Horizon coordinates:

We correct a typographical error on the second line of Eq. 5.45 of [4] (it should have $\cos A$, not $\sin A$). We also note that while our latitudinal coordinate is just the altitude $a$ in this system, our longitudinal coordinate increases counterclockwise, and thus corresponds to the negative of the azimuth $A$ as defined by [4]. So we have:

$$a = \arcsin(\sin\delta\sin\phi + \cos\delta\cos\phi\cos h) \; ,$$ (29.44)

$$-A = \arctan\!2(\cos\delta\sin h, \sin\delta\cos\phi - \cos\delta\sin\phi\cos h) \; ,$$ (29.45)

where $\delta$ is the declination (geographic latitude) of the direction being transformed, $\phi$ is the geographic latitude of the observer's zenith (i.e. the observer's geodetic latitude), and $h$ is the hour angle of the direction being transformed. This is defined as:

$$h = \lambda_\mathrm{zenith} - \lambda$$

$$= \mathrm{LMST} - \alpha$$

where LMST is the local mean sidereal time at the point of observation. The inverse transformation is:

$$\delta = \arcsin(\sin a\sin\phi + \cos a\cos A\cos\phi) \; ,$$ (29.46)

$$h = \arctan\!2[\cos a\sin(-A), \sin a\cos\phi - \cos a\cos A\sin\phi] \; .$$ (29.47)

As explained in CelestialCoordinates.c, the function $\arctan\!2(y,x)$ returns the argument of the complex number $x+iy$.

r0.35

$\parbox{0.3\textwidth}{\caption{The difference between geodetic and geocentric latitude.}}$