Celestial coordinate system

Definition

Let $S$ be the celestial sphere of an observer. A celestial coordinate system on $S$ is given by three data:

1. a fundamental plane through the centre of $S$, with a chosen pole $P$ on the normal line;
2. a primary direction $X$, a point of the great circle cut out by the fundamental plane;
3. a sense of rotation along that great circle, the sense of increasing longitude.

Take the orthonormal frame with $e_1$ pointing to $X$ and $e_3$ pointing to $P$. The vector $e_2$ lies on the great circle, a quarter turn from $X$ in the chosen sense. Each point $u \in S$ other than $\pm P$ is written uniquely as $$u = (\cos b \cos l)\, e_1 + (\cos b \sin l)\, e_2 + (\sin b)\, e_3,$$ with latitude $b \in (-\pi/2, \pi/2)$ and longitude $l \in [0, 2\pi)$. The poles $\pm P$ have latitude $\pm\pi/2$, and their longitude is undefined.