In a

spherical coordinate system In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point ...

, a colatitude is the complementary angle of a given

latitude In geography, latitude is a geographic coordinate system, geographic 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 t ...

, i.e. the difference between a right angle and the latitude. In geography, 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 mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinate system, coordinates. These are * the radial distance along the line connecting the poin ...

in spherical coordinates, as opposed to the latitude as used in

cartography Cartography (; from , '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 imagined reality) can ...

Examples

Latitude and colatitude sum up to 90°.

Astronomical use

The colatitude is most useful in

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

because it refers to the zenith distance of the

celestial pole The north and south celestial poles are the two points in the sky where Earth's axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to observers at ...

s. For example, at latitude 42°N, for

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

(approximately on the North celestial pole), the distance from the

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

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

horizon The horizon is the apparent curve 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 curve divides all viewing directions based on whethe ...

culmination In observational astronomy, culmination is the passage of a celestial object (such as the Sun, the Moon, a planet, a star, constellation or a deep-sky object) across the observer's local meridian. These events are also known as meridian tran ...

or upper transit). For example, if

Alpha Centauri Alpha Centauri (, α Cen, or Alpha Cen) is a star system in the southern constellation of Centaurus (constellation), Centaurus. It consists of three stars: Rigil Kentaurus (), Toliman (), and Proxima Centauri (). Proxima Centauri ...

is seen with upper culmination altitude of 72° north (or 108° south) w.r.t. the observer and its declination is known (60°S), then it can be determined that the observer's colatitude is (i.e. the observer's latitude is ). Stars whose declination absolute value exceed the observer's colatitude in the corresponding hemisphere (see

) are called circumpolar because they will never set as seen from that latitude. If the star's declination absolute value exceed the observer's colatitude in the opposite hemisphere, then it will never be seen from that location. For example,

will always be visible at night from

Perth, Western Australia Perth () is the list of Australian capital cities, capital city of Western Australia. It is the list of cities in Australia by population, fourth-most-populous city in Australia, with a population of over 2.3 million within Greater Perth . The ...

(32°S) because the colatitude of Perth is , and the declination of Alpha Centauri (-60°) has an absolute value 60 which is greater than 58 in the corresponding hemisphere; on the other hand, the star will never rise in Juneau, Alaska (58°N) because Alpha Centauri's declination absolute value of 60 is more than observer's colatitude (32°) in the opposite hemisphere. Additionally, colatitude is used as part of the Schwarzschild metric in general relativity.

References

{{Reflist Spherical geometry Orientation (geometry)