reference plane. West to east for a globe.

x-axis

  Axis orthogonal to y and z-axis, on the reference plane. For a globe, this will be a line from behind the globe through the origin towards us, the viewer.

Inclination angle

  The angle between the OP and the z-axis. This can also be called the polar angle, or the co-latitude.

Azimuth angle

  or azimuthal angle or longitude. The angle between the projection of OP onto the reference plane and the x-axis

## The physics convention

The radius is \(r\), the inclination angle is \(\theta\) and the azimuth angle is \(\phi\). Spherical coordinates are specified by the tuple of \((r, \theta, \phi)\) in that order.

Here is a good illustration we made from the scripts kindly provided by Jorge Stolfi on Wikipedia.

The formulae relating Cartesian coordinates \((x, y, z)\) to \(r, \theta, \phi\) are:

\[ \begin{align}\begin{aligned}r=\sqrt{x^2+y^2+z^2}\\\theta=\arccos\frac{z}{\sqrt{x^2+y^2+z^2}}\\\phi = \operatorname{atan2}(y,x)\end{aligned}\end{align} \]

and from \((r, \theta, \phi)\) to \((x, y, z)\):

\[ \begin{align}\begin{aligned}x=r \, \sin\theta \, \cos\phi\\y=r \, \sin\theta \, \sin\phi\\z=r \, \cos\theta\end{aligned}\end{align} \]

## The mathematics convention

See the Wikipedia spherical coordinate system. The mathematics convention reverses the meaning of \(\theta\) and \(\phi\) so that \(\theta\) refers to the azimuthal angle and \(\phi\) refers to the inclination angle.

## Matlab convention

Matlab has functions `sph2cart` and `cart2sph`. These use the terms `theta` and `phi`, but with a different meaning again from the standard physics and mathematics conventions. Here `theta` is the azimuth angle, as for the mathematics convention, but `phi` is the angle between the reference plane and OP. This implies different formulae for the conversions between Cartesian and spherical coordinates that are easy to derive.