Latitude

Last updated
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the equator are lines of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example meridians are spaced at 6deg intervals and parallels at 4deg intervals. Division of the Earth into Gauss-Krueger zones - Globe.svg
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the equator are lines of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example meridians are spaced at 6° intervals and parallels at 4° intervals.

In geography, latitude is a geographic coordinate that specifies the northsouth position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North or South) at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular (or normal ) to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Geography The science that studies the terrestrial surface, the societies that inhabit it and the territories, landscapes, places or regions that form it

Geography is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the word γεωγραφία was Eratosthenes. Geography is an all-encompassing discipline that seeks an understanding of Earth and its human and natural complexities—not merely where objects are, but also how they have changed and come to be.

Geographic coordinate system Coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

North one of the four cardinal directions

North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. North is a noun, adjective, or adverb indicating direction or geography.

Contents

Preliminaries

Two levels of abstraction are employed in the definition of latitude and longitude. In the first step the physical surface is modeled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere, but the geoid is more accurately modeled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard. [lower-alpha 1]

Geoid irregular surface approximating the mean sea level

The geoid is the shape that the ocean surface would take under the influence of the gravity and rotation of Earth alone, if other influences such as winds and tides were absent. This surface is extended through the continents. According to Gauss, who first described it, it is the "mathematical figure of the Earth", a smooth but irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of geodesy and geophysics, it has been defined to high precision only since advances in satellite geodesy in the late 20th century.

Sea level Average level for the surface of the ocean at any given geographical position on the planetary surface

Mean sea level (MSL) is an average level of the surface of one or more of Earth's oceans from which heights such as elevation may be measured. MSL is a type of vertical datum – a standardised geodetic datum – that is used, for example, as a chart datum in cartography and marine navigation, or, in aviation, as the standard sea level at which atmospheric pressure is measured to calibrate altitude and, consequently, aircraft flight levels. A common and relatively straightforward mean sea-level standard is the midpoint between a mean low and mean high tide at a particular location.

Sphere round geometrical and circular object in three-dimensional space; special case of spheroid

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

Since there are many different reference ellipsoids, the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.

Reference ellipsoid an ellipsoid that approximates the figure of the Earth

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

Global Positioning System American satellite navigation system

The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Air Force. It is a global navigation satellite system that provides geolocation and time information to a GPS receiver anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites. Obstacles such as mountains and buildings block the relatively weak GPS signals.

In English texts the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi (φ or ϕ). It is measured in degrees, minutes and seconds or decimal degrees, north or south of the equator.

Degree (angle) angle unit; π/180 radians

A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

Decimal degrees (DD) express latitude and longitude geographic coordinates as decimal fractions and are used in many geographic information systems (GIS), web mapping applications such as OpenStreetMap, and GPS devices. Decimal degrees are an alternative to using degrees, minutes, and seconds (DMS). As with latitude and longitude, the values are bounded by ±90° and ±180° respectively.

The precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy.

Theodolite surveying instrument that measures azimuth and elevation between points

A theodolite is a precision optical instrument for measuring angles between designated visible points in the horizontal and vertical planes. The traditional use has been for land surveying, but they are also used extensively for building and infrastructure construction, and some specialized applications such as meteorology and rocket launching.

Figure of the Earth mathematical descriptions of Earths complex shape

Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. The sphere is an approximation of the figure of the Earth that is satisfactory for many purposes. Several models with greater accuracy have been developed so that coordinate systems can serve the precise needs of navigation, surveying, cadastre, land use, and various other concerns.

Geodesy The science of the geometric shape, orientation in space, and gravitational field of Earth

Geodesy, is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space, and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measurements for other planets. Geodynamical phenomena include crustal motion, tides, and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques, and relying on datums and coordinate systems.

This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature.

Latitude on the sphere

A perspective view of the Earth showing how latitude (
ph
{\displaystyle \phi }
) and longitude (
l
{\displaystyle \lambda }
) are defined on a spherical model. The graticule spacing is 10 degrees. Latitude and longitude graticule on a sphere.svg
A perspective view of the Earth showing how latitude () and longitude () are defined on a spherical model. The graticule spacing is 10 degrees.

The graticule on the sphere

The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to the rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians; and the angle between any one meridian plane and that through Greenwich (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North (written 90° N or +90°), and the South Pole has a latitude of 90° South (written 90° S or −90°). The latitude of an arbitrary point is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radius vector.

Geographical pole Points on a rotating astronomical body where the axis of rotation intersects the surface

A geographical pole is either of the two points on a rotating body where its axis of rotation intersects its surface. As with Earth's North and South Poles, they are usually called that body's "north pole" and "south pole", one lying 90 degrees in one direction from the body's equator and the other lying 90 degrees in the opposite direction from the equator.

Meridian (geography) line between the poles with the same longitude

A (geographic) meridian is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km.

Equator Intersection of a spheres surface with the plane perpendicular to the spheres axis of rotation and midway between the poles

An equator of a rotating spheroid is its zeroth circle of latitude (parallel). It is the imaginary line on the spheroid, equidistant from its poles, dividing it into northern and southern hemispheres. In other words, it is the intersection of the spheroid with the plane perpendicular to its axis of rotation and midway between its geographical poles.

The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article.

Named latitudes on the Earth

The orientation of the Earth at the December solstice. December solstice geometry.svg
The orientation of the Earth at the December solstice.

Besides the equator, four other parallels are of significance:

Arctic Circle 66° 34′ (66.57°) N
Tropic of Cancer 23° 26′ (23.43°) N
Tropic of Capricorn 23° 26′ (23.43°) S
Antarctic Circle 66° 34′ (66.57°) S

The plane of the Earth's orbit about the Sun is called the ecliptic, and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i. The latitude of the tropical circles is equal to i and the latitude of the polar circles is its complement (90° - i). The axis of rotation varies slowly over time and the values given here are those for the current epoch. The time variation is discussed more fully in the article on axial tilt. [lower-alpha 2]

The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December solstice when the Sun is overhead at some point of the Tropic of Capricorn. The south polar latitudes below the Antarctic Circle are in daylight, whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice, when the Sun is overhead at the Tropic of Cancer. Only at latitudes in between the two tropics is it possible for the Sun to be directly overhead (at the zenith).

On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used Mercator projection and the Transverse Mercator projection. On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves.

Normal MercatorTransverse Mercator
MercNormSph enhanced.png

\

MercTranSph enhanced.png

Meridian distance on the sphere

On the sphere the normal passes through the centre and the latitude (φ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m(φ) then

where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for R since higher-precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km (69.1 statute miles) (60.0 nautical miles). The length of 1 minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see nautical mile).

Latitude on the ellipsoid

Ellipsoids

In 1687 Isaac Newton published the Philosophiæ Naturalis Principia Mathematica , in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. [1] (This article uses the term ellipsoid in preference to the older term spheroid.) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc.) An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed triaxial.)

Many different reference ellipsoids have been used in the history of geodesy. In pre-satellite days they were devised to give a good fit to the geoid over the limited area of a survey but, with the advent of GPS, it has become natural to use reference ellipsoids (such as WGS84) with centre at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within 100 m (330 ft) of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid.

The geometry of the ellipsoid

A sphere of radius a compressed to an oblate ellipsoid of revolution. Ellipsoid parametric euler mono.svg
A sphere of radius a compressed to an oblate ellipsoid of revolution.

The shape of an ellipsoid of revolution is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the semi-major axis, a. The other parameter is usually (1) the polar radius or semi-minor axis, b; or (2) the (first) flattening, f; or (3) the eccentricity, e. These parameters are not independent: they are related by

Many other parameters (see ellipse, ellipsoid) appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set a, b, f and e. Both f and e are small and often appear in series expansions in calculations; they are of the order 1/300 and 0.08 respectively. Values for a number of ellipsoids are given in Figure of the Earth. Reference ellipsoids are usually defined by the semi-major axis and the inverse flattening, 1/f. For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are [2]

from which are derived

The difference between the semi-major and semi-minor axes is about 21 km (13 miles) and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from a 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening.

Geodetic and geocentric latitudes

The definition of geodetic latitude (ph) and longitude (l) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles. Latitude and longitude graticule on an ellipsoid.svg
The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles.

The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing:

The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the Eiffel Tower has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on the datum ED50 define a point on the ground which is 140 metres (460 feet) distant from the tower.[ citation needed ] A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified.

Length of a degree of latitude

In Meridian arc and standard texts [3] [4] [5] it is shown that the distance along a meridian from latitude φ to the equator is given by (φ in radians)

where M(φ) is the meridional radius of curvature.

The distance from the equator to the pole is

For WGS84 this distance is 10001.965729 km.

The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length of the meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a small meridian arc is given by [4] [5]

Δ1
lat
Δ1
long
110.574 km111.320 km
15°110.649 km107.550 km
30°110.852 km96.486 km
45°111.132 km78.847 km
60°111.412 km55.800 km
75°111.618 km28.902 km
90°111.694 km0.000 km

When the latitude difference is 1 degree, corresponding to π/180 radians, the arc distance is about

The distance in metres (correct to 0.01 metre) between latitudes   0.5 degrees and  + 0.5 degrees on the WGS84 spheroid is

The variation of this distance with latitude (on WGS84) is shown in the table along with the length of a degree of longitude (east-west distance):

A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency (NGA). [6]

The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude.

The definition of geodetic latitude (ph) and geocentric latitude (th). WGS84 angle to distance conversion.svg
The definition of geodetic latitude (φ) and geocentric latitude (θ).

The nautical mile

Historically a nautical mile was defined as the length of one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However for all practical purposes distances are measured from the latitude scale of charts. As the Royal Yachting Association says in its manual for day skippers: "1 (minute) of Latitude = 1 sea mile", followed by "For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile". [7]

Auxiliary latitudes

There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and the theory of map projections:

The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional geographic coordinate system as discussed below. The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of the reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower.

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. (For inverses see below.) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. [8] Derivations of these expressions may be found in Adams [9] and online publications by Osborne [4] and Rapp. [5]

Geocentric latitude

The definition of geodetic latitude (ph) and geocentric latitude (th). Geocentric coords 03.svg
The definition of geodetic latitude (φ) and geocentric latitude (θ).

The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (θ) and the geodetic latitude (φ) is derived in the above references as

The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′. [lower-alpha 3]

Parametric (or reduced) latitude

Definition of the parametric latitude (b) on the ellipsoid. Ellipsoid reduced angle definition.svg
Definition of the parametric latitude (β) on the ellipsoid.

The parametric or reduced latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude φ. It was introduced by Legendre [10] and Bessel [11] who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, u(φ), is also used in the current literature. The parametric latitude is related to the geodetic latitude by: [4] [5]

The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the ellipse is:

The Cartesian coordinates of the point are parameterized by

Cayley suggested the term parametric latitude because of the form of these equations. [12]

The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, (Vincenty, Karney [13] ).

Rectifying latitude

The rectifying latitude, μ, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or π/2 radians:

where the meridian distance from the equator to a latitude φ is (see Meridian arc)

and the length of the meridian quadrant from the equator to the pole (the polar distance) is

Using the rectifying latitude to define a latitude on a sphere of radius

defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale. The sphere may then be projected to the plane with an equirectangular projection to give a double projection from the ellipsoid to the plane such that all meridians have true length and uniform meridian scale. An example of the use of the rectifying latitude is the Equidistant conic projection. (Snyder, Section 16). [8] The rectifying latitude is also of great importance in the construction of the Transverse Mercator projection.

Authalic latitude

The authalic (Greek for same area) latitude, ξ, gives an area-preserving transformation to a sphere.

where

and

and the radius of the sphere is taken as

An example of the use of the authalic latitude is the Albers equal-area conic projection. [8] :§14

Conformal latitude

The conformal latitude, χ, gives an angle-preserving (conformal) transformation to the sphere.

where gd(x) is the Gudermannian function. (See also Mercator projection.)

The conformal latitude defines a transformation from the ellipsoid to a sphere of arbitrary radius such that the angle of intersection between any two lines on the ellipsoid is the same as the corresponding angle on the sphere (so that the shape of small elements is well preserved). A further conformal transformation from the sphere to the plane gives a conformal double projection from the ellipsoid to the plane. This is not the only way of generating such a conformal projection. For example, the 'exact' version of the Transverse Mercator projection on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).

Isometric latitude

The isometric latitude, ψ, is used in the development of the ellipsoidal versions of the normal Mercator projection and the Transverse Mercator projection. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of ψ and longitude λ give rise to equal distance displacements along the meridians and parallels respectively. The graticule defined by the lines of constant ψ and constant λ, divides the surface of the ellipsoid into a mesh of squares (of varying size). The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude, tending to infinity at the poles. The conventional notation is given in Snyder (page 15): [8]

For the normal Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is E (units of length or pixels) then the distance, y, of a parallel of latitude φ from the equator is

The isometric latitude ψ is closely related to the conformal latitude χ:

Inverse formulae and series

The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and parametric latitudes may be inverted directly but this is impossible in the four remaining cases: the rectifying, authalic, conformal, and isometric latitudes. There are two methods of proceeding. The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are fixed-point iteration and Newton–Raphson root finding. The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of Lagrange reversion. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity. [9] Osborne [4] derives series to arbitrary order by using the computer algebra package Maxima [14] and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must use the conformal latitude in an intermediate step.

Numerical comparison of auxiliary latitudes

The following plot shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles) for the case of the WGS84 ellipsoid. In every case the auxiliary latitude is the less (in magnitude) than the geodetic latitude. The differences shown on the plot are in arc minutes. The horizontal resolution of the plot fails to make clear that the maxima of the curves are not at 45° but calculation shows that they are within a few arc minutes of 45°. Some representative data points are given in the table following the plot. Note the closeness of the conformal and geocentric latitudes. This was exploited in the days of hand calculators to expedite the construction of map projections. [8] :108

To first order in the flattening f, the auxiliary latitudes can be expressed as ζ = φCf sin 2φ where the constant C takes on the values [12, 23, 34, 1, 1] for ζ = [β, ξ, μ, χ, θ].

Types of latitude difference.svg
Approximate difference from geodetic latitude (φ)
φParametric
βφ
Authalic
ξφ
Rectifying
μφ
Conformal
χφ
Geocentric
θφ
0.00′0.00′0.00′0.00′0.00′
15°−2.88′−3.84′−4.32′−5.76′−5.76′
30°−5.00′−6.66′−7.49′−9.98′−9.98′
45°−5.77′−7.70′−8.66′−11.54′−11.55′
60°−5.00′−6.67′−7.51′−10.01′−10.02′
75°−2.89′−3.86′−4.34′−5.78′−5.79′
90°0.00′0.00′0.00′0.00′0.00′

Latitude and coordinate systems

The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and parametric latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.

Geodetic coordinates

Geodetic coordinates P(F,l,h) Geodetic coordinates.svg
Geodetic coordinates P(ɸ,λ,h)

At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(ɸ,λ,h) are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth.

Spherical polar coordinates

Geocentric coordinate related to spherical polar coordinates P(r,th',l) Geocentric coords 02.svg
Geocentric coordinate related to spherical polar coordinates P(r,θ′,λ)

The geocentric latitude θ is the complement of the polar angle θ′ in conventional spherical polar coordinates in which the coordinates of a point are P(r,θ′,λ) where r is the distance of P from the centre O, θ′ is the angle between the radius vector and the polar axis and λ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points P' on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

Ellipsoidal coordinates

Ellipsoidal coordinates P(u,b,l) Ellipsoidal coordinates.svg
Ellipsoidal coordinates P(u,β,λ)

The parametric latitude can also be extended to a three-dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F′) with the reference ellipsoid: the necessary condition is that the product ae of semi-major axis and eccentricity is the same for both ellipsoids. Let u be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the parametric latitude of P on the auxiliary ellipsoid. The set (u,β,λ) define the ellipsoid coordinates. [3] :§4.2.2 These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body.

Coordinate conversions

The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in Geographic coordinate conversion. The relation of Cartesian and spherical polars is given in Spherical coordinate system. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge. [3]

Astronomical latitude

Astronomical latitude (Φ) is the angle between the equatorial plane and the true vertical at a point on the surface. The true vertical, the direction of a plumb line, is also the direction of the gravity acceleration, the resultant of the gravitational acceleration (mass-based) and the centrifugal acceleration at that latitude. [3] Astronomic latitude is calculated from angles measured between the zenith and stars whose declination is accurately known.

In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The angle between the astronomic and geodetic normals is usually a few seconds of arc but it is important in geodesy. [3] [15] The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape "at mean sea level". Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out.

Astronomical latitude is not to be confused with declination, the coordinate astronomers use in a similar way to specify the angular position of stars north/south of the celestial equator (see equatorial coordinates), nor with ecliptic latitude, the coordinate that astronomers use to specify the angular position of stars north/south of the ecliptic (see ecliptic coordinates).

See also

Related Research Articles

Longitude A geographic coordinate that specifies the east-west position of a point on the Earths surface

Longitude, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question.

Mercator projection map projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects, the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.

Spherical coordinate system 3-dimensional 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 measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

Azimuth the angle between a reference plane and a point

An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

Earth radius mean distance from the Earths center to its surface

Earth radius is the distance from the center of Earth to a point on its surface. Its value ranges from 6,378 kilometres at the equator to 6,357 kilometres at a pole.

Rhumb line arc crossing all meridians of longitude at the same angle

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

Great-circle distance shortest distance between two points along the surface of a sphere

The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.

Transverse Mercator projection map projection

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Geodetic datum reference frame used in geodesy, surveying, chartography and navigation

A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used to locate places on the Earth. An approximate definition of sea level is the datum WGS 84, an ellipsoid, whereas a more accurate definition is Earth Gravitational Model 2008 (EGM2008), using at least 2,159 spherical harmonics. Other datums are defined for other areas or at other times; ED50 was defined in 1950 over Europe and differs from WGS 84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there.

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected.

Universal Transverse Mercator coordinate system coordinate system

The Universal Transverse Mercator (UTM) conformal projection uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, i.e. it is used to identify locations on the Earth independently of altitude. However, it differs from that method in several respects.

Cassini projection map projection

The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:

Great ellipse

A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about , the length of the great ellipse connecting the points is close to the geodesic distance.

Geographical distance

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

The article Transverse Mercator projection restricts itself to general features of the projection. This article describes in detail one of the (two) implementations developed by Louis Krüger in 1912; that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Lee in 1946, by Redfearn in 1948, and by Thomas in 1952. They are often referred to as the Redfearn series, or the Thomas series. This implementation is of great importance since it is widely used in the U.S. State Plane Coordinate System, in national and also international mapping systems, including the Universal Transverse Mercator coordinate system (UTM). They are also incorporated into the Geotrans coordinate converter made available by the United States National Geospatial-Intelligence Agency. When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.

In 1989 Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy. Bowring rewrote the fourth order Redfearn series in a more compact notation by replacing the spherical terms, i.e. those independent of ellipticity, by the exact expressions used in the spherical transverse Mercator projection. There was no gain in accuracy since the elliptic terms were still truncated at the 1mm level. Such modifications were of possible use when computing resources were minimal.

Geodesics on an ellipsoid application of geodesics on ellipsoid geometries

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.

References

Footnotes

  1. The current full documentation of ISO 19111 may be purchased from http://www.iso.org but drafts of the final standard are freely available at many web sites, one such is available at the following CSIRO
  2. The value of this angle today is 23°26′12.4″ (or 23.43678°). This figure is provided by Template:Circle of latitude.
  3. An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes.

Citations

  1. Newton, Isaac. "Book III Proposition XIX Problem III". Philosophiæ Naturalis Principia Mathematica. Translated by Motte, Andrew. p. 407.
  2. "TR8350.2". National Geospatial-Intelligence Agency publication. p. 3-1.[ permanent dead link ]
  3. 1 2 3 4 5 Torge, W. (2001). Geodesy (3rd ed.). De Gruyter. ISBN   3-11-017072-8.
  4. 1 2 3 4 5 Osborne, Peter (2013). "Chapters 5,6". The Mercator Projections. doi:10.5281/zenodo.35392. for LaTeX code and figures.
  5. 1 2 3 4 Rapp, Richard H. (1991). "Chapter 3". Geometric Geodesy, Part I. Columbus, OH: Dept. of Geodetic Science and Surveying, Ohio State Univ.
  6. "Length of degree calculator". National Geospatial-Intelligence Agency.
  7. Hopkinson, Sara (2012). RYA day skipper handbook - sail. Hamble: The Royal Yachting Association. p. 76. ISBN   9781-9051-04949.
  8. 1 2 3 4 5 Snyder, John P. (1987). Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395. Washington, DC: United States Government Printing Office.
  9. 1 2 Adams, Oscar S. (1921). Latitude Developments Connected With Geodesy and Cartography (with tables, including a table for Lambert equal area meridional projection (PDF). Special Publication No. 67. US Coast and Geodetic Survey. (Note: Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).)
  10. Legendre, A. M. (1806). "Analyse des triangles tracés sur la surface d'un sphéroïde". Mém. Inst. Nat. Fr. 1st semester: 130–161.
  11. Bessel, F. W. (1825). "Über die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen". Astron. Nachr. 4 (86): 241–254. arXiv: 0908.1824 . Bibcode:2010AN....331..852K. doi:10.1002/asna.201011352.
    Translation:Karney, C. F. F.; Deakin, R. E. (2010). "The calculation of longitude and latitude from geodesic measurements". Astron. Nachr. 331 (8): 852–861. arXiv: 0908.1824 . Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.
  12. Cayley, A. (1870). "On the geodesic lines on an oblate spheroid". Phil. Mag. 40 (4th ser.): 329–340.
  13. Karney, C. F. F. (2013). "Algorithms for geodesics". J. Geodesy. 87 (1): 43–55. arXiv: 1109.4448 . Bibcode:2013JGeod..87...43K. doi:10.1007/s00190-012-0578-z.
  14. "Maxima computer algebra system". Sourceforge .
  15. Hofmann-Wellenhof, B.; Moritz, H. (2006). Physical Geodesy (2nd ed.). ISBN   3-211-33544-7.