# Latitude

**Latitude**, usually denoted by the Greek letter phi (φ) gives the location of a place on Earth (or other planetary body) north or south of the equator. **Lines of Latitude** are the horizontal lines shown running east-to-west on maps (particularly so in the Mercator projection). Technically, latitude is an angular measurement in degrees (marked with °) ranging from 0° at the equator (low latitude) to 90° at the poles (90° N or +90° for the North Pole and 90° S or −90° for the South Pole). The latitude is approximately the angle between straight up at the surface (the zenith) and the sun at an equinox. The complementary angle of a latitude is called the **colatitude**.

## Contents

## Circles of latitude

All locations of a given latitude are collectively referred to as a *circle of latitude* or *line of latitude* or *parallel*, because they are coplanar, and all such planes are parallel to the equator. Lines of latitude other than the Equator are approximately small circles on the surface of the Earth; they are not geodesics since the shortest route between two points at the same latitude involves a path that bulges toward the nearest pole, first moving farther away from and then back toward the equator (see great circle).

A specific latitude may then be combined with a specific longitude to give a precise position on the Earth's surface (see satellite navigation system).

### Important named circles of latitude

Besides the equator, four other lines of latitude are named because of the role they play in the geometrical relationship with the Earth and the Sun:

- Arctic Circle: 66° 33′ 39″ N
- Tropic of Cancer: 23° 26′ 21″ N
- Tropic of Capricorn: 23° 26′ 21″ S
- Antarctic Circle: 66° 33′ 39″ S

Only at latitudes between the Tropics is it possible for the sun to be at the zenith. Only north of the Arctic Circle or south of the Antarctic Circle is the midnight sun possible.

The reason that these lines have the values that they do lies in the axial tilt of the Earth with respect to the sun, which is 23° 26′ 21.41″.

Note that the Arctic Circle and Tropic of Cancer are colatitudes, since the sum of their angles is 90°—similarly for the Antarctic Circle and Tropic of Capricorn.

## Subdivisions

A degree is divided into 60 minutes. One minute can be further divided into 60 seconds. An example of a latitude specified in this way is 13°19'43″ N (for greater precision, a decimal fraction can be added to the seconds). An alternative representation uses only degrees and minutes, where the seconds are expressed as a decimal fraction of minutes: the above example would be expressed as 13°19.717' N. Degrees can also be expressed singularly, with both the minutes and seconds incorporated as a decimal number and rounded as desired (decimal degree notation): 13.32861° N. Sometimes, the north/south suffix is replaced by a negative sign for south (−90° for the South Pole).

## Effect of latitude

A region's latitude has a great effect on its climate and weather (see *Effect of sun angle on climate*). Latitude more loosely determines tendencies in polar auroras, prevailing winds, and other physical characteristics of geographic locations.

Researchers at Harvard's Center for International Development (CID) found in 2001 that only three tropical economies — Hong Kong, Singapore, and Taiwan — were classified as high-income by the World Bank, while all countries within regions zoned as temperate had either middle- or high-income economies. ^{[1]} The validity of the Harvard report may be questioned because a different threshold is used for the tropical regions and the World Bank list fails to include Qatar's, United Arab Emirates', and Kuwait's economies. Further, countries such as Brazil have far better incomes than much of the Former Soviet Union and Iron Curtain states^{[citation needed]}.

## Elliptic parameters

Because most planets (including Earth) are *ellipsoids of revolution*, or spheroids, rather than spheres, both the radius and the length of arc varies with latitude. This variation requires the introduction of elliptic parameters based on an ellipse's **angular eccentricity**, (which equals , where and are the equatorial and polar radii; is the first eccentricity squared, ; and or is the flattening, ). Utilized in creating the integrands for curvature is the inverse of the principal elliptic integrand, :

## Degree length

On Earth, the length of an arcdegree of north–south latitude difference, , is about 60 nautical miles, 111 kilometres or 69 statute miles at any latitude. The length of an arcdegree of east-west longitude difference, , is about the same at the equator as the north-south, reducing to zero at the poles.

In the case of a spheroid, a meridian and its anti-meridian form an ellipse, from which an exact expression for the length of an arcdegree of latitude difference is:

This radius of arc (or "arcradius") is in the plane of a meridian, and is known as the *meridional radius of curvature*, .^{[2]}^{[3]}

Similarly, an exact expression for the length of an arcdegree of longitude difference is:

The arcradius contained here is in the plane of the prime vertical, the east-west plane perpendicular (or "normal") to both the plane of the meridian and the plane tangent to the surface of the ellipsoid, and is known as the *normal radius of curvature*, .^{[2]}^{[3]}

Along the equator (east-west), equals the equatorial radius. The radius of curvature at a right angle to the equator (north-south), , is 43 km shorter, hence the length of an arcdegree of latitude difference at the equator is about 1 km less than the length of an arcdegree of longitude difference at the equator. The radii of curvature are equal at the poles where they are about 64 km greater than the north-south equatorial radius of curvature *because* the polar radius is 21 km less than the equatorial radius. The shorter polar radii indicate that the northern and southern hemispheres are flatter, making their radii of curvature longer. This flattening also 'pinches' the north-south equatorial radius of curvature, making it 43 km less than the equatorial radius. Both radii of curvature are perpendicular to the plane tangent to the surface of the ellipsoid at all latitudes, directed toward a point on the polar axis in the opposite hemisphere (except at the equator where both point toward Earth's center). The east-west radius of curvature reaches the axis, whereas the north-south radius of curvature is shorter at all latitudes except the poles.

The WGS84 ellipsoid, used by all GPS devices, uses an equatorial radius of 6378137.0 m and an inverse flattening, (1/f), of 298.257223563, hence its polar radius is 6356752.3142 m and its first eccentricity squared is 0.00669437999014.^{[4]} The more recent but little used IERS 2003 ellipsoid provides equatorial and polar radii of 6378136.6 and 6356751.9 m, respectively, and an inverse flattening of 298.25642.^{[5]} Lengths of degrees on the WGS84 and IERS 2003 ellipsoids are the same when rounded to six significant digits. An appropriate calculator for any latitude is provided by the U.S. government's National Geospatial-Intelligence Agency (NGA).^{[6]}

Latitude | N-S radius of curvature |
Surface distance per 1° change in latitude |
E-W radius of curvature |
Surface distance per 1° change in longitude | |
---|---|---|---|---|---|

0° | 6335.44 km | 110.574 km | 6378.14 km | 111.320 km | |

15° | 6339.70 km | 110.649 km | 6379.57 km | 107.551 km | |

30° | 6351.38 km | 110.852 km | 6383.48 km | 96.486 km | |

45° | 6367.38 km | 111.132 km | 6388.84 km | 78.847 km | |

60° | 6383.45 km | 111.412 km | 6394.21 km | 55.800 km | |

75° | 6395.26 km | 111.618 km | 6398.15 km | 28.902 km | |

90° | 6399.59 km | 111.694 km | 6399.59 km | 0.000 km |

## Types of latitude

With a spheroid that is slightly flattened by its rotation, cartographers refer to a variety of auxiliary latitudes to precisely adapt spherical projections according to their purpose.

For planets other than Earth, such as Mars, geographic and geocentric latitude are called "planetographic" and "planetocentric" latitude, respectively. Most maps of Mars since 2002 use planetocentric coordinates.

### Common "latitude"

In common usage, "latitude" refers to **geodetic** or **geographic latitude** and is the angle between the equatorial plane and a line that is normal to the reference ellipsoid, which approximates the shape of Earth to account for flattening of the poles and bulging of the equator. This value usually differs from the geocentric latitude.

The expressions following assume elliptical polar sections and that all sections parallel to the equatorial plane are circular. Geographic latitude (with longitude) then provides a Gauss map. As defined earlier in this article, is the angular eccentricity of a meridian.

### Reduced latitude

- On a spheroid, lines of
**reduced**or**parametric latitude**, , form circles whose radii are the same as the radii of circles formed by the corresponding lines of latitude on a sphere with radius equal to the equatorial radius of the spheroid.

### Authalic latitude

**Authalic latitude**, , gives an area-preserving transform to the sphere.

### Rectifying latitude

**Rectifying latitude**, , is the surface distance from the equator, scaled so the pole is 90°, but involves elliptic integration:

### Conformal latitude

**Conformal latitude**, , gives an angle-preserving (conformal) transform to the sphere.

### Geocentric latitude

- The
**geocentric latitude**, , is the angle between the equatorial plane and a line from the center of Earth.

- It is the size of the central angle between the equator and the point of interest, as measured along a meridian. This value usually differs from the geographic latitude, as so:

### Astronomical latitude

A more obscure measure of latitude is the **astronomical latitude**, which is the angle between the equatorial plane and the normal to the geoid (ie a plumb line). It originated as the angle between horizon and pole star. It differs from the geodetic latitude only slightly, due to the slight deviations of the geoid from the reference ellipsoid.

Astronomical latitude is not to be confused with declination, the coordinate astronomers use to describe the locations of stars north/south of the celestial equator (see equatorial coordinates), nor with ecliptic latitude, the coordinate that astronomers use to describe the locations of stars north/south of the ecliptic (see ecliptic coordinates).

### Palaeolatitude

Continents move over time, due to continental drift, taking whatever fossils and other features of interest they may have with them. Particularly when discussing fossils, it's often more useful to know where the fossil was when it was laid down, than where it is when it was dug up: this is called the *palæolatitude* of the fossil. The Palæolatitude can be constrained by palæomagnetic data. If tiny magnetisable grains are present when the rock is being formed, these will align themselves with Earth's magnetic field like compass needles. A magnetometer can deduce the orientation of these grains by subjecting a sample to a magnetic field, and the magnetic declination of the grains can be used to infer the latitude of deposition.

### Comparison of selected types

The following plot shows the differences between the types of latitude. The data used are found in the table following the plot. Please note that the values in the table are in minutes, not degrees, and the plot reflects this as well. Also observe that the conformal symbols are hidden behind the geocentric due to being very close in value. Finally it is important to mention also that these differences don't mean that the use of one specific latitude will necessarily cause more distortions than the other (the real fact is that each latitude type is optimized for achieving a different goal).

Approximate difference from geographic latitude ("Lat") Lat

Reduced

Authalic

Rectifying

Conformal

Geocentric

0° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′ 5° 1.01′ 1.35′ 1.52′ 2.02′ 2.02′ 10° 1.99′ 2.66′ 2.99′ 3.98′ 3.98′ 15° 2.91′ 3.89′ 4.37′ 5.82′ 5.82′ 20° 3.75′ 5.00′ 5.62′ 7.48′ 7.48′ 25° 4.47′ 5.96′ 6.70′ 8.92′ 8.92′ 30° 5.05′ 6.73′ 7.57′ 10.09′ 10.09′ 35° 5.48′ 7.31′ 8.22′ 10.95′ 10.96′ 40° 5.75′ 7.66′ 8.62′ 11.48′ 11.49′ 45° 5.84′ 7.78′ 8.76′ 11.67′ 11.67′ 50° 5.75′ 7.67′ 8.63′ 11.50′ 11.50′ 55° 5.49′ 7.32′ 8.23′ 10.97′ 10.98′ 60° 5.06′ 6.75′ 7.59′ 10.12′ 10.13′ 65° 4.48′ 5.97′ 6.72′ 8.95′ 8.96′ 70° 3.76′ 5.01′ 5.64′ 7.52′ 7.52′ 75° 2.92′ 3.90′ 4.39′ 5.85′ 5.85′ 80° 2.00′ 2.67′ 3.00′ 4.00′ 4.01′ 85° 1.02′ 1.35′ 1.52′ 2.03′ 2.03′ 90° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′

### Corrections for altitude

When converting from geodetic ("common") latitude to other types of latitude, corrections must be made for altitude for systems which do not measure the angle from the normal of the spheroid. For example, in the figure at right, point *H* (located on the surface of the spheroid) and point *H'* (located at some greater elevation) have different *geocentric* latitudes (angles *β* and *γ* respectively), even though they share the same *geodetic* latitude (angle *α*). Note that the flatness of the spheroid and elevation of point *H'* in the image is significantly greater than what is found on the Earth, exaggerating the errors inherent in such calculations if left uncorrected. Note also that the reference ellipsoid used in the geodetic system is itself just an approximation of the true geoid, and therefore introduces its own errors, though the differences are less severe. (See Astronomical latitude, above.)

## Further reading

- John P. Snyder
*Map Projections: a working manual*excerpts

## See also

- American Practical Navigator
- Cardinal direction
- Geographic coordinate system
- Geodetic system
- Geodesy
- Geotagging
- Great-circle distance
- Horse latitudes
- List of cities by latitude
- List of cities by longitude
- Longitude
- Natural Area Code
- Navigation
- World Geodetic System
- Orders of magnitude (length)

## Footnotes

- ↑ Location, Location, Location. The relationship of climate to, and the effect of disease and agricultural productivity on, the economic success of a city or region.
- ↑
^{2.0}^{2.1}The Math Forum - ↑
^{3.0}^{3.1}John P. Snyder,*Map Projections—A Working Manual*(1987) 24-25 - ↑ NIMA TR8350.2 page 3-1.
- ↑ IERS Conventions (2003) (Chp. 1, page 12)
- ↑ Length of degree calculator - National Geospatial-Intelligence Agency

## External links

- libproj4: A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft)PDF (2.18 MB)
- Free GeoCoder
- GEONets Names Server, access to the National Geospatial-Intelligence Agency's (NGA) database of foreign geographic feature names.
- Look-up Latitude and Longitude
- Resources for determining your latitude and longitude
- Convert decimal degrees into degrees, minutes, seconds - Info about decimal to sexagesimal conversion
- Convert decimal degrees into degrees, minutes, seconds
- Latitude and longitude converter – Convert latitude and longitude from degree, decimal form to degree, minutes, seconds form and vice versa. Also included a farthest point and a distance calculator.
- Worldwide Index - Tageo.com – contains 2,700,000 coordinates of places including US towns
- for each city it gives the satellite map location, country, province, coordinates (dd,dms), variant names and nearby places.

- Distance calculation based on latitude and longitude - JavaScript version
- Zoomable version of the mapPDF (3.47 MB)
- Average Latitude & Longitude of Countries
- Get the latitude and longitude of any place in the World
- Latitude / Longitude Converter – convert latitude / longitude between DMS and decimal formats.
- Determination of Latitude by Francis Drake on the Coast of California in 1579