= Horizontal coordinate system
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A <horizontal coordinate system> describes a viewing direction using <altitude> $\gamma$ above the observer's <astronomical horizon> and <azimuth> $\eta$ around it. Here <azimuth> starts at true north and increases through east: north is $0^\circ$, east $90^\circ$, south $180^\circ$, and west $270^\circ$, with $0\leq\eta<360^\circ$. True north is the northward intersection of the <celestial meridian> with the <astronomical horizon>, rather than an uncorrected magnetic-compass reading. Other origins and directions are legitimate, but their formulae must be converted consistently. https://www.astro.uvic.ca/~tatum/celmechs/celm6.pdf[Jeremy Tatum: Celestial mechanics, section 6.2, altazimuth coordinates and azimuth conventions]{ref}.
Construct a local <orthonormal basis> of east $\mathbf e_E$, north $\mathbf e_N$, and upward $\mathbf e_U$, with $\mathbf e_E\times\mathbf e_N=\mathbf e_U$. The upward <unit vector> points to the <zenith>; the horizontal <plane> is perpendicular to it. For a ground instrument, a freely hanging weight establishes the local vertical through <gravity>. In an elementary spherical-<Earth> model, this vertical is radial; precision work distinguishes it from the normal to a reference surface. With the observer as origin, write a direction's <unit vector> as
$$
\mathbf n=\cos\gamma\sin\eta\,\mathbf e_E+\cos\gamma\cos\eta\,\mathbf e_N+\sin\gamma\,\mathbf e_U.
$$
Its local components $(E,N,U)$ obey $E^2+N^2+U^2=1$. Conversely, $\gamma=\arcsin U$ and $\eta=\operatorname{atan2}(E,N)$ modulo one turn, except where $E=N=0$. The quadrant-aware <arctangent> takes east as its first argument and north as its second. A synthetic direction with $\eta=60^\circ$ and $\gamma=30^\circ$ has $(E,N,U)=(3/4,\sqrt3/4,1/2)$. Thus its horizontal component has <norm> $\sqrt3/2$; its <altitude> is not the <angle> of the projected arrow in the horizontal <plane>.
\Image[/horizontal-sky-coordinates-local-frame.png]
{title=Analytical illustration of north-through-east azimuth and altitude in a local orthogonal frame; synthetic direction}
A <horizontal coordinate system> depends on the observer's site and the observation <time>. The <Earth>'s <rotation> changes the local axes relative to a fixed distant direction, while changing the site changes the <zenith> and the <astronomical horizon>. This makes the <horizontal coordinate system> directly useful for pointing an <optical telescope> and checking whether terrain blocks a target. A pair of <angles> without a site, <time>, <azimuth> convention, and geometric or refracted interpretation is an incomplete pointing instruction.
For a check against an <equatorial coordinate system>, let $\theta$ be the observer's <geographical latitude>, $\delta$ the direction's <declination>, and $h$ its west-positive <hour angle>, obtained from local <sidereal time> minus <right ascension>. In the spherical-<Earth>, unrefracted fixed-direction model, projection onto the local <orthonormal basis> gives
$$
E=-\cos\delta\sin h,\qquad N=\cos\theta\sin\delta-\sin\theta\cos\delta\cos h,\qquad U=\sin\theta\sin\delta+\cos\theta\cos\delta\cos h.
$$
Taking the <dot product> with the upward <unit vector> gives the last formula; the other two follow by the same projection. Recover <altitude> and <azimuth> from $(E,N,U)$ as above, using <radians> inside software <trigonometric functions>. The minus sign in $E$ is essential: a negative <hour angle> puts this synthetic direction east of the <celestial meridian>. https://aa.usno.navy.mil/faq/alt_az[US Naval Observatory: Computing altitude and azimuth, local hour angle and quadrant-aware conversion]{ref}.
For illustrative <geographical latitude> $\theta=45^\circ$ and invented <declination> $\delta=15^\circ$, <hour angle> $h=-60^\circ$ gives $\gamma\simeq31.636^\circ$ and $\eta\simeq100.729^\circ$. At $h=+60^\circ$, the same <altitude> accompanies $\eta\simeq259.271^\circ$: the direction is now westward. At $h=0$, it is due south with <altitude> $60^\circ$, agreeing with $90^\circ-|\theta-\delta|$ at upper <meridian transit>. These are analytical checks, not present positions of any <star>. For nearby objects, the direction must also be referred to the observer; simply rotating a geocentric direction omits the change of viewing position. Catalogue-to-observation refinements are described under <equatorial coordinate system>.
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