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Easting and northing From Wikipedia, the free encyclopedia Jump to: navigation, search It has been suggested that this article be merged into Grid reference. (Discuss  Proposed since June 2015. !he terms easting and northing are geographic "artesian coordinates for a point. #asting refers to the east$ard%measured distance (or the x%coordinate, $hile northing refers to the north$ard%measured distance (or the y%coordinate. &!'%ones #asting and northing coordinates are commonly measured in metres from a horiontal datum. )o$ever, imperial units (e.g., survey feet are also used. !he coordinates are most commonly associated $ith the &niversal !ransverse 'ercator coordinate system (&!', $hich has uni*ue ones that cover the #arth to provide detailed referencing. !he term northing has also been used by e+plorers to describe a general progress to$ard the  orth -ole. Isaac Israel )ayes  used this te rm in an /0 address to the e$%1o rk 2eographical and 3tatistical 3ociety saying, 4!he $ant of steam po$er curtailed my northing.4 56 Notation and conventions[edit] 7ocations can be found using easting8northing (or  x, y pairs. !he pair is usually represented conventionally $ith easting first, northing second. For e+ample, the peak of 'ount 9ssiniboine (at ;< => ;? <@A> ;@?W ;./0ABB< .0;/@<W in &!' Cone is repres ented by 11U 594934 5636174. ther conventions can also be used, such as a truncated grid reference , 5=6  $hich $ould shorten the e+ample coordinates to 949-361.  egative northing and easting values indicate a position due south and $est of the origin, respectively. Universal Transverse Mercator coordinate system
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Easting and northing

From Wikipedia, the free encyclopedia

Jump to: navigation, search It has been suggested that this article be merged into Grid reference. (Discuss

 Proposed since June 2015.

!he terms easting and northing are geographic "artesian coordinates for a point. #asting

refers to the east$ard%measured distance (or the x%coordinate, $hile northing refers to the

north$ard%measured distance (or the y%coordinate.

&!'%ones

#asting and northing coordinates are commonly measured in metres from a horiontal datum.

)o$ever, imperial units (e.g., survey feet are also used. !he coordinates are most commonly

associated $ith the &niversal !ransverse 'ercator coordinate system (&!', $hich has

uni*ue ones that cover the #arth to provide detailed referencing.

!he term northing has also been used by e+plorers to describe a general progress to$ard the orth -ole. Isaac Israel )ayes used this term in an /0 address to the e$%1ork

2eographical and 3tatistical 3ociety saying, 4!he $ant of steam po$er curtailed my

northing.456

Notation and conventions[edit]

7ocations can be found using easting8northing (or x, y pairs. !he pair is usually represented

conventionally $ith easting first, northing second.

For e+ample, the peak of 'ount 9ssiniboine (at

;<=>;? <@A>;@?W ;./0ABB< .0;/@<W in &!' Cone is represented by 11U

594934 5636174. ther conventions can also be used, such as a truncated grid reference,5=6 

$hich $ould shorten the e+ample coordinates to 949-361.

 egative northing and easting values indicate a position due south and $est of the origin,

respectively.

Universal Transverse Mercator coordinate system

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From Wikipedia, the free encyclopedia

 Jump to: navigation, search 

Geodesy

Fundamentals

• Geodesy

• Geodynamics

• Geomatics

• Cartography

• istory

Concepts

• !atum

• Geographical distance

• Geoid

• Figure of the "arth

• Geodetic system

• Geodesic

• Geographic coordinate system

• ori#ontal position representation

$atitude % $ongitude• Map pro&ection

• 'eference ellipsoid

• (atellite geodesy

• (patial reference system

Technologies

• Glo)al *avigation (atellite (ystem +G*((

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• Glo)al -ositioning (ystem +G-(

• G$.*/(( +'ussian

• Galileo +"uropean

• 0ndian 'egional *avigation

(atellite (ystem +0'*((

• 1ei!ou +1!( +Chinese

Standards

"!23 "uropean !atum 4523

(/!65 (outh /merican !atum 4565

G'( 73 Geodetic 'eference (ystem 4573

*/!78 *orth /merican !atum 4578

WG(79 World Geodetic (ystem 4579

*/!77 *; /merican ertical !atum 4577

"T'(75 "uropean Terrestrial 'eference(ystem 4575

GCJ<3= Chinese encrypted datum =33=

• (patial 'eference (ystem 0denti>er +('0!

• Universal Transverse Mercator (UTM)

History

• *G!=5 +(ea $evel !atum 45=5

• v

t

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• e

!he Universal Transverse Mercator (UTM conformal proEection uses a =%dimensional 

"artesian coordinate system to give locations on the surface of the #arth. 7ike the traditionalmethod of latitude and longitude, it is a horiontal position representation, i.e. it is used to

identify locations on the #arth independently of vertical position. )o$ever, it differs from

that method in several respects.

!he &!' system is not a single map proEection. !he system instead divides the #arth into

si+ty ones, each being a si+%degree band of longitude, and uses a secant transverse 'ercator

 proEection in each one.

 The UTM grid;

Contents

 ?hide@

• 4 istory

• = !e>nitions 

o =;4 UTM #one

o =;= .verlapping grids

• 8 $atitude )ands 

o 8;4 $atitude )ands

o 8;= *otation

o 8;8 "Aceptions

• 9 $ocating a position using UTM coordinates 

o 9;4 (impli>ed formulas 

9;4;4 From latitude, longitude +B, to UTM coordinates +", *

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9;4;= From UTM coordinates +", *, Done, emi to latitude,longitude +B,

• 2 (ee also

6 'eferences

• E Further reading

• 7 "Aternal links

istory?edit@

!he &niversal !ransverse 'ercator coordinate system $as developed by the &nited 3tates

9rmy "orps of #ngineers in the AB;s.56 !he system $as based on an ellipsoidal model of

#arth. For areas $ithin the contiguous &nited 3tates the "larke #llipsoid of /005=6 $as used.

For the remaining areas of #arth, including )a$aii, the International #llipsoid5@6 $as used.

!he W23/B ellipsoid is no$ generally used to model the #arth in the &!' coordinate

system, $hich means current &!' northing at a given point can be =;; meters different

from the old. For different geographic regions, other datum systems (e.g.: #D;, 9D/@

can be used.

-rior to the development of the &niversal !ransverse 'ercator coordinate system, several

#uropean nations demonstrated the utility of grid%based conformal maps by mapping their

territory during the inter$ar period. "alculating the distance bet$een t$o points on these

maps could be performed more easily in the field (using the -ythagorean theorem than $as possible using the trigonometric formulas re*uired under the graticule%based system of

latitude and longitude. In the post%$ar years, these concepts $ere e+tended into the &niversal

!ransverse 'ercator 8 &niversal -olar 3tereographic (&!'8&-3 coordinate system, $hich

is a global (or universal system of grid%based maps.

!he transverse 'ercator proEection is a variant of the 'ercator proEection, $hich $as

originally developed by the Flemish geographer and cartographer 2erardus 'ercator , in

G;. !his proEection is conformal, so it preserves angles and appro+imates shape but distorts

distance and area. &!' involves non%linear  anisotropic scaling in both easting and northing 

to ensure the proEected map of the ellipsoid is conformal.

!e>nitions?edit@

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UTM zone[edit

(impli>ed vie of U( UTM #ones, pro&ected ith $am)ert conformal conic;

!he &!' system divides the #arth bet$een /;<3 and /B< latitude into 0; ones, each 0< of

longitude in $idth. Cone covers longitude /;< to GB< WH one numbering increases

east$ard to one 0; that covers longitude GB to /; #ast.

#ach of the 0; ones uses a transverse 'ercator  proEection that can map a region of large

north%south e+tent $ith lo$ distortion. y using narro$ ones of 0< of longitude (up to

/;; km in $idth, and reducing the scale factor along the central meridian to ;.AAA0 (a

reduction of :=;;, the amount of distortion is held belo$ part in ,;;; inside each one.

Distortion of scale increases to .;;; at the one boundaries along the e*uator .

In each one the scale factor of the central meridian reduces the diameter of the transverse

cylinder to produce a secant proEection $ith t$o standard lines, or lines of true scale, about

/; km on each side of, and about parallel to, the central meridian (9rc cos ;.AAA0 .0=< atthe #*uator. !he scale is less than inside the standard lines and greater than outside them,

 but the overall distortion is minimied.

!verlapping grids[edit

Distortion of scale increases in each &!' one as the boundaries bet$een the &!' ones

are approached. )o$ever, it is often convenient or necessary to measure a series of locations

on a single grid $hen some are located in t$o adEacent ones. 9round the boundaries of large

scale maps (:;;,;;; or larger coordinates for both adEoining &!' ones are usually

 printed $ithin a minimum distance of B; km on either side of a one boundary. Ideally, the

coordinates of each position should be measured on the grid for the one in $hich they arelocated, but because the scale factor is still relatively small near one boundaries, it is

 possible to overlap measurements into an adEoining one for some distance $hen necessary.

$atitude )ands?edit@

7atitude bands are not a part of &!', but rather a part of '2K3.5B6 !hey are ho$ever

sometimes used.

"atitude #ands[edit

#ach one is segmented into =; latitude bands. #ach latitude band is / degrees high, and islettered starting from 4"4 at /;<3, increasing up the #nglish alphabet until 4L4, omitting the

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letters 4I4 and 44 (because of their similarity to the numerals one and ero. !he last latitude

 band, 4L4, is e+tended an e+tra B degrees, so it ends at /B< latitude, thus covering the

northernmost land on #arth. 7atitude bands 494 and 44 do e+ist, as do bands 414 and 4C4.

!hey cover the $estern and eastern sides of the 9ntarctic and 9rctic regions respectively. 9

convenient mnemonic to remember is that the letter 44 is the first letter in 4northernhemisphere4, so any letter coming before 44 in the alphabet is in the southern hemisphere,

and any letter 44 or after is in the northern hemisphere.

$otation[edit

!he combination of a one and a latitude band defines a grid one. !he one is al$ays $ritten

first, follo$ed by the latitude band. For e+ample (see image, top right, a position in !oronto,

"anada, $ould find itself in one G and latitude band 4!4, thus the full grid one reference is

4G!4. !he grid ones serve to delineate irregular &!' one boundaries. !hey also are an

integral part of the military grid reference system.

9 note of caution: 9 method also is used that simply adds or 3 follo$ing the one number

to indicate orth or 3outh hemisphere (the easting and northing coordinates along $ith the

one number supplying everything necessary to geolocate a position e+cept $hich

hemisphere. )o$ever, this method has caused some confusion since, for instance, 4;34 can

mean southern hemisphere but also grid zone 4;34 in the northern hemisphere.56 !here are

many possible $ays to disambiguate bet$een the t$o methods, t$o of $hich are

demonstrated later in this article.

%&ceptions[edit

!hese grid ones are uniform over the globe, e+cept in t$o areas. n the south$est coast of

 or$ay, grid one @=M (A< of longitude in $idth is e+tended further $est, and grid one

@M (@< of longitude in $idth is correspondingly shrunk to cover only open $ater. 9lso, in

the region around 3valbard, the four grid ones @L (A< of longitude in $idth, @@L (=< of

longitude in $idth, @L (=< of longitude in $idth, and @GL (A< of longitude in $idth are

e+tended to cover $hat $ould other$ise have been covered by the seven grid ones @L to

@GL. !he three grid ones @=L, @BL and @0L are not used.

#urope

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9frica

3outh 9merica

ering 3ea $ith 9laska

-icture gallery: 2rid ones in various parts of the $orld

$ocating a position using UTM coordinates?edit@

9 position on the #arth is given by the &!' one number and the easting and northing 

coordinate pair in that one. !he point of origin of each &!' one is the intersection of the

e*uator and the oneNs central meridian, but to avoid dealing $ith negative numbers the

central meridian of each one is set at ;;,;;; meters #ast. In any one a point that has an

easting of B;;,;;; meters is ;; km $est of the central meridian, measured on the transverse

'ercator proEection (or slightly more than ;; km measured on the actual surface of the

#arth. &!' eastings range from about 0G,;;; meters (near the poles to /@@,;;; meters at

the e*uator. In the northern hemisphere positions are measured north$ard from ero at the

e*uatorH the ma+imum 4northing4 value is about A,@;;,;;; meters at latitude /B degrees

 orth, the north end of the &!' ones. In the southern hemisphere northings decrease

south$ard from the e*uator to about ,;;,;;; metres at /; degrees 3outh, the south end of

the &!' onesH the northing at the e*uator is set at ;,;;;,;;; meters so no point has a

negative northing value.

!he " !o$er  is at

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9887H88;=9I* E5=8H48;EIW % 98;69=266E* E5;87E485W, hich is in UTM

#one 4E, and the grid position is 683379m east, 9788987m north; To points in

Done 4E have these coordinates, one in the northern hemisphere and one in the

south one of to conventions is used to say hich:

4; /ppend a hemisphere designator to the #one num)er, K*K or K(K, thusK4E* 683379 9788987K; This supplies the minimum information to de>nethe position uniLuely;

=; (upply the grid #one, i;e;, the latitude )and designator appended to the#one num)er, thus K4ET 683379 9788987K; The provision of the latitude)and along ith northing supplies redundant information +hich may, as aconseLuence, )e contradictory;

ecause latitude band 434 is in the northern hemisphere, a designation such as 4@/34 is

unclear. !he 434 might refer to the latitude band (@=< O B;< or it might mean 43outh4. It is

therefore important to specify $hich convention is being used, e.g., by spelling out the

hemisphere, 4orth4 or 43outh4, or using different symbols, such as P for south and for

north.

Simpli'ed ormulas[edit

!hese formulas are truncated version of !ransverse 'ercator: flattening series, $hich $ere

originally derived by Johann )einrich 7ouis QrRger  in A=.506 !hey are accurate to around a

millimeter  $ithin @,;;; km of the central meridian.5G6 "oncise commentaries for their

derivation have also been given.5/65A6

!he W23 /B spatial reference system describes #arth as an oblate spheroid along north%south

a+is $ith an e*uatorial radius of km and an inverse flattening of 

. 7etNs take a point of latitude and of longitude and compute

its &!' coordinates as $ell as point scale factor   and meridian convergence  using a

reference meridian of longitude . y convention, in the northern hemisphere  km

and in the southern hemisphere  km. y convention also and

km.

In the follo$ing formulas, the distances are in kilometers. In advance letNs compute some preliminary values:

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From latitude, longitude +B, to UTM coordinates +", *?edit@

First letNs compute some intermediate values:

!he final formulas are:

From UTM coordinates +", *, Done, emi to latitude, longitude +B, ?edit@

 ote: )emi for orthern, )emi% for 3outhern

First letNs compute some intermediate values:

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"ongitude ()

$ines of longitude appear vertical ith varying

curvature in this pro&ection, )ut are actually

halves of great ellipses, ith identical radii at agiven latitude;

"atitude (*)

$ines of latitude appear hori#ontal ith varying

curvature in this pro&ection )ut are actually

circular ith dierent radii; /ll locations ith a

given latitude are collectively referred to as a

circle of latitude;

 The e+uator divides the planet into a *orthern

emisphere and a (outhern emisphere, and

has a latitude of 3;

• v

• t

• e

Geodesy

Fundamentals

• Geodesy

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• Geodynamics

• Geomatics

• Cartography

• istory

Concepts

• !atum

• Geographical distance

• Geoid

• Figure of the "arth

• Geodetic system

• Geodesic

Geographic coordinate system• ori#ontal position representation

• $atitude % "ongitude

• Map pro&ection

• 'eference ellipsoid

• (atellite geodesy

• (patial reference system

Technologies

• Glo)al *avigation (atellite (ystem +G*((

• Glo)al -ositioning (ystem +G-(

• G$.*/(( +'ussian

• Galileo +"uropean

• 0ndian 'egional *avigation

(atellite (ystem +0'*((

• 1ei!ou +1!( +Chinese

Standards

"!23 "uropean !atum 4523

(/!65 (outh /merican !atum 4565

G'( 73 Geodetic 'eference (ystem 4573

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*/!78 *orth /merican !atum 4578

WG(79 World Geodetic (ystem 4579

*/!77 *; /merican ertical !atum 4577

"T'(75 "uropean Terrestrial 'eference(ystem 4575

GCJ<3= Chinese encrypted datum =33=

• (patial 'eference (ystem 0denti>er +('0!

• Universal Transverse Mercator +UTM

History

• *G!=5 +(ea $evel !atum 45=5

• v

• t

• e

Longitude (8 l nd tEu d8ˈ ɒ ʒɨ ː  or 8 l nd tu d8ˈ ɒ ʒɨ ː , ritish also 8 l S tEu d8ˈ ɒ ɡɨ ː ,56 is a geographic

coordinate that specifies the east%$est position of a point on the #arthNs surface. It is an

angular measurement, usually e+pressed in degrees and denoted by the 2reek letter  lambda 

(T. -oints $ith the same longitude lie in lines running from the orth -ole to the 3outh -ole.

y convention, one of these, the -rime 'eridian, $hich passes through the Koyalbservatory, 2reen$ich, #ngland, $as intended to establish the position of ero degrees

longitude. !he longitude of other places $as to be measured as the angle east or $est from

the -rime 'eridian, ranging from ;< at the -rime 'eridian to /;< east$ard and P/;<

$est$ard. 3pecifically, it is the angle bet$een a plane containing the -rime 'eridian and a

 plane containing the orth -ole, 3outh -ole and the location in *uestion. (!his forms a right%

handed coordinate system $ith the z  a+is (right hand thumb pointing from the #arthNs center

to$ard the orth -ole and the x a+is (right hand inde+ finger e+tending from #arthNs center

through the e*uator at the -rime 'eridian.

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9 locationNs northOsouth position along a meridian is given by its latitude, $hich is (not *uite

e+actly the angle bet$een the local vertical and the plane of the #*uator.

If the #arth $ere perfectly spherical and homogeneous, then longitude at a point $ould Eust

 be the angle bet$een a vertical northOsouth plane through that point and the plane of the

2reen$ich meridian. #very$here on #arth the vertical northOsouth plane $ould contain the

#arthNs a+is. ut the #arth is not homogeneous, and has mountainsU$hich have gravity and

so can shift the vertical plane a$ay from the #arthNs a+is. !he vertical northOsouth plane still

intersects the plane of the 2reen$ich meridian at some angleH that angle is astronomical

longitude, the longitude you calculate from star observations. !he longitude sho$n on maps

and 2-3 devices is the angle bet$een the 2reen$ich plane and a not%*uite%vertical plane

through the pointH the not%*uite%vertical plane is perpendicular to the surface of the spheroid

chosen to appro+imate the #arthNs sea%level surface, rather than perpendicular to the sea%level

surface itself.

Contents

 ?hide@

• 4 istory

• = *oting and calculating longitude 

o =;4 (ingularity and discontinuity of longitude

• 8 -late movement and longitude

• 9 $ength of a degree of longitude

• 2 $ongitude on )odies other than "arth

• 6 (ee also

• E 'eferences

• 7 "Aternal links

istory?edit@

!ain arti"le# Histor$ of longitude

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/merigo espucciNs means of determining longitude

!he measurement of longitude is important both to cartography and for ocean navigation.

'ariners and e+plorers for most of history struggled to determine longitude. Finding a

method of determining longitude took centuries, resulting in the history of longitude

recording the effort of some of the greatest scientific minds.

7atitude $as calculated by observing $ith *uadrant or astrolabe the altitude of the sun or of

charted stars above the horion, but longitude is harder.

9merigo Mespucci $as perhaps the first #uropean to proffer a solution, after devoting a great

deal of time and energy studying the problem during his soEourns in the e$ World:

 As to longitude, I declare that I found so uch difficulty in deterining it that I !as put to

 great pains to ascertain the east"!est distance I had co#ered. $he final result of y la%ours

!as that I found nothing %etter to do than to !atch for and ta&e o%ser#ations at night of the

con'unction of one planet !ith another, and especially of the con'unction of the oon !ith

the other planets, %ecause the oon is s!ifter in her course than any other planet. I

copared y o%ser#ations !ith an alanac. After I had ade experients any nights, one

night, the t!enty"third of August 1()), there !as a con'unction of the oon !ith *ars, !hich

according to the alanac !as to occur at idnight or a half hour %efore. I found that...at

idnight *ars+s position !as three and a half degrees to the east .5=6

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 John arrison solved the greatest pro)lem of his day;?8@

y comparing the positions of the moon and 'ars $ith their anticipated positions, Mespucci

$as able to crudely deduce his longitude. ut this method had several limitations: First, it

re*uired the occurrence of a specific astronomical event (in this case, 'ars passing through

the same right ascension as the moon, and the observer needed to anticipate this event via an

astronomical almanac. ne needed also to kno$ the precise time, $hich $as difficult to

ascertain in foreign lands. Finally, it re*uired a stable vie$ing platform, rendering the

techni*ue useless on the rolling deck of a ship at sea. 3ee 7unar distance (navigation.

In 0= 2alileo 2alilei demonstrated that $ith sufficiently accurate kno$ledge of the orbits

of the moons of Jupiter one could use their positions as a universal clock and this $ould

make possible the determination of longitude, but the method he devised $as impracticable

for navigators on ships because of their instability.5B6 In GB the ritish government passed

the 7ongitude 9ct $hich offered large financial re$ards to the first person to demonstrate a practical method for determining the longitude of a ship at sea. !hese re$ards motivated

many to search for a solution.

!raing of "arth ith longitudes

John )arrison, a self%educated #nglish clockmaker , invented the marine chronometer , a key

 piece in solving the problem of accurately establishing longitude at sea, thus revolutionising

and e+tending the possibility of safe long distance sea travel. 5@6 !hough the oard of7ongitude re$arded John )arrison for his marine chronometer in GG@, chronometers

remained very e+pensive and the lunar distance method continued to be used for decades.

Finally, the combination of the availability of marine chronometers and $ireless telegraph 

time signals put an end to the use of lunars in the =;th century.

&nlike latitude, $hich has the e*uator as a natural starting position, there is no natural

starting position for longitude. !herefore, a reference meridian had to be chosen. It $as a

 popular practice to use a nationNs capital as the starting point, but other locations $ere also

used. While ritish cartographers had long used the 2reen$ich meridian in 7ondon, other

references $ere used else$here, including: #l )ierro, Kome, "openhagen, Jerusalem, 3aint

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-etersburg, -isa, -aris, -hiladelphia, and Washington D.". In //B the International 'eridian

"onference adopted the 2reen$ich meridian as the uni#ersal Prie *eridian or zero point of 

longitude.

*oting and calculating longitude?edit@

7ongitude is given as an angular measurement ranging from ;< at the -rime 'eridian to

/;< east$ard and P/;< $est$ard. !he 2reek letter T (lambda,56506 is used to denote the

location of a place on #arth east or $est of the -rime 'eridian.

#ach degree of longitude is sub%divided into 0; minutes, each of $hich is divided into 0;

seconds. 9 longitude is thus specified in se+agesimal notation as 2- 2/ 0  . For higher

 precision, the seconds are specified $ith a decimal fraction. 9n alternative representation

uses degrees and minutes, $here parts of a minute are e+pressed in decimal notation $ith a

fraction, thus: 2- 2.500/  . Degrees may also be e+pressed as a decimal fraction:

2.(5-  . For calculations, the angular measure may be converted to radians, so longitude

may also be e+pressed in this manner as a signed fraction of V ( pi, or an unsigned fraction of

=V.

For calculations, the West8#ast suffi+ is replaced by a negative sign in the $estern

hemisphere. "onfusingly, the convention of negative for #ast is also sometimes seen. !he

 preferred conventionUthat #ast be positiveUis consistent $ith a right%handed "artesian

coordinate system, $ith the orth -ole up. 9 specific longitude may then be combined $ith a

specific latitude (usually positive in the northern hemisphere to give a precise position on the

#arthNs surface.

7ongitude at a point may be determined by calculating the time difference bet$een that at its

location and "oordinated &niversal !ime (&!". 3ince there are =B hours in a day and @0;

degrees in a circle, the sun moves across the sky at a rate of degrees per hour (@0;<8=B

hours < per hour. 3o if the time one a person is in is three hours ahead of &!" then that

 person is near B< longitude (@ hours < per hour B<. !he $ord near  $as used because

the point might not be at the center of the time oneH also the time ones are defined

 politically, so their centers and boundaries often do not lie on meridians at multiples of <.

In order to perform this calculation, ho$ever, a person needs to have a chronometer  ($atchset to &!" and needs to determine local time by solar or astronomical observation. !he

details are more comple+ than described here: see the articles on &niversal !ime and on the

e*uation of time for more details.

Singularity and discontinuity o longitude[edit

 ote that the longitude is singular  at the -oles and calculations that are sufficiently accurate

for other positions, may be inaccurate at or near the -oles. 9lso the discontinuity at the X/;<

meridian must be handled $ith care in calculations. 9n e+ample is a calculation of east

displacement by subtracting t$o longitudes, $hich gives the $rong ans$er if the t$o

 positions are on either side of this meridian. !o avoid these comple+ities, consider replacinglatitude and longitude $ith another horiontal position representation in calculation.

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-late movement and longitude?edit@

!he #arthNs tectonic plates move relative to one another in different directions at speeds on

the order of ; to ;;mm per year .5G6 3o points on the #arthNs surface on different plates are

al$ays in motion relative to one another, for e+ample, the longitudinal difference bet$een a

 point on the #*uator in &ganda, on the 9frican -late, and a point on the #*uator in #cuador,

on the 3outh 9merican -late, is increasing by about ;.;;B arcseconds per year. !hese

tectonic movements like$ise affect latitude.

If a global reference frame such as W23/B is used, the longitude of a place on the surface

$ill change from year to year. !o minimie this change, $hen dealing Eust $ith points on a

single plate, a different reference frame can be used, $hose coordinates are fi+ed to a

 particular plate, such as 9D/@ for orth 9merica or #!K3/A for #urope.

$ength of a degree of longitude?edit@

!he length of a degree of longitude depends only on the radius of a circle of latitude. For a

sphere of radius a that radius at latitude Y is (cos Y times a, and the length of a one%degree

(or V8/; radians arc along a circle of latitude is

3 443;2E9 km 444;8=3 km

42 443;695 km 43E;224 km

83 443;72= km 56;976 km

92 444;48= km E7;79E km

63 444;94= km 22;733 km

E2 444;647 km =7;53= km

53 444;659 km 3;333 km

When the #arth is modelled by an ellipsoid this arc length becomes5/65A6

$here e, the eccentricity of the ellipsoid, is related to the maEor and minor a+es (the

e*uatorial and polar radii respectively by

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9n alternative formula is

here

"os Y decreases from at the e*uator to ero at the poles, so the length of a degree of

longitude decreases like$ise. !his contrasts $ith the small (Z increase in the length of a

degree of latitude, e*uator to pole. !he table sho$s both for the W23/B ellipsoid $ith a 

0,@G/,@G.; m and %  0,@0,G=.@B= m. ote that the distance bet$een t$o points degree

apart on the same circle of latitude, measured along that circle of latitude, is slightly more

than the shortest (geodesic distance bet$een those pointsH the difference is less than ;.0 m.

$ongitude on )odies other than "arth?edit@

See also# %ri&e &eridian (planets)

-lanetary co%ordinate systems are defined relative to their mean a+is of rotation and various

definitions of longitude depending on the body. !he longitude systems of most of those

 bodies $ith observable rigid surfaces have been defined by references to a surface feature

such as a crater . !he north pole is that pole of rotation that lies on the north side of the

invariable plane of the solar system (near the ecliptic. !he location of the -rime 'eridian as

$ell as the position of bodyNs north pole on the celestial sphere may vary $ith time due to

 precession of the a+is of rotation of the planet (or satellite. If the position angle of the bodyNs-rime 'eridian increases $ith time, the body has a direct (or   prograde rotationH other$ise

the rotation is said to be retrograde.

In the absence of other information, the a+is of rotation is assumed to be normal to the mean

orbital planeH 'ercury and most of the satellites are in this category. For many of the

satellites, it is assumed that the rotation rate is e*ual to the mean orbital period. In the case of

the giant planets, since their surface features are constantly changing and moving at various

rates, the rotation of their magnetic fields is used as a reference instead. In the case of the

3un, even this criterion fails (because its magnetosphere is very comple+ and does not reallyrotate in a steady fashion, and an agreed%upon value for the rotation of its e*uator is used

instead.

For planetographic longitude, $est longitudes (i.e., longitudes measured positively to the

$est are used $hen the rotation is prograde, and east longitudes (i.e., longitudes measured

 positively to the east $hen the rotation is retrograde. In simpler terms, imagine a distant,

non%orbiting observer vie$ing a planet as it rotates. 9lso suppose that this observer is $ithin

the plane of the planetNs e*uator. 9 point on the #*uator that passes directly in front of this

observer later in time has a higher planetographic longitude than a point that did so earlier in

time.

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)o$ever, planetocentric longitude is al$ays measured positively to the east, regardless of

$hich $ay the planet rotates. ast  is defined as the counter%clock$ise direction around the

 planet, as seen from above its north pole, and the north pole is $hichever pole more closely

aligns $ith the #arthNs north pole. 7ongitudes traditionally have been $ritten using 4#4 or

4W4 instead of 44 or 4P4 to indicate this polarity. For e+ample, the follo$ing all mean thesame thing:

• O54

• 54W

• P=65

• =65";

!he reference surfaces for some planets (such as #arth and 'ars are ellipsoids of revolutionfor $hich the e*uatorial radius is larger than the polar radiusH in other $ords, they are oblate

spheroids. 3maller bodies (Io, 'imas, etc. tend to be better appro+imated by tria+ial

ellipsoidsH ho$ever, tria+ial ellipsoids $ould render many computations more complicated,

especially those related to map proEections. 'any proEections $ould lose their elegant and

 popular properties. For this reason spherical reference surfaces are fre*uently used in

mapping programs.

!he modern standard for maps of 'ars (since about =;;= is to use planetocentric

coordinates. !he meridian of 'ars is located at 9iry%; crater.5;6

!idally%locked  bodies have a natural reference longitude passing through the point nearest to

their parent body: ;< the center of the primary%facing hemisphere, A;< the center of the

leading hemisphere, /;< the center of the anti%primary hemisphere, and =G;< the center of

the trailing hemisphere.56 )o$ever, libration due to non%circular orbits or a+ial tilts causes

this point to move around any fi+ed point on the celestial body like an analemma.

$atitude

From Wikipedia, the free encyclopedia

 Jump to: navigation, search 

This arti"le is about the geographi"al referen"e s$ste&. For other uses, see

Latitude (disa&biguation).

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/ graticule on the "arth as a sphere or an ellipsoid; The lines from pole to pole

are lines of constant longitude, or meridians; The circles parallel to the eLuator 

are lines of constant latitude, or parallels; The graticule determines the latitude

and longitude of points on the surface; 0n this eAample meridians are spaced at

6 intervals and parallels at 9 intervals;

In geography, latitude (Y is a geographic coordinate that specifies the north%south position

of a point on the #arthNs surface. 7atitude is an angle (defined belo$ $hich ranges from ;< at

the #*uator  to A;< (orth or 3outh at the poles. 7ines of constant latitude, or parallels, run

east%$est as circles parallel to the e*uator. 7atitude is used together $ith longitude to specify

the precise location of features on the surface of the #arth. !$o levels of abstraction are

employed in the definition of these coordinates. In the first step the physical surface is

modelled by the geoid, a surface $hich appro+imates the mean sea level over the oceans and

its continuation under the land masses. !he second step is to appro+imate the geoid by a

mathematically simpler reference surface. !he simplest choice for the reference surface is a

sphere, but the geoid is more accurately modelled by an ellipsoid. !he definitions of latitude

and longitude on such reference surfaces are detailed in the follo$ing sections. 7ines of

constant latitude and longitude together constitute a graticule on the reference surface. !he

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 $hich passes

through the point on the physical surface. 7atitude and longitude together $ith some

specification of height constitute a geographic coordinate system as defined in the

specification of the I3 A standard.56

3ince there are many different reference ellipsoids the latitude of a feature on the surface is

not uni*ue: this is stressed in the I3 standard $hich states that 4$ithout the full

specification of the coordinate reference system, coordinates (that is latitude and longitude

are ambiguous at best and meaningless at $orst4. !his is of great importance in accurate

applications, such as 2-3, but in common usage, $here high accuracy is not re*uired, the

reference ellipsoid is not usually stated.

In #nglish te+ts the latitude angle, defined belo$, is usually denoted by the 2reek lo$er%case

letter phi (Y or ɸ. It is measured in degrees, minutes and seconds or decimal degrees, north

or south of the e*uator.

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'easurement of latitude re*uires an understanding of the gravitational field of the #arth,

either for setting up theodolites or for determination of 2-3 satellite orbits. !he study of the

figure of the #arth together $ith its gravitational field is the science of geodesy. !hese topics

are not discussed in this article. (3ee for e+ample the te+tbooks by !orge5=6 and )ofmann%

Wellenhof and 'orit.

5@6

!his article relates to coordinate systems for the #arth: it may be e+tended to cover the 'oon,

 planets and other celestial obEects by a simple change of nomenclature.

!he follo$ing lists are available:

• $ist of cities )y latitude

• $ist of countries )y latitude

Contents

 ?hide@

• 4 $atitude on the sphere 

o 4;4 The graticule on the sphere

o 4;= *amed latitudes

o 4;8 Map pro&ections from the sphere

o 4;9 Meridian distance on the sphere

• = $atitude on the ellipsoid 

o =;4 "llipsoids

o =;= The geometry of the ellipsoid

o =;8 Geodetic and geocentric latitudes

o =;9 $ength of a degree of latitude

• 8 /uAiliary latitudes 

o 8;4 Geocentric latitude

o 8;= 'educed +or parametric latitude

o 8;8 'ectifying latitude

o 8;9 /uthalic latitude

o 8;2 Conformal latitude

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o 8;6 0sometric latitude

o 8;E 0nverse formulae and series

o 8;7 *umerical comparison of auAiliary latitudes

• 9 $atitude and coordinate systems 

o 9;4 Geodetic coordinates

o 9;= (pherical polar coordinates

o 9;8 "llipsoidal coordinates

o 9;9 Coordinate conversions

• 2 /stronomical latitude

• 6 (ee also

• E Footnotes

• 7 "Aternal links

$atitude on the sphere?edit@

/ perspective vie of the "arth shoing ho latitude +B and longitude + arede>ned on a spherical model; The graticule spacing is 43 degrees;

The graticule on the sphere[edit

!he graticule formed by the lines of constant latitude and constant longitude is constructed

$ith reference to the rotation a+is of the #arth. !he primary reference points are the  poles 

$here the a+is of rotation of the #arth intersects the reference surface. -lanes $hich contain

the rotation a+is intersect the surface in the meridians and the angle bet$een any one

meridian plane and that through 2reen$ich (the -rime 'eridian defines the longitude:

meridians are lines of constant longitude. !he plane through the centre of the #arth and

orthogonal to the rotation a+is intersects the surface in a great circle called the e*uator . -lanes parallel to the e*uatorial plane intersect the surface in circles of constant latitudeH these are

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the parallels. !he e*uator has a latitude of ;<, the  orth -ole has a latitude of A;< north

($ritten A;< or A;<, and the 3outh -ole has a latitude of A;< south ($ritten A;< 3 or

PA;<. !he latitude of an arbitrary point is the angle bet$een the e*uatorial plane and the

radius to that point.

!he latitude that is defined in this $ay for the sphere is often termed the spherical latitude to

avoid ambiguity $ith au+iliary latitudes defined in subse*uent sections.

$amed latitudes[edit

 The orientation of the "arth at the !ecem)er solstice;

esides the e*uator, four other parallels are of significance:

/rctic Circle 66 88H 85I *

 Tropic of Cancer =8 =6H =4I *

 Tropic of Capricorn =8 =6H =4I (

/ntarctic Circle 66 88H 85K (

!he plane of the #arthNs orbit about the sun is called the ecliptic. !he plane perpendicular to

the rotation a+is of the #arth is the e*uatorial plane. !he angle bet$een the ecliptic and the

e*uatorial plane is called the inclination of the ecliptic, denoted by in the figure. !he current

value of this angle is =@< =0> =?.5B6 It is also called the a+ial tilt of the #arth since it is e*ual to

the angle bet$een the a+is of rotation and the normal to the ecliptic.

!he figure sho$s the geometry of a cross section of the plane normal to the ecliptic and

through the centres of the #arth and the 3un at the December solstice $hen the sun is

overhead at some point of the !ropic of "apricorn. !he south polar latitudes belo$ the

9ntarctic "ircle are in daylight $hilst the north polar latitudes above the 9rctic "ircle are in

night. !he situation is reversed at the June solstice $hen the sun is overhead at the !ropic of

"ancer. !he latitudes of the tropics are e*ual to the inclination of the ecliptic and the polar

circles are at latitudes e*ual to its complement. nly at latitudes in bet$een the t$o tropics isit possible for the sun to be directly overhead (at the enith.

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!he named parallels are clearly indicated on the 'ercator proEections sho$n belo$.

Map pro,ections rom the sphere[edit

n map proEections there is no simple rule as to ho$ meridians and parallels should appear.

For e+ample, on the spherical 'ercator proEection the parallels are horiontal and the

meridians are vertical $hereas on the !ransverse 'ercator proEection there is no correlation

of parallels and meridians $ith horiontal and verticalH both are complicated curves. !he red

lines are the named latitudes of the previous section.

$ormal Mercator Transverse Mercator

For map proEections of large regions, or the $hole $orld, a spherical #arth model is

completely satisfactory since the variations attributable to ellipticity are negligible on the

final printed maps.

Meridian distance on the sphere[edit

n the sphere the normal passes through the centre and the latitude (Y is therefore e*ual to

the angle subtended at the centre by the meridian arc from the e*uator to the point concerned.

If the meridian distance is denoted by (Y then

$here K denotes the mean radius of the #arth. K is e*ual to 0,@G km or @,AA miles. o

higher accuracy is appropriate for K since higher precision results necessitate an ellipsoid

model. With this value for K the meridian length of degree of latitude on the sphere is

.= km or 0A miles. !he length of minute of latitude is ./@ km, or . miles. (3ee

nautical mile.

$atitude on the ellipsoid?edit@

 This article duplicates the scope o other articles, speci>cally,

'eference ellipsoidQ"llipsoid parameters; -lease discuss this issue onthe talk page and conform ith WikipediaNs Manual of (tyle )y

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replacing the section ith a link and a summary of the repeated

material, or )y spinning o the repeated teAt into an article in its on

right; (June *)

%llipsoids[editIn 0/G Isaac e$ton published the Philosophi3 4aturalis Principia *atheatica in $hich

he proved that a rotating self%gravitating fluid body in e*uilibrium takes the form of an oblate

ellipsoid.56 (!his article uses the term ellipsoid in preference to the older term spheroid .

 e$tonNs result $as confirmed by geodetic measurements in the eighteenth century. (3ee

'eridian arc. 9n oblate ellipsoid is the three%dimensional surface generated by the rotation

of an ellipse about its shorter a+is (minor a+is. 4blate ellipsoid of revolution4 is abbreviated

to ellipsoid in the remainder of this article. (#llipsoids $hich do not have an a+is of

symmetry are termed tri%a+ial.

'any different reference ellipsoids have been used in the history of geodesy. In pre%satellite

days they $ere devised to give a good fit to the geoid over the limited area of a survey but,

$ith the advent of 2-3, it has become natural to use reference ellipsoids (such as W23/B

$ith centres at the centre of mass of the #arth and minor a+is aligned to the rotation a+is of

the #arth. !hese geocentric ellipsoids are usually $ithin ;;m of the geoid. 3ince latitude is

defined $ith respect to an ellipsoid, the position of a given point is different on each

ellipsoid: one canNt e+actly specify the latitude and longitude of a geographical feature

$ithout specifying the ellipsoid used. 'any maps maintained by national agencies are based

on older ellipsoids so it is necessary to kno$ ho$ the latitude and longitude values are

transformed from one ellipsoid to another. 2-3 handsets include soft$are to carry out datumtransformations $hich link W23/B to the local reference ellipsoid $ith its associated grid.

The geometry o the ellipsoid[edit

!he shape of an ellipsoid of revolution is determined by the shape of the ellipse $hich is

rotated about its minor (shorter a+is. !$o parameters are re*uired. ne is invariably the

e*uatorial radius, $hich is the semi%maEor a+is, a. !he other parameter is usually ( the polar 

radius or semi%minor a+is, %H or (= the (first flattening, f H or (@ the eccentricity, e. !hese

 parameters are not independent: they are related by

'any other parameters (see ellipse, ellipsoid appear in the study of geodesy, geophysics and

map proEections but they can all be e+pressed in terms of one or t$o members of the set a, %,

 f  and e. oth f  and e are small and often appear in series e+pansions in calculationsH they are

of the order 8@;; and ;.;/, respectively. Malues for a number of ellipsoids are given in

Figure of the #arth. Keference ellipsoids are usually defined by the semi%maEor a+is and the

in#erse flattening, 1f . For e+ample, the defining values for the W23/B ellipsoid, used by all

2-3 devices, are506

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• a +eLuatorial radius: 6,8E7,48E;3 m eAactly

• +f  +inverse Rattening: =57;=2E==8268 eAactly

from $hich are derived

• b +polar radius: 6,826,E2=;849= m

• e= +eccentricity sLuared: 3;3366598E555349

!he difference of the maEor and minor semi%a+es is about = km and as fraction of the semi%

maEor a+is it e*uals the flatteningH on a computer the ellipsoid could be sied as @;;p+ by

=AAp+. !his $ould be indistinguishable from a sphere sho$n as @;;p+ by @;;p+, so

illustrations al$ays e+aggerate the flattening.

Geodetic and geocentric latitudes[edit

 The definition of geodetic latitude +B and longitude + on an ellipsoid; The

normal to the surface does not pass through the centre, eAcept at the eLuator

and at the poles;

!he graticule on the ellipsoid is constructed in e+actly the same $ay as on the sphere. !he

normal at a point on the surface of an ellipsoid does not pass through the centre, e+cept for

 points on the e*uator or at the poles, but the definition of latitude remains unchanged as the

angle bet$een the normal and the e*uatorial plane. !he terminology for latitude must be

made more precise by distinguishing

Geodetic latitude- the angle )eteen the normal and the eLuatorial

plane; The standard notation in "nglish pu)lications is B; This is the

de>nition assumed hen the ord latitude is used ithout Luali>cation;

 The de>nition must )e accompanied ith a speci>cation of the ellipsoid;

Geocentric latitude- the angle )eteen the radius +from centre to the

point on the surface and the eLuatorial plane; +Figure )elo; There is no

standard notation: eAamples from various teAts include S, , BN, Bc, Bg; This

article uses S;

Spherical latitude- the angle )eteen the normal to a sphericalreference surface and the eLuatorial plane;

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Geographic latitude must )e used ith care; (ome authors use it as a

synonym for geodetic latitude hilst others use it as an alternative to the

astronomical latitude;

"atitude +unLuali>ed should normally refer to the geodetic latitude;

!he importance of specifying the reference datum may be illustrated by a simple e+ample.

n the reference ellipsoid for W23/B, the centre of the #iffel !o$er  has a geodetic latitude

of B/< > =A? , or B/.//@< and longitude of =< G> B;? # or =.=ABB<#. !he same

coordinates on the datum #D; define a point on the ground $hich is B; m distant from

!o$er.5citation needed 6 9 $eb search may produce several different values for the latitude of the

!o$erH the reference ellipsoid is rarely specified.

"ength o a degree o latitude[edit

!ain arti"le# !eridian ar"

0t has )een suggested that portions of this section )e moved into

!eridian ar"; +!iscuss

In 'eridian arc and standard te+ts5=65G65/6 it is sho$n that the distance along a meridian from

latitude Y to the e*uator is given by (Y in radians

!he function in the first integral is the meridional radius of curvature.

!he distance from the e*uator to the pole is

For W23/B this distance is ;,;;.A0G=A km.

!he evaluation of the meridian distance integral is central to many studies in geodesy and

map proEection. It can be evaluated by e+panding the integral by the binomial series and

integrating term by term: see 'eridian arc for details. !he length of the meridian arc bet$eent$o given latitudes is given by replacing the limits of the integral by the latitudes concerned.

!he length of a sall  meridian arc is given by5G65/6

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When the latitude difference is degree, corresponding to 8/; radians, the arc distance is

about

!he distance in metres (correct to ;.; metre bet$een latitudes ( deg and (

 deg on the W23/B spheroid is

!he variation of this distance $ith latitude (on W23/B is sho$n in the table along $ith the

length of a degree of longitude:

9 calculator for any latitude is provided by the &.3. governmentNs ational 2eospatial%

Intelligence 9gency (29.5A6

/uAiliary latitudes?edit@

!here are si+ auxiliary latitudes that have applications to special problems in geodesy,

geophysics and the theory of map proEections:

• Geocentric latitude

• 'educed +or parametric latitude

• 'ectifying latitude

• /uthalic latitude

Conformal latitude

3 443;2E9 km 444;8=3 km

42 443;695 km 43E;223 km

83 443;72= km 56;976 km

92 444;48= km E7;79E km

63 444;94= km 22;733 km

E2 444;647 km =7;53= km

53 444;659 km 3;333 km

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• 0sometric latitude

!he definitions given in this section all relate to locations on the reference ellipsoid but the

first t$o au+iliary latitudes, like the geodetic latitude, can be e+tended to define a three%

dimensional geographic coordinate system as discussed belo$. !he remaining latitudes are

not used in this $ayH they are used only as intermediate constructs in map proEections of the

reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. !heir

numerical values are not of interest. For e+ample, no one $ould need to calculate the authalic

latitude of the #iffel !o$er.

!he e+pressions belo$ give the au+iliary latitudes in terms of the geodetic latitude, the semi%

maEor a+is, a, and the eccentricity, e. (For inverses see belo$. !he forms given are, apart

from notational variants, those in the standard reference for map proEections, namely 4'ap

 proEections: a $orking manual4 by J. -. 3nyder .5;6 Derivations of these e+pressions may be

found in 9dams56

 and online publications by sborne5G6

 and Kapp.5/6

Geocentric latitude[edit

 The de>nition of geodetic +or geographic and geocentric latitudes;

!he geocentric latitude is the angle bet$een the e*uatorial plane and the radius from the

centre to a point on the surface. !he relation bet$een the geocentric latitude ([ and the

geodetic latitude (Y is derived in the above references as

!he geodetic and geocentric latitudes are e*ual at the e*uator and poles. !he value of thes*uared eccentricity is appro+imately ;.;;0G (depending on the choice of ellipsoid and the

ma+imum difference of (Y%[ is appro+imately . minutes of arc at a geodetic latitude of

B<>.

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.educed (or parametric) latitude[edit

!efinition of the reduced latitude + on the ellipsoid;

!he reduced or parametric latitude, \, is defined by the radius dra$n from the centre of the

ellipsoid to that point ] on the surrounding sphere (of radius a $hich is the proEection parallel to the #arthNs a+is of a point - on the ellipsoid at latitude . It $as introduced by

7egendre5=6 and essel5@6 $ho solved problems for geodesics on the ellipsoid by transforming

them to an e*uivalent problem for spherical geodesics by using this smaller latitude. esselNs

notation, , is also used in the current literature. !he reduced latitude is related to the

geodetic latitude by:5G65/6

!he alternative name arises from the parameteriation of the e*uation of the ellipsedescribing a meridian section. In terms of "artesian coordinates p, the distance from the

minor a+is, and z , the distance above the e*uatorial plane, the e*uation of the ellipse is:

!he "artesian coordinates of the point are parameteried by

"ayley suggested the term paraetric latitude because of the form of these e*uations.5B6

!he reduced latitude is not used in the theory of map proEections. Its most important

application is in the theory of ellipsoid geodesics. (Mincenty, Qarney.56

.ectiying latitude[edit

See also# -e"tif$ing radius

!he rectifying latitude, ^, is the meridian distance scaled so that its value at the poles is

e*ual to A; degrees or V8= radians:

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$here the meridian distance from the e*uator to a latitude Y is (see 'eridian arc

and the length of the meridian *uadrant from the e*uator to the pole (the  polar distance is

&sing the rectifying latitude to define a latitude on a sphere of radius

defines a proEection from the ellipsoid to the sphere such that all meridians have true length

and uniform scale. !he sphere may then be proEected to the plane $ith an e*uirectangular

 proEection to give a double proEection from the ellipsoid to the plane such that all meridians

have true length and uniform meridian scale. 9n e+ample of the use of the rectifying latitude

is the #*uidistant conic proEection. (3nyder, 3ection 0.5;6 !he rectifying latitude is also of

great importance in the construction of the !ransverse 'ercator proEection.

/uthalic latitude[editSee also# uthali" radius

!he authalic (2reek for same area latitude, _, gives an area%preserving transformation to a

sphere.

$here

and

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and the radius of the sphere is taken as

9n e+ample of the use of the authalic latitude is the 9lbers e*ual%area conic proEection.(3nyder ,5;6 3ection B.

Conormal latitude[edit

!he conformal latitude, `, gives an angle%preserving (conformal transformation to the

sphere.

,

$here gd( x is the 2udermannian function. (3ee also 'ercator proEection. !he conformal

latitude defines a transformation from the ellipsoid to a sphere of ar%itrary radius such that

the angle of intersection bet$een any t$o lines on the ellipsoid is the same as thecorresponding angle on the sphere (so that the shape of sall  elements is $ell preserved. 9

further conformal transformation from the sphere to the plane gives a conformal double

 proEection from the ellipsoid to the plane. !his is not the only $ay of generating such a

conformal proEection. For e+ample, the Ne+actN version of the !ransverse 'ercator proEection 

on the ellipsoid is not a double proEection. (It does, ho$ever, involve a generalisation of the

conformal latitude to the comple+ plane.

0sometric latitude[edit

!he isometric latitude is conventionally denoted by [ (not to be confused $ith the

geocentric latitude: it is used in the development of the ellipsoidal versions of the normal

'ercator proEection and the !ransverse 'ercator proEection. !he name 4isometric4 arises

from the fact that at any point on the ellipsoid e*ual increments of [ and longitude T give rise

to e*ual distance displacements along the meridians and parallels respectively. !he graticule 

defined by the lines of constant [ and constant T, divides the surface of the ellipsoid into a

mesh of s*uares (of varying sie. !he isometric latitude is ero at the e*uator but rapidly

diverges from the geodetic latitude, tending to infinity at the poles. !he conventional notation

is given in 3nyder (page :5;6

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For the noral  'ercator proEection (on the ellipsoid this function defines the spacing of the

 parallels: if the length of the e*uator on the proEection is # (units of length or pi+els then the

distance, y, of a parallel of latitude Y from the e*uator is

!he isometric latitude is closely related to the conformal latitude:

0nverse ormulae and series[edit

!he formulae in the previous sections give the au+iliary latitude in terms of the geodetic

latitude. !he e+pressions for the geocentric and reduced latitudes may be inverted directly but

this is impossible in the four remaining cases: the rectifying, authalic, conformal, and

isometric latitudes. !here are t$o methods of proceeding. !he first is a numerical inversion

of the defining e*uation for each and every particular value of the au+iliary latitude. !he

methods available are fi+ed%point iteration and e$ton%Kaphson root finding. !he other,

more useful, approach is to e+press the au+iliary latitude as a series in terms of the geodeticlatitude and then invert the series by the method of  7agrange reversion. 3uch series are

 presented by 9dams $ho uses !aylor series e+pansions and gives coefficients in terms of the

eccentricity.56 sborne5G6 derives series to arbitrary order by using the computer algebra

 package 'a+ima506 and e+presses the coefficients in terms of both eccentricity and flattening.

!he series method is not applicable to the isometric latitude and one must use the conformal

latitude in an intermediate step.

$umerical comparison o au&iliary latitudes[edit

!he follo$ing plot sho$s the magnitude of the difference bet$een the geodetic latitude,

(denoted as the 4common4 latitude on the plot, and the au+iliary latitudes other than the

isometric latitude ($hich diverges to infinity at the poles. In every case the geodetic latitude

is the greater. !he differences sho$n on the plot are in arc minutes. !he horiontal resolution

of the plot fails to make clear that the ma+ima of the curves are not at B< but calculation

sho$s that they are $ithin a fe$ arc minutes of B<. 3ome representative data points are

given in the table follo$ing the plot. ote the closeness of the conformal and geocentric

latitudes. !his $as e+ploited in the days of hand calculators to e+pedite the construction of

map proEections. (3nyder ,5;6 page ;/.

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/ppro&imate di1erence rom geodetic latitude ( )

.educed /uthalic .ectiying Conormal Geocentric

3 3;33H 3;33H 3;33H 3;33H 3;33H

42 =;54H 8;75H 9;8EH 2;7=H 2;7=H

83 2;32H 6;E8H E;2EH 43;35H 43;35H

92 2;79H E;E7H 7;E6H 44;6EH 44;6EH

63 2;36H 6;E2H E;25H 43;4=H 43;48H

E2 =;5=H 8;53H 9;85H 2;72H 2;72H

53 3;33H 3;33H 3;33H 3;33H 3;33H

$atitude and coordinate systems?edit@

!he geodetic latitude, or any of the au+iliary latitudes defined on the reference ellipsoid,

constitutes $ith longitude a t$o%dimensional coordinate system on that ellipsoid. !o define

the position of an arbitrary point it is necessary to e+tend such a coordinate system into threedimensions. !hree latitudes are used in this $ay: the geodetic, geocentric and reduced

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latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal

coordinates respectively.

Geodetic coordinates[edit

Geodetic coordinates -+,,h

9t an arbitrary point - consider the line - $hich is normal to the reference ellipsoid. !he

geodetic coordinates -( ,T,ɸ h are the latitude and longitude of the point on the ellipsoid

and the distance -. !his height differs from the height above the geoid or a reference height

such as that above mean sea level at a specified location. !he direction of - $ill also differ

from the direction of a vertical plumb line. !he relation of these different heights re*uires

kno$ledge of the shape of the geoid and also the gravity field of the #arth.

Spherical polar coordinates[edit

Geocentric coordinate related to spherical polar coordinates -+r , V,

!he geocentric latitude [ is the complement of the polar angle in conventional spherical

 polar coordinates in $hich the coordinates of a point are -(r , , T $here r  is the distance of

 P  from the centre , is the angle bet$een the radius vector and the polar a+is and T is

longitude. 3ince the normal at a general point on the ellipsoid does not pass through the

centre it is clear that points on the normal, $hich all have the same geodetic latitude, $ill

have differing geocentic latitudes. 3pherical polar coordinate systems are used in the analysis

of the gravity field.

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%llipsoidal coordinates[edit

"llipsoidal coordinates -+u,,

!he reduced latitude can also be e+tended to a three%dimensional coordinate system. For a

 point - not on the reference ellipsoid (semi%a+es 9 and construct an au+iliary ellipsoid$hich is confocal (same foci F, FN $ith the reference ellipsoid: the necessary condition is that

the product ae of semi%maEor a+is and eccentricity is the same for both ellipsoids. 7et u be the

semi%minor a+is (D of the au+iliary ellipsoid. Further let \ be the reduced latitude of - on

the au+iliary ellipsoid. !he set (u,\,T define the ellipsoid coordinates. (!orge5=6 3ection B.=.=.

!hese coordinates are the natural choice in models of the gravity field for a uniform

distribution of mass bounded by the reference ellipsoid.

Coordinate conversions[edit

!he relations bet$een the above coordinate systems, and also "artesian coordinates are not

 presented here. !he transformation bet$een geodetic and "artesian coordinates may be foundin 2eodetic system. !he relation of "artesian and spherical polars is given in 3pherical

coordinate system. !he relation of "artesian and ellipsoidal coordinates is discussed in !orge.5=6

/stronomical latitude?edit@

Astronomical latitude ( is the angle bet$een the e*uatorial plane and the true vertical at a

 point on the surface. !he 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 (see !orge.5=6

 9stronomic latitude is calculated fromangles measured bet$een the enith and stars $hose declination is accurately kno$n.

In general the true vertical at a point on the surface does not e+actly coincide $ith either the

normal to the reference ellipsoid or the normal to the geoid. !he angle bet$een the

astronomic and geodetic normals is usually a fe$ seconds of arc but it is important in

geodesy.5=65@6 !he reason $hy it differs from the normal to the geoid is, because the geoid is an

idealied, theoretical shape 4at mean sea level4. -oints on the real surface of the earth are

usually above or belo$ this idealied geoid surface and here the true vertical can vary

slightly. 9lso, the true vertical at a point at a specific time is influenced by tidal forces, $hich

the theoretical geoid averages out.


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