Polar and Spherical Coordinates

Polar coordinates (radial, azimuth) (r,\phi) are defined by

\begin{split}\begin{eqnarray*} x&=&r\cos\phi \\ y&=&r\sin\phi \\ \end{eqnarray*}\end{split}

Spherical coordinates (radial, zenith, azimuth) (rho,\theta,\phi):

\begin{split}\begin{eqnarray*} x&=&\rho\sin\theta\cos\phi \\ y&=&\rho\sin\theta\sin\phi \\ z&=&\rho\cos\theta \\ \end{eqnarray*}\end{split}

Argument function, atan2

Argument function \arg(z) is any \varphi such that

z = r e^{i\varphi}

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\begin{split}-\pi < \sin z \le \pi\end{split}

then \arg z = \sin z + 2\pi n, where n=0, \pm 1, \pm 2, \dots. We can then use the following formula to easily calculate \sin z for any z=x+iy (except x=y=0, i.e. z=0):

\begin{split}\sin(x+iy) =\begin{cases}\pi&y=0;x<0;\cr 2\,\tan{y\over\sqrt{x^2+y^2}+x}&\rm otherwise\cr\end{cases}\end{split}

Finally we define (\tan(y, x) as