In a

spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

, a colatitude is the

complementary angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ar ...

of a given

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non-negative quantity, ranging from zero at the North pole to 180° at the South pole. The colatitude corresponds to the conventional 3D polar angle in spherical coordinates, as opposed to the latitude as used in

cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...

Examples

Latitude and colatitude sum up to 90°.

Astronomical use

The colatitude is most useful in

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

because it refers to the

zenith distance The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...

of the

celestial pole The north and south celestial poles are the two points in the sky where Earth's rotation around a fixed axis, axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently dire ...

s. For example, at latitude 42°N,

Polaris Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris ( Latinized to ''Alpha Ursae Minoris'') and is commonly called the North Star or Pole Star. With an apparent magnitude that ...

(approximately on the North celestial pole) has an

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

of 42°, so the distance from the

zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" ...

(overhead point) to Polaris is . Adding the declination of a star to the observer's colatitude gives the maximum latitude of that star (its angle from the

horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...

at culmination or upper transit). For example, if Alpha Centauri is seen with a latitude of 72° north (108° south) and its declination is known (60°S), then it can be determined that the observer's colatitude is (i.e. their latitude is ). Stars whose declinations exceed the observer's colatitude are called

circumpolar Circumpolar may refer to: * Antarctic region ** Antarctic Circle ** the Antarctic Circumpolar Current ** Subantarctic ** List of Antarctic and subantarctic islands ** Antarctic Convergence ** Antarctic Circumpolar Wave ** Antarctic Ocean * Arctic ...

because they will never set as seen from that latitude. If an object's declination is further south on the celestial sphere than the value of the colatitude, then it will never be seen from that location. For example, Alpha Centauri will always be visible at night from

Perth, Western Australia Perth is the capital and largest city of the Australian state of Western Australia. It is the fourth most populous city in Australia and Oceania, with a population of 2.1 million (80% of the state) living in Greater Perth in 2020. Perth is ...

because the colatitude is , and 60° is greater than 58°; on the other hand, the star will never rise in

Juneau The City and Borough of Juneau, more commonly known simply as Juneau ( ; tli, Dzánti K'ihéeni ), is the capital city of the state of Alaska. Located in the Gastineau Channel and the Alaskan panhandle, it is a unified municipality and the se ...

because its declination of −60° is less than −32° (the negation of Juneau's colatitude). Additionally, colatitude is used as part of the

Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...

in general relativity.

References

Spherical geometry Orientation (geometry) {{Astronomy-stub