A brass planispheric astrolabe maps the sky onto a plate by stereographic projection from the south celestial pole: circles stay circles, so every altitude and azimuth becomes an exact circular arc. Measure a star's altitude with the alidade, set the rete, and read the time.
Choose a star, then flip the astrolabe over and aim the alidade at it.
Compares the closed-form altitude formula with a vector rotation, tests that projected almucantars and azimuth lines are exact circles, and re-derives the star's altitude from its position on the plate.
The science
Stereographic projection. Every point of the celestial sphere is projected from the south celestial pole onto the plane of the celestial equator. A point at declination δ lands at radius
r = Req · tan((90° − δ) / 2)
The projection is conformal and maps every circle on the sphere (not through the projection pole) to a circle on the plate. The plate edge is the Tropic of Capricorn, so Rcap / Req = tan(45° + ε/2) = 1.5235 for ε = 23.44°.
The tympan is engraved for one latitude φ. The zenith has declination φ; a circle of constant altitude h (an almucantar) crosses the meridian at plate heights yS = Req tan((90°−φ+z)/2) and yN = Req tan((90°−φ−z)/2) with z = 90°−h, so its centre is (yS+yN)/2 and its radius (yS−yN)/2. Azimuth lines are circles through the zenith and the nadir. Altitudes above the horizon fill the horizon circle; the dashed circles are the −6°, −12° and −18° twilight limits.
The rete carries the ecliptic (an off-centre circle touching the two tropics) and pointers for bright stars at their (α, δ). Turning the rete by 15° per sidereal hour reproduces the sky's rotation (counter-clockwise on this plate, which is drawn as seen from inside the sphere, east on the right); the right ascension under the meridian line is the local sidereal time LST.
Altitude. The exact relation between horizon and equatorial coordinates is
sin h = sinφ sinδ + cosφ cosδ cos H
cos H = (sin h − sinφ sinδ) / (cosφ cosδ)
H = LST − α, Tsolar = 12 h + (LST − α☉)
Measuring h with the alidade and putting the star on that almucantar (east side if it is rising, west side if setting) fixes H, hence LST and the solar time; the rule through the Sun's point on the ecliptic reads it on the 24-hour limb. The sign of H is not in the altitude, which is why you must know which side of the meridian the star is on.
Error propagation. dh = −cosφ cosδ sin H dH / cos h, so a star near the prime vertical (|sin H| near 1, azimuth near east or west) gives the best time; near the meridian dH blows up because the altitude barely changes. The experiment checks this with random sightings.
Honest simplifications. Star positions are rounded J2000 values (precession since 2000, about 0.36°, is ignored); the Sun uses a low-precision formula (about 0.01°) and apparent solar time, not clock time (the equation of time is shown, longitude is not modelled); no atmospheric refraction (about 1 arcminute at 45° altitude, 0.5° at the horizon, which a real observer must subtract), no dip of the horizon, no parallax. The plate drawing is exact; a real brass astrolabe is engraved by hand to roughly half a degree. Latitude is limited to 0–70° N.
3D models generated with Meshy AI and processed in Blender (astrolabe ring and desk); the rule, alidade and stand are modelled in Blender 5.2; the plates, rete and all mathematics run live in Three.js r160.