Editing Astrolabe (section)

=== Designing a tympanum through stereographic projection ===
[[File:Parts of a tympan of an astrolabe.png|thumb|upright=1.4|Parts of an Astrolabe tympanum]]

The tympanum captures the celestial coordinate axes upon which the ''rete'' will rotate. It is the component that will enable the precise determination of a star's position at a specific time of [[Earth's rotation|day]] and [[Earth's orbit|year]].

Therefore, it should project:

# The [[zenith]], which will vary depending on the [[latitude]] of the astrolabe user.
# The [[Horizon|horizon line]] and [[almucantar]] or circles parallel to the horizon, which will allow for the determination of a celestial body's [[Horizontal coordinate system|altitude]] (from the horizon to the zenith).
# The [[Meridian (astronomy)|celestial meridian]] (north-south meridian, passing through the zenith) and secondary meridians (circles intersecting the north-south meridian at the zenith), which will enable the measurement of [[azimuth]] for a celestial body.
# The three main [[Circle of latitude|circles of latitude]] ([[Tropic of Capricorn|Capricorn]], [[Equator]], and [[Tropic of Cancer|Cancer]]) to determine the exact moments of [[solstice]]s and [[equinox]]es throughout the year.

==== The tropics and the equator define the tympanum ====
[[File:Proyección estereográfica de trópicos y ecuador en un astrolabio.png|thumb|upright=2.8|Stereographic projection of Earth's tropics and equator from the South Pole.]]
On the right side of the image above:

# {{color box|#C4F5F5}} The blue sphere represents the [[celestial sphere]].
# {{color box|#0325F1}} The blue arrow indicates the direction of true north (the [[Polaris|North Star]]).
# {{color box|#0325F1}} The central blue point represents Earth (the observer's location).
# {{color box|#001919}} The geographic south of the celestial sphere acts as the [[Stereographic projection|projection pole]].
# {{color box|#B8B8B8}} The celestial equatorial plane serves as the [[projection plane]].
# Three parallel circles represent the projection on the celestial sphere of Earth's main [[Circle of latitude|circles of latitude]]: 
#* {{color box|#DA5E1B}} In orange, the celestial [[Tropic of Cancer]]. 
#* {{color box|#8533DB}} In purple, the celestial [[equator]].
#* {{color box|#006713}} In green, the celestial [[Tropic of Capricorn]].

When projecting onto the celestial equatorial plane, three concentric circles correspond to the celestial sphere's three [[Circle of latitude|circles of latitude]] (left side of the image). The largest of these, the projection on the celestial equatorial plane of the celestial [[Tropic of Capricorn]], defines the size of the astrolabe's tympanum. The center of the tympanum (and the center of the three circles) is actually the north-south axis around which Earth rotates, and therefore, the ''rete'' of the astrolabe will rotate around this point as the hours of the day pass (due to [[Earth's rotation|Earth's rotational motion]]).

The three concentric circles on the tympanum are useful for determining the exact moments of [[solstice]]s and [[equinox]]es throughout the year: if the sun's altitude at noon on the ''rete'' is known and coincides with the outer circle of the tympanum (Tropic of Capricorn), it signifies the [[winter solstice]] (the sun will be at the [[zenith]] for an observer at the Tropic of Capricorn, meaning summer in the southern hemisphere and winter in the northern hemisphere). If, on the other hand, its altitude coincides with the inner circle (Tropic of Cancer), it indicates the [[summer solstice]]. If its altitude is on the middle circle (equator), it corresponds to one of the two [[equinox]]es.

==== The horizon and the measurement of altitude ====
[[File:Proyección estereográfica del horizonte sobre el tímpano de un astrolabio.png|thumb|upright=2.8|Stereographic projection of an observer's horizon at a specific latitude]]
On the right side of the image above:

# {{color box|#0325F1}} The blue arrow indicates the direction of true north (the [[Polaris|North Star]]).
# {{color box|#0325F1}} The central blue point represents Earth (the observer's location).
# {{color box|#001919}} The black arrow represents the [[zenith]] direction for the observer (which would vary depending on the observer's [[latitude]]).
# {{color box|#001919}} The two black circles represent the [[horizon]] surrounding the observer, which is perpendicular to the zenith vector and defines the portion of the [[celestial sphere]] visible to the observer, and its projection on the celestial equatorial plane. 
# {{color box|#001919}} The geographic south of the celestial sphere acts as the [[Stereographic projection|projection pole]].
# {{color box|#B8B8B8}} The celestial equatorial plane serves as the [[projection plane]].

When projecting the [[horizon]] onto the celestial equatorial plane, it transforms into an ellipse upward-shifted relatively to the center of the tympanum (both the observer and the projection of the north-south axis). This implies that a portion of the celestial sphere will fall outside the outer circle of the tympanum (the projection of the celestial [[Tropic of Capricorn]]) and, therefore, won't be represented.

[[File:Proyección estereográfica del horizonte y almucantar sobre el tímpano de un astrolabio.png|thumb|Stereographic projection of the horizon and an almucantar.|upright=2.8]]

Additionally, when drawing circles parallel to the horizon up to the zenith ([[almucantar]]), and projecting them on the celestial equatorial plane, as in the image above, a grid of consecutive ellipses is constructed, allowing for the determination of a [[Horizontal coordinate system|star's altitude]] when its ''rete'' overlaps with the designed tympanum.

==== The meridians and the measurement of azimuth ====
[[File:Proyección estereográfica del horizonte y azimut sobre el tímpano de un astrolabio.png|thumb|upright=2.8|Stereographic projection of the north-south meridian and a meridian 40° E on the tympanum of an astrolabe]]

On the right side of the image above:

# {{color box|#0325F1}} The blue arrow indicates the direction of true north (the [[Polaris|North Star]]).
# {{color box|#0325F1}} The central blue point represents Earth (the observer's location).
# {{color box|#001919}} The black arrow represents the [[zenith]] direction for the observer (which would vary depending on the observer's [[latitude]]).
# {{color box|#001919}} The two black circles represent the [[horizon]] surrounding the observer, which is perpendicular to the zenith vector and defines the portion of the [[celestial sphere]] visible to the observer, and its projection on the celestial equatorial plane.
# {{color box|#D92929}} The five red dots represent the [[zenith]], the [[nadir (astronomy)|nadir]] (the point on the [[celestial sphere]] opposite the zenith with respect to the observer), their projections on the celestial equatorial plane, and the center (with no physical meaning attached) of the circle obtained by projecting the secondary meridian (see below) on the celestial equatorial plane.
# {{color box|#DA5E1B}} The orange circle represents the [[Meridian (astronomy)|celestial meridian]] (or meridian that goes, for the observer, from the north of the horizon to the south of the horizon passing through the zenith).
# {{color box|#D92929}} The two red circles represent a secondary meridian with an [[azimuth]] of 40° East relative to the observer's horizon (which, like all secondary meridians, intersects the principal meridian at the zenith and nadir), and its projection on the celestial equatorial plane.
# {{color box|#001919}} The geographic south of the celestial sphere acts as the [[Stereographic projection|projection pole]].
# {{color box|#B8B8B8}} The celestial equatorial plane serves as the [[projection plane]].

When projecting the [[Meridian (astronomy)|celestial meridian]], it results in a straight line that overlaps with the vertical axis of the tympanum, where the [[zenith]] and [[nadir (astronomy)|nadir]] are located. However, when projecting the 40° E meridian, another circle is obtained that passes through both the zenith and nadir projections, so its center is located on the perpendicular [[bisection]] of the segment connecting both points. In deed, the projection of the celestial meridian can be considered as a circle with an infinite radius (a straight line) whose center is on this bisection and at an infinite distance from these two points.

If successive meridians that divide the celestial sphere into equal sectors (like "orange slices" radiating from the zenith) are projected, a family of curves passing through the zenith projection on the tympanum is obtained. These curves, once overlaid with the ''rete'' containing the major stars, allow for determining the [[azimuth]] of a star located on the ''rete'' and rotated for a specific time of day.